Schur Products

Abstract

We call Schur products (in analogy to crossed products which inspired the present construction) a C*-algebra obtained from a given unital C*-algebra E and a discrete group T. This new C*-algebra consists of operators acting on the Hilbert right E-module E ⊗ l²(T) and are defined by a twisted representation of T.

Main article text

INTRODUCTION, NOTATION, AND TERMINOLOGY

The projective representation of groups was introduced in 1904 by Issai Schur (1875–1941) in his paper [7]. It differs from the normal representation of groups (introduced by his tutor Ferdinand Georg Frobenius (1849–1917) at the suggestion of Richard Dedekind (1831–1916)) by a twisting factor, which we call Schur function in this Article and which is called sometimes normalized factor set in the literature (other names are also used). It starts with a discrete group T and a Schur function f for T. This is a scalar valued function on T × T satisfying the conditions f(1, 1) = 1 and

$$|f(s,t)|=1,f(r,s)f(rs,t)=f(r,st)f(s,t)$$

for all r, s, t ∈ T. The projective representation of T twisted by f is a unital C*-subalgebra of the C*-algebra $\mathcal{L}(l^2(T))$ of operators on the Hilbert space l²(T). This representation can be used in order to construct many examples of C*-algebras (see e.g. [1] Chapter 7). By replacing the scalars ℝ or ℂ with an arbitrary unital (real or complex) C*-algebra E the field of applications is enhanced in an essential way. In this case l²(T) is replaced by the Hilbert right E-module $\underset{t\in T}{⦶}E\approx E\otimes l^2(T)$ and $\mathcal{L}(l^2(T))$ is replaced by ${\mathcal{L}}_E(E\otimes l^2(T))$, the C*-algebra of adjointable operators of $\mathcal{L}(E\otimes l^2(T))$. We call Schur product of E and T the resulting C*-algebra. It opens a way to create new K-theories ([4]).

In a Preliminaries we introduce some results which are needed for this construction, which is developed in the Schur Products. In the Examples we present examples of C*-algebras obtained by this method. The classical Clifford Algebras (including the infinite dimensional ones) are C*-algebras which can be obtained by projective representations of certain groups ([1] Section 7.2). The last Section 4 of this Article is dedicated to the generalization of these Clifford Algebras as an example of Schur products.

Throughout this Article we use the following notation: T is a group, 1 is its neutral element, K ≔ l²(T), 1_K ≔ id_K ≔ identity map of K, E is a unital C*-algebra (resp. a W*-algebra), 1_E is its unit, $\mathit{Ĕ}$ denotes the set E endowed with its canonical structure of a Hilbert right E-module ([1] Proposition 5.6.1.5),

$$H≔\mathit{Ĕ}\otimes K\approx \hspace{0.17em}\underset{t\in T}{⦶}\mathit{Ĕ},(\text{resp}.H≔\mathit{Ĕ}\bar{\otimes }K\approx \hspace{0.17em}\underset{t\in T}{\overset{W}{⦶}}\mathit{Ĕ})$$

([C3] Proposition 2.1, (resp. [C3] Corollary 2.2)). In some examples, in which T is additive, 1 will be replaced by 0.

The map

$${\mathcal{L}}_E(\mathit{Ĕ})⟶E,u⟼\langle u{1}_E|1_E\rangle =u{1}_E$$

is an isomorphism of C*-algebras with inverse

$$E⟶{\mathcal{L}}_E(\mathit{Ĕ}),x⟼x\cdot .$$

We identify E with ${\mathcal{L}}_E(\mathit{Ĕ})$ using these isomorphisms.

In general we use the notation of [1]. For tensor products of C*-algebras we use [9], for W*-tensor products of W*-algebras we use [8], for tensor products of Hilbert right C^*-modules we use [6], and for the exterior W*-tensor products of selfdual Hilbert right W^*-modules we use [2] and [3].

In the sequel we give a list of notation (mainly introduced in [1]) which are used in this Article.

• 1) $𝕂$ denotes the field of real numbers (≔ ℝ) or the field of complex numbers (≔ ℂ). In general the C*-algebras will be complex or real. ℍ denotes the field of quaternions, ℕ denotes the set of natural numbers (0 ∉ ℕ), and for every n ∈ ℕ ∪ { 0 } we put

  $${\mathbb{N}}_n≔\left\{m\in \mathbb{N}|m\le n\right\}.$$

  ℤ denotes the group of integers and for every n ∈ ℕ we put ℤ_n ≔ ℤ / (nℤ).

• 2) For every set $A,\mathfrak{P}(A)$ denotes the set of subsets of $A,{\mathfrak{P}}_f(A)$ the set of finite subsets of A, and Card A denotes the cardinal number of A. If f is a function defined on A and B is a subset of A then f|B denotes the restriction of f to B.

• 3) If A, B are sets then A^B denotes the set of maps of B in A.

• 4) For all i, j we denote by δ_i,j Kronecker’s symbol:

  $${\mathit{\delta }}_{i,j}≔\{\begin{array}{lll}1\hfill & \text{if}\hfill & i=j\hfill \\ 0\hfill & \text{if}\hfill & i\ne j\hfill \end{array}.$$

• 5) If A, B are topological spaces then $\mathcal{C}(A,B)$ denotes the set of continuous maps of A into B. If A is locally compact space and E is a C*-algebra then $\mathcal{C}(A,E)$ (resp. ${\mathcal{C}}_0(A,E)$) denotes the C*-algebra of continuous maps A ⟶ E, which are bounded (resp. which converge to 0 at the infinity).

• 6) For every set I and for every J ⊂ I we denote by $e_J≔e_J^I$ the characteristic function of J i.e. the function on I equal to 1 on J and equal to 0 on I / J. For i ∈ I we put e_i ≔ (δ_i,j)_{j ∈ I} ∈ l²(I).

• 7) If F is an additive group and S is a set then

  $$F^{(S)}≔\left\{x\in F^S|\left\{s\in S|x_s\ne 0\right\}\text{is}\text{finite}\right\}.$$

• 8) If E, F are vector spaces in duality then E_F denotes the vector space E endowed with the locally convex topology of pointwise convergence on F, i.e. with the weak topology σ(E, F).

• 9) If E is a normed vector space then E′ denotes its dual and E^# denotes its unit ball:

  $$E^{\#}≔\left\{x\in E|\Vert x\Vert \le 1\right\}.$$

  Moreover if E is an ordered Banach space then E₊ denotes the convex cone of its positive elements. If E has a unique predual (up to isomorphisms), then we denote by Ë this predual and so by $E_{\mathit{Ë}}$ the vector space E endowed with the locally convex topology of pointwise convergence on Ë.

• 10) The expressions of the form “ … C*- … (resp. … W*- …)”, which appear often in this Article, will be replaced by expressions of the form “… C**- …”.

• 11) If F is a unital C*-algebra and A is a subset of F then we denote by 1_F the unit of F, by Pr F the set of orthogonal projections of F, by

  $$A^c≔\left\{x\in F|y\in A\Rightarrow xy=yx\right\},ReF≔\left\{x\in F|x=x^{*}\right\},$$

  and by Un F the set of unitary elements of F. If F is a real C*-algebra then $\stackrel{\circ }{F}$ denotes its complexification.

• 12) If F is a C*-algebra then we denote for every n ∈ ℕ by F_n,n the C*-algebra of n × n matrices with entries in F. If T is finite then F_T,T has a corresponding signification.

• 13) Let F be a C*-algebra and H, K Hilbert right F-modules. We denote by ${\mathcal{L}}_F(H,K)$ the Banach subspace of $\mathcal{L}(H,K)$ of adjointable operators, by 1_H the identity map H ⟶ H which belongs to

  $${\mathcal{L}}_F(H)≔{\mathcal{L}}_F(H,H).$$

  For (ξ, η) ∈ H × K we put

  $$\mathit{\eta }\langle \cdot |\mathit{\xi }\rangle :H⟶K,\mathit{\zeta }⟼\mathit{\eta }\langle \mathit{\zeta }|\mathit{\xi }\rangle$$

  and denote by ${\mathcal{K}}_F(H)$ the closed vector subspace of ${\mathcal{L}}_F(H)$ generated by $\{\mathit{\eta }\langle \cdot |\mathit{\xi }\rangle |\mathit{\xi },\mathit{\eta }\in H\}$.

• 14) Let F be a W*-algebra and H, K Hilbert right F-modules. We put for $a\in \ddot{F}$ and (ξ, η) ∈ H × K,

  $$\begin{array}{c}\widetilde{(a,\mathit{\xi })}:H⟶𝕂,\mathit{\zeta }⟼\langle \langle \mathit{\zeta }|\mathit{\xi }\rangle ,a\rangle ,\\ \widetilde{(a,\mathit{\xi },\mathit{\eta })}:{\mathcal{L}}_F(H,K)⟶𝕂,u⟼\langle \langle u\mathit{\xi }|\mathit{\eta }\rangle ,a\rangle \end{array}$$

  and denote by $\ddot{H}$ the closed vector subspace of the dual H′ of H generated by

  $$\left\{\widetilde{(a,\mathit{\xi })}|a\in \ddot{F},\mathit{\xi }\in H\right\}$$

  and by $\stackrel{⃛}{H}$ the closed vector subspace of ${\mathcal{L}}_F(H,K)\prime$ generated by

  $$\left\{\widetilde{(a,\mathit{\xi },\mathit{\eta })}|(a,\mathit{\xi },\mathit{\eta })\in \ddot{F}\times H\times K\right\}.$$

  If H is selfdual then $\stackrel{⃛}{H}$ is the predual of ${\mathcal{L}}_F(H)$ ([1] Theorem 5.6.3.5 b)) and $\ddot{H}$ is the predual of H ([1] Proposition 5.6.3.3). Moreover a map defined on F is called W*-continuous if it is continuous on $F_{\ddot{F}}$. If G is a W*-algebra a C*-homomorphism $\mathit{\phi }:F⟶G$ is called a W*-homomorphism if the map $\mathit{\phi }:F_{\ddot{F}}⟶G_{\ddot{G}}$ is continuous; in this case $\ddot{\mathit{\phi }}$ denotes the pretranspose of φ.

• 15) If F is a C**-algebra and (H_i)_{i ∈ I} a family of Hilbert right F-modules then we put

  $$\underset{i\in I}{⦶}{H}_i≔\left\{\mathit{\xi }\in {\displaystyle \prod _{i\in I}{H}_i}|\text{the}\text{family}{\langle {\mathit{\xi }}_i|{\mathit{\xi }}_i\rangle }_{i\in I}\text{is}\text{summable}\text{in}F\right\}$$

  respectively

  $$\underset{i\in I}{\overset{W}{⦶}}{H}_i≔\left\{\mathit{\xi }\in {\displaystyle \prod _{i\in I}{H}_i}|\text{the}\text{family}{\langle {\mathit{\xi }}_i|{\mathit{\xi }}_i\rangle }_{i\in I}\text{is}\text{summable}\text{in}{F}_{\ddot{F}}\right\}.$$

• 16) ⊙ denotes the algebraic tensor product of vector spaces.

• 17) If F, G are W*-algebras and H (resp. K) is a selfdual Hilbert right F-module (resp. G-module) then we denote by H ⊗̅ K the W*-tensor product of H and K, which is a selfdual Hilbert right H ⊗̅ G-module ([2] Definition 2.3).

• 18) ≈ denotes isomorphic.

If T is finite then (by [1] Theorem 5.6.6.1 f))

$${\mathcal{L}}_E(H)=E_{T,T}={𝕂}_{T,T}\otimes E={\mathcal{K}}_E(H).$$

PRELIMINARIES

1.1. Schur functions

We list in this subsection some properties of the Schur functions needed later.

DEFINITION 1.1.1 A Schur E-function for T is a map

$$f:T\times T⟶Un{E}^c$$

such that f(1, 1) = 1_E and

$$f(r,s)f(rs,t)=f(r,st)f(s,t)$$

for all r,s,t ∈ T. We denote by $\mathcal{ℱ}(\mathit{T},\mathit{E})$ the set of Schur E-functions for T and put

$$\begin{array}{c}\tilde{f}:T⟶Un{E}^c,t⟼f(t,t^{-1}{)}^{*},\\ \hat{f}:T\times T⟶Un{E}^c,(s,t)⟼f(t^{-1},s^{-1})\end{array}$$

for every $f\in \mathit{ℱ}(T,E)$.

Schur functions are also called normalized factor set or multiplier or two-co-cycle (for T with values in Un E^c) in the literature. We present in this section only some elementary properties (which will be used in the sequel) in order to fix the notation and the terminology. By the way, Un E^c can be replaced in this section by an arbitrary commutative multiplicative group (with * replaced by −1).

PROPOSITION 1.1.2 Let $f\in \mathit{ℱ}(T,E)$.

• a) For every t ∈ T,

  $$f(t,1)=f(1,t)=1_E,f(t,t^{-1})=f(t^{-1},t),\tilde{f}(t)=\tilde{f}(t^{-1}).$$

• b) For all s, t ∈ T,

  $$f(s,t)\tilde{f}(s)=f(s^{-1},st){*}^{},f(s,t)\tilde{f}(t)=f(st,t^{-1}){*}^{}.$$

a) Putting s = 1 in the equation of f we obtain

$$f(r,1)f(r,t)=f(r,t)f(1,t)$$

so

$$f(r,1)=f(1,t)$$

for all r, t ∈ T. Hence

$$f(t,1)=f(1,t)=f(1,1)=1_E.$$

Putting r = t and s = t⁻¹ in the equation of f we get

$$f(t,t^{-1})f(1,t)=f(t,1)f(t^{-1},t).$$

By the above,

$$f(t,t^{-1})=f(t^{-1},t),\tilde{f}(t)=\tilde{f}(t^{-1}).$$

b) Putting r = s⁻¹ in the equation of f, by a),

$$\begin{array}{c}f(s,t)f(s^{-1},st)=f(s^{-1},s)f(1,t)=\tilde{f}(s{)}^{*},\\ f(s,t)\tilde{f}(s)=f(s^{-1},st{)}^{*}.\end{array}$$

Putting now t = s⁻¹ in the equation of f, by a) again,

$$\begin{array}{c}f(r,s)f(rs,s^{-1})=f(r,1)f(s,s^{-1})=\tilde{f}(s{)}^{*},\\ f(r,s)\tilde{f}(s)=f(rs,s^{-1}{)}^{*},f(s,t)\tilde{f}(t)=f(st,t^{-1}{)}^{*}.\end{array}$$

DEFINITION 1.1.3 We put

$$\Lambda (T,E)≔\{\mathit{\lambda }:T⟶Un{E}^c|\mathit{\lambda }(1)=1_E\}$$

and

$$\begin{array}{c}\hat{\mathit{\lambda }}:T⟶Un{E}^c,t⟼\mathit{\lambda }(t^{-1}),\\ \mathit{\delta }\mathit{\lambda }:T\times T⟶Un{E}^c,(s,t)⟼\mathit{\lambda }(s)\mathit{\lambda }(t)\mathit{\lambda }(st{)}^{*}\end{array}$$

for every λ ∈ Λ (T, E).

PROPOSITION 1.1.4

• a) $\mathit{ℱ}(T,E)$ is a subgroup of the commutative multiplicative group (Un E^c)^T×T such that f* is the inverse of f for every $f\in \mathit{ℱ}(T,E)$.

• b) $\hat{f}\in \mathit{ℱ}(T,E)$ for every $f\in \mathit{ℱ}(T,E)$ and the map

  $$\mathit{ℱ}(T,E)⟶\mathit{ℱ}(T,E),f⟼\hat{f}$$

  is an involutive group automorphism.

• c) Λ(T, E) is a subgroup of the commutative multiplicative group (Un E^c)^T, $\mathit{\delta }\mathit{\lambda }\in \mathit{ℱ}(T,E)$ for every λ ∈ Λ (T, E), and the map

  $$\mathit{\delta }:\Lambda (T,E)⟶\mathit{ℱ}(T,E),\mathit{\lambda }⟼\mathit{\delta }\mathit{\lambda }$$

  is a group homomorphism with kernel

  $$\left\{\mathit{\lambda }\in \Lambda (T,E)|\mathit{\lambda }\text{is a group homomorphism}\right\}$$

  such that $\widehat{\mathit{\delta }\mathit{\lambda }}=\mathit{\delta }\hat{\mathit{\lambda }}$ for every λ ∈ Λ(T, E).

a) is obvious.

b) For r, s, t ∈ T,

$$\begin{array}{c}\hat{f}(r,s)\hat{f}(rs,t)=f(s^{-1},r^{-1})f(t^{-1},s^{-1}{r}^{-1})=\\ =f(t^{-1},s^{-1})f(t^{-1}{s}^{-1},r^{-1})=\hat{f}(r,st)\hat{f}(s,t),\end{array}$$

so $\hat{f}\in \mathit{ℱ}(T,E)$. For $f,g\in \mathit{ℱ}(T,E)$,

$$\begin{array}{c}\widehat{fg}(s,t)=(fg)(t^{-1},s^{-1})=f(t^{-1},s^{-1})g(t^{-1},s^{-1})=\hat{f}(s,t)\hat{g}(s,t)=(\hat{f}\hat{g})(s,t),\\ \widehat{fg}=\hat{f}\hat{g},\\ \hat{f}{*}^{}(s,t)=\hat{f}(s,t{)}^{*}=f(t^{-1},s^{-1}{)}^{*}=f^{*}(t^{-1},s^{-1})=\widehat{f^{*}}(s,t),(\hat{f}{)}^{*}=\widehat{f^{*}}.\end{array}$$

c) For r, s, t ∈ T,

$$\begin{array}{c}\mathit{\delta }\mathit{\lambda }(r,s)\mathit{\delta }\mathit{\lambda }(rs,t)=\mathit{\lambda }(r)\mathit{\lambda }(s)\mathit{\lambda }(rs{)}^{*}\mathit{\lambda }(rs)\mathit{\lambda }(t)\mathit{\lambda }(rst{)}^{*}=\mathit{\lambda }(r)\mathit{\lambda }(s)\mathit{\lambda }(t)\mathit{\lambda }(rst{)}^{*},\\ \mathit{\delta }\mathit{\lambda }(r,st)\mathit{\delta }\mathit{\lambda }(s,t)=\mathit{\lambda }(r)\mathit{\lambda }(st)\mathit{\lambda }(rst{)}^{*}\mathit{\lambda }(s)\mathit{\lambda }(t)\mathit{\lambda }(st{)}^{*}=\mathit{\lambda }(r)\mathit{\lambda }(s)\mathit{\lambda }(t)\mathit{\lambda }(rst{)}^{*}\end{array}$$

so $\mathit{\delta }\mathit{\lambda }\in \mathit{ℱ}(T,E)$. For $\mathit{\lambda },\mathit{\mu }\in \mathit{ℱ}(T,E)$ and s,t ∈ T,

$$\begin{array}{c}\mathit{\delta }\mathit{\lambda }(s,t)\mathit{\delta }\mathit{\mu }(s,t)=\mathit{\lambda }(s)\mathit{\lambda }(t)\mathit{\lambda }(st{)}^{*}\mathit{\mu }(s)\mathit{\mu }(t)\mathit{\mu }(st{)}^{*}=\\ =(\mathit{\lambda }\mathit{\mu })(s)(\mathit{\lambda }\mathit{\mu })(t)(\mathit{\lambda }\mathit{\mu })(st{)}^{*}=\mathit{\delta }(\mathit{\lambda }\mathit{\mu })(s,t),\\ (\mathit{\delta }\mathit{\lambda })(\mathit{\delta }\mathit{\mu })=\mathit{\delta }(\mathit{\lambda }\mathit{\mu }),\\ \mathit{\delta }{\mathit{\lambda }}^{*}(s,t)={\mathit{\lambda }}^{*}(s){\mathit{\lambda }}^{*}(t)\mathit{\lambda }(st)={(\mathit{\delta }\mathit{\lambda }(s,t))}^{*}={(\mathit{\delta }\mathit{\lambda })}^{*}(s,t),\mathit{\delta }{\mathit{\lambda }}^{*}={(\mathit{\delta }\mathit{\lambda })}^{*},\end{array}$$

so δ is a group homomorphism. The other assertions are obvious.

PROPOSITION 1.1.5 Let t ∈ T, m, n ∈ ℤ, and $f\in \mathit{ℱ}(T,E)$.

• a) f(t^m, tⁿ) = f(tⁿ, t^m).

• b) $m\in \mathbb{N}\Rightarrow f(t^m,t^n)=\left({\displaystyle \prod _{j=0}^{m-1}f(t^{n+j},t)}\right)\left({\displaystyle \prod _{k=1}^{m-1}f(t^k,t{)}^{*}}\right)$.

• c) We define

  $$\mathit{\lambda }:\mathbf{\mathbb{Z}}⟶Un{E}^c,\hspace{1em}n⟼\{\begin{array}{lll}{\displaystyle \prod _{j=1}^{n-1}f}(t^j,t){*}^{},\hfill & if\hfill & n\in \mathbb{N}\hfill \\ {\displaystyle \prod _{j=1}^{-n}f}(t^{-j},t)\hfill & if\hfill & n\notin \mathbb{N}\hfill \end{array}.$$

  If t^p≠1 for every p ∈ ℕ then

  $$f(t^m,t^n)=\mathit{\lambda }(m)\mathit{\lambda }(n)\mathit{\lambda }{(m+n)}^{*}$$

  for all m, n ∈ ℤ.

a) We may assume m ∈ ℕ because otherwise we can replace t by t⁻¹. Put

$$\begin{array}{c}P(m,n):\iff f(t^m,t^n)=f(t^n,t^m),\\ Q(m):\iff P(m,n)\text{holds for all}n\in \mathbf{\mathbb{Z}}.\end{array}$$

From

$$f(t^m,t^{n-m})f(t^n,t^m)=f(t^m,t^n)f(t^{n-m},t^m)$$

it follows

$$P(m,n)\iff P(m,n-m)\iff P(m,n-km)$$

for all k ∈ ℤ.

We prove the assertion by induction. P(m, 0) follows from Proposition 1.1.2 a). By the above

$$P(1,0)\iff P(1,k)$$

for all k ∈ ℤ. Thus Q(1) holds.

Assume Q(p) holds for all p ∈ ℕ_m−1. Then P(m, p) holds for all p ∈ ℕ_m−1 ∪ {0}. Let. n ∈ ℤ. There is a k ∈ ℤ such that

$$p≔n-km\in {\mathbb{N}}_{\text{m}-1}\cup \left\{0\right\}.$$

By the above P(m, n) holds. Thus Q(m) holds and this finishes the inductive proof.

b) We prove the formula by induction with respect to m. By a), the formula holds for m = 1. Assume the formula holds for an m ∈ ℕ. Since

$$f(t^m,t)f(t^{m+1},t^n)=f(t^m,t^{n+1})f(t,t^n)$$

we get by a),

$$\begin{array}{c}f(t^{m+1},t^n)=f(t^m,t^{n+1})f(t,t^n)f(t^m,t{)}^{*}=\\ =\left({\displaystyle \prod _{j=0}^{m-1}f}(t^{n+1+j},t)\right)\left({\displaystyle \prod }_{k=1}^{m-1}f(t^k,t){*}^{}\right)f(t^n,t)f(t^m,t{)}^{*}=\\ =\left({\displaystyle \prod }_{j=0}^{m}f(t^{n+j},t)\right)\left({\displaystyle \prod }_{k=1}^{m}f(t^k,t){*}^{}\right).\end{array}$$

Thus the formula holds also for m + 1.

c) If m, n ∈ ℕ then by b),

$$\begin{array}{c}\mathit{\lambda }(m)\mathit{\lambda }(n)\mathit{\lambda }(m+n{)}^{*}=\\ =\left({\displaystyle \prod }_{k=1}^{m-1}f(t^k,t){*}^{}\right)\left({\displaystyle \prod }_{j=1}^{n-1}f(t^j,t){*}^{}\right)\left({\displaystyle \prod }_{j=1}^{m+n-1}f(t^j,t)\right)=\\ =\left({\displaystyle \prod }_{j=0}^{m-1}f(t^{n+j},t)\right)\left({\displaystyle \prod }_{k=1}^{m-1}f(t^k,t){*}^{}\right)=f(t^m,t^n).\end{array}$$

If m, n ∈ ℕ, n ≤ m − 1 then by b),

$$\begin{array}{c}\mathit{\lambda }(m)\mathit{\lambda }(-n)\mathit{\lambda }(m-n{)}^{*}=\\ =\left({\displaystyle \prod }_{j=1}^{m-1}f(t^j,t){*}^{}\right)\left({\displaystyle \prod }_{j=1}^{n}f(t^{-j},t)\right)\left({\displaystyle \prod }_{j=1}^{m-n-1}f(t^j,t)\right)=\\ =\left({\displaystyle \prod }_{j=0}^{m-1}f(t^{-n+j},t)\right)\left({\displaystyle \prod }_{k=1}^{m-1}f(t^k,t){*}^{}\right)=f(t^m,t^{-n}).\end{array}$$

If m, n ∈ ℕ, n ≥ m then by b),

$$\begin{array}{c}\mathit{\lambda }(m)\mathit{\lambda }(-n)\mathit{\lambda }(m-n{)}^{*}=\\ =\left({\displaystyle \prod }_{k=1}^{m-1}f(t^k,t){*}^{}\right)\left({\displaystyle \prod }_{j=1}^{n}f(t^{-j},t)\right)\left({\displaystyle \prod }_{j=1}^{n-m}f(t^{-j},t){*}^{}\right)=\\ =\left({\displaystyle \prod }_{j=n-m+1}^{n}f(t^{-j},t)\right)\left({\displaystyle \prod }_{k=1}^{m-1}f(t^k,t){*}^{}\right)=f(t^m,t^{-n}).\end{array}$$

For all m, n ∈ ℕ put

$$R(m,n):\iff f(t^{-m},t^{-n})=\mathit{\lambda }(-m)\mathit{\lambda }(-n)\mathit{\lambda }(-m-n){*}^{}.$$

By the above and by Proposition 1.1.2 a), b),

$$\mathit{\lambda }(-1)\mathit{\lambda }(-1)\mathit{\lambda }{(-2)}^{*}=f(t^{-1},t)f{(t^{-2},t)}^{*}=\tilde{f}{(t^{-1})}^{*}f{(t,t^{-2})}^{*}=f(t^{-1},t^{-1}),$$

so R(1, 1) holds. Let now m, n ∈ ℕ and assume R(m, n) holds. Then

$$\begin{array}{c}\mathit{\lambda }(-m)\mathit{\lambda }(-n-1)\mathit{\lambda }(-m-n-1{)}^{*}=\\ =\left({\displaystyle \prod }_{j=1}^{m}f(t^{-j},t)\right)\left({\displaystyle \prod }_{j=1}^{n+1}f(t^{-j},t)\right)\left({\displaystyle \prod }_{j=1}^{m+n+1}f(t^{-j},t){*}^{}\right)=\\ =f(t^{-m},t^{-n})f(t^{-n-1},t)f(t^{-m-n-1},t{)}^{*}=f(t^{-m},t^{-n-1}),\end{array}$$

so R(m, n) ⇒ R(m, n + 1). By symmetry and a), R(m, n) holds for all m, n ∈ ℕ.

COROLLARY 1.1.6 The map

$$\Lambda (\mathbf{\mathbb{Z}},E)⟶\mathit{ℱ}(\mathbf{\mathbb{Z}},E),\mathit{\lambda }⟼\mathit{\delta }\mathit{\lambda }$$

is a surjective group homomorphism with kernel

$$\left\{\mathit{\lambda }\in \Lambda (\mathbf{\mathbb{Z}},E)|n\in \mathbf{\mathbb{Z}}\Rightarrow \mathit{\lambda }(n)=\mathit{\lambda }{(1)}^n\right\}.$$

By Proposition 1.1.4 c), only the surjectivity of the above map has to be proved and this follows from Proposition 1.1.5 c).

1.2. E-C*-algebras

By replacing the scalars with the unital C*-algebra E we restrict the category of C*-algebras to the subcategory of those C*-algebras which are connected in a certain way with E. The category of unital C*-algebras is replaced by the category of E-C*-algebras, while the general category of C*-algebras is replaced by the category of adapted E-modules. The Schur products are always adapted E-modules.

DEFINITION 1.2.1 We call in this Article E-module a C*-algebra F endowed with the bilinear maps

$$\begin{array}{c}E\times F⟶F,(\mathit{\alpha },x)⟼\mathit{\alpha }x,\\ F\times E⟶F,(x,\mathit{\alpha })⟼x\mathit{\alpha }\end{array}$$

such that for all α, β ∈ E and x, y ∈ F,

$$\begin{array}{c}(\mathit{\alpha }\mathit{\beta })x=\mathit{\alpha }(\mathit{\beta }x),\mathit{\alpha }(x\mathit{\beta })=(\mathit{\alpha }x)\mathit{\beta },x(\mathit{\alpha }\mathit{\beta })=(x\mathit{\alpha })\mathit{\beta },\\ \text{\hspace{1em};}\mathit{\alpha }(xy)=(\mathit{\alpha }x)y,(xy)\mathit{\alpha }=x(y\mathit{\alpha }),\mathit{\alpha }\in E^c\Rightarrow \mathit{\alpha }x=x\mathit{\alpha },\\ (\mathit{\alpha }x{)}^{*}=x^{*}{\mathit{\alpha }}^{*},(x\mathit{\alpha }{)}^{*}={\mathit{\alpha }}^{*}{x}^{*},1_{E}x=x{1}_E=x.\end{array}$$

If F, G are E-modules then a C*-homomorphism φ : F ⟶ G is called E-linear if for all (α, x) ∈ E × F,

$$\mathit{\phi }(\mathit{\alpha }x)=\mathit{\alpha }(\mathit{\phi }x),\mathit{\phi }(x\mathit{\alpha })=(\mathit{\phi }x)\mathit{\alpha }.$$

For all (α, x) ∈ E × F,

$${\Vert \mathit{\alpha }x\Vert }^2=\Vert x^{*}{\mathit{\alpha }}^{*}\mathit{\alpha }x\Vert \le {\Vert x\Vert }^2{\Vert \mathit{\alpha }\Vert }^2,{\Vert x\mathit{\alpha }\Vert }^2=\Vert {\mathit{\alpha }}^{*}{x}^{*}x\mathit{\alpha }\Vert \le {\Vert \mathit{\alpha }\Vert }^2{\Vert x\Vert }^2$$

so

$$\Vert \mathit{\alpha }x\Vert \le \Vert \mathit{\alpha }\Vert \Vert x\Vert ,\Vert x\mathit{\alpha }\Vert \le \Vert x\Vert \Vert \mathit{\alpha }\Vert .$$

DEFINITION 1.2.2 An E-C**-algebra is a unital C**-algebra F for which E is a canonical unital C**-subalgebra such that E^c defined with respect to E coincides with E^c defined with respect to F i.e. for every x ∈ E, if xy = yx for all y ∈ E then xy = yx for all y ∈ F. Every closed ideal of an E-C*-algebra is canonically an E-module.

Let F, G be E-C**-algebras. A map φ : F ⟶ G is called an E-C**-homomorphism if it is an E-linear C**-homomorphism. If in addition φ is a C*-isomorphism then we say that φ is an E-C*-isomorphism and we use in this case the notation ≈_E. A C**-subalgebra F₀ of F is called E-C**-subalgebra of F if E ⊂ F₀.

With the notation of the above Definition (α − φ α) φx = 0 for all α ∈ E and x ∈ F. Thus φ is unital iff φα = α for every α ∈ E. The example

$$𝕂⟶𝕂\times 𝕂,x⟼(x,0)$$

shows that an E-C*-homomorphism need not be unital.

If we put $𝕋≔\{z\in ℂ|\left|z\right|=1\},E≔\mathcal{C}(𝕋,ℂ)$, and

$$x:𝕋⟶ℂ,z⟼z$$

and if we denote by λ the Lebesgue measure on $𝕋$ then L^∞(λ) is an E-C*-algebra, x ∈ Un E, and x is homotopic to 1_E in Un L^∞(λ) but not in $Un\mathcal{C}(𝕋,ℂ)$.

DEFINITION 1.2.3 We denote by ℭ_E (resp. by ${\mathbf{ℭ}}_E^1$) the category of E-C*-algebras for which the morphisms are the E-C*-homomorphisms (resp. the unital E-C*-homomorphisms).

PROPOSITION 1.2.4 Let F be an E-module.

• a) We denote by $\check{\mathit{F}}$ the vector space E × F endowed with the bilinear map

  $$(E\times F)\times (E\times F)⟶E\times F,((\alpha ,x),(\beta ,y))⟶(\alpha \beta ,\alpha y+x\beta +xy)$$

  and with the conjugate linear map

  $$E\times F⟶E\times F,(\mathit{\alpha },x)⟼({\mathit{\alpha }}^{*},x^{*}).$$

  $\check{F}$ is an involutive unital algebra with (1_E, 0) as unit.

• b)The maps

  $$\mathit{\pi }:\check{F}⟶E,(\mathit{\alpha },x)⟼\mathit{\alpha },$$

  $$\mathit{\lambda }:E⟶\check{F},\mathit{\alpha }⟼(\mathit{\alpha },0),$$

  $$\iota :F⟶\check{F},x⟼(0,x)$$

  are involutive algebra homomorphisms such that π ο λ is the identity map of E, λ and ι are injective, and λ and π are unital. If there is a norm on $\check{F}$ with respect to which it is a C*-algebra (in which case such a norm is unique), then we call F adapted. We denote by ${{M}}_{\mathit{E}}$ the category of adapted E-modules for which the morphism are the E-linear C*-homomorphisms.

• c) If F is adapted then $\check{F}$ is an E-C*-algebra by using canonically the injection λ and for all α ∈ E and x ∈ F,

  $$\Vert \mathit{\alpha }\Vert \le \Vert (\mathit{\alpha },x)\Vert \le \Vert \mathit{\alpha }\Vert +\Vert x\Vert ,\Vert (0,x)\Vert =\Vert x\Vert \le 2\Vert (\mathit{\alpha },x)\Vert ,$$

  $$\Vert (\mathit{\alpha },0)(0,x)\Vert \le \Vert \mathit{\alpha }\Vert \Vert x\Vert ,\Vert (0,x)(\mathit{\alpha },0)\Vert \le \Vert x\Vert \Vert \mathit{\alpha }\Vert .$$

  In particular F (identified with ι(F)) is a closed ideal of $\check{F}$.

• d) If E and F are C*-subalgebras of a C*-algebra G in such a way that the structure of E-module of F is inherited from G then

  $$\mathit{\phi }:\check{F}⟶E\times G,(\mathit{\alpha },x)⟼(\mathit{\alpha },\mathit{\alpha }+x)$$

  is an injective involutive algebra homomorphism, $\mathit{\phi }(\check{F})$ is closed, F is adapted, and for all α ∈ E and x ∈ F,

  $${\Vert (\mathit{\alpha },x)\Vert }_{E\times F}=\text{sup}\{\Vert \mathit{\alpha }\Vert ,\Vert \mathit{\alpha }+x\Vert \}.$$

  In particular every closed ideal of an E-C*-algebra is adapted and ℭ_E is a full subcategory of ${\mathfrak{M}}_E$.

• e) A closed ideal G of an adapted E-module F, which is at the same time an E-submodule of F, is adapted.

• f) If F is unital then it is adapted and

  $$\check{F}⟶{ℝ}_{+},(\mathit{\alpha },x)⟼\text{sup}\{\Vert \mathit{\alpha }\Vert ,\Vert \mathit{\alpha }{1}_F+x\Vert \}$$

  is the C*-norm of $\check{F}$.

• g) If

  $$\underset{y,\mathfrak{F}}{\text{lim}}\Vert \mathit{\alpha }y-y\mathit{\alpha }\Vert =0$$

  for all α ∈ E₊, where $\mathfrak{F}$ denotes the canonical approximate unit of F, then F is adapted and

  $$\check{F}⟶{ℝ}_{+},(\mathit{\alpha },x)⟼\text{sup}\left\{\Vert \mathit{\alpha }\Vert ,\underset{y,\mathfrak{F}}{\text{lim}\text{sup}}\Vert \mathit{\alpha }y+x\Vert \right\}$$

  is the C*-norm of $\check{F}$. In particular F is adapted if E is commutative.

• h) If F is an adapted E-module then (with the notation of b))

  $$0⟶F\stackrel{\iota }{⟶}\check{F}\begin{array}{c}\stackrel{\mathit{\pi }}{⟶}\\ \stackrel{\mathit{\lambda }}{⟵}\end{array}E⟶0$$

  is a split exact sequence in the category ${\mathfrak{M}}_E$.

a) and b) are easy to see.

c) Since λ and ι are injective and,

$$\mathit{\pi }(\mathit{\alpha },x)=\mathit{\alpha },(\mathit{\alpha },x)=(\mathit{\alpha },0)+(0,x),$$

$$(\mathit{\alpha },0)(0,x)=(0,\mathit{\alpha }x),(0,x)(\mathit{\alpha },0)=(0,x\mathit{\alpha })$$

we get the first and the last two inequalities as well as the identity ‖(0, x)‖ = ‖x‖. It follows

$$\Vert (0,x)\Vert \le \Vert (\mathit{\alpha },x)\Vert +\Vert (\mathit{\alpha },0)\Vert =\Vert (\mathit{\alpha },x)\Vert +\Vert \mathit{\lambda }\mathit{\pi }(\mathit{\alpha },x)\Vert \le$$

$$\le \Vert (\mathit{\alpha },x)\Vert +\Vert (\mathit{\alpha },x)\Vert =2\Vert (\mathit{\alpha },x)\Vert .$$

d) It is easy to see that φ is an injective involutive algebra homomorphism. Let $(\mathit{\alpha },x)\in \overline{\mathit{\phi }(\check{F})}$. There are sequences (α_n)_{n ∈ ℕ} and (x_n)_{n ∈ ℕ} in E and F, respectively, such that

$$\underset{n⟶\infty }{\text{lim}}({\mathit{\alpha }}_n,{\mathit{\alpha }}_n+x_n)=(\mathit{\alpha },x).$$

it follows

$$\mathit{\alpha }=\underset{n⟶\infty }{\text{lim}}{\mathit{\alpha }}_n\in E,x-\mathit{\alpha }=\underset{n⟶\infty }{\text{lim}}{x}_n\in F,(\mathit{\alpha },x)=\mathit{\phi }(\mathit{\alpha },x-\mathit{\alpha })\in \mathit{\phi }(\check{F}).$$

Thus $\mathit{\phi }(\check{F})$ is closed, which proves the assertion by pulling back the norm of E × G.

e) By c), F is a closed ideal of $\check{F}$ so G is a closed ideal of $\check{F}$ (use an approximate unit of F). Since G is an E-submodule of F its structure of E-module is inherited from $\check{F}$. By d), G is adapted.

f) The map

$$\check{F}⟶E\times F,(\mathit{\alpha },x)⟼(\mathit{\alpha },\mathit{\alpha }{1}_F+x)$$

is an isomorphism of involutive algebras and so we can pull back the norm of E × F.

g) It is easy to see that the above map is a norm. Since

$$\text{sup}\{\Vert \mathit{\alpha }\Vert ,\frac{1}{2}\Vert x\Vert \}\le \Vert (\mathit{\alpha },x)\Vert \le \Vert \mathit{\alpha }\Vert +\Vert x\Vert$$

for all (α, x) ∈ E × F, $\check{F}$ endowed with this norm is complete. For (α, x) ∈ E × F,

$${(\mathit{\alpha },x)}^{*}(\mathit{\alpha },x)=({\mathit{\alpha }}^{*}\mathit{\alpha },{\mathit{\alpha }}^{*}x+x^{*}\mathit{\alpha }+x^{*}x),$$

$$\Vert {(\mathit{\alpha },x)}^{*}(\mathit{\alpha },x)\Vert =\text{sup}\{{\Vert \mathit{\alpha }\Vert }^2,\underset{y,\mathfrak{F}}{\text{lim}\text{sup}}\Vert {\mathit{\alpha }}^{*}\mathit{\alpha }y+{\mathit{\alpha }}^{*}x+x^{*}\mathit{\alpha }+x^{*}x\Vert \}.$$

For $y\in F_{+}^{\#}$,

$$\Vert (\mathit{\alpha }{y}^{\frac{1}{2}}+x){*}^{}(\mathit{\alpha }{y}^{\frac{1}{2}}+x)-({\mathit{\alpha }}^{*}\mathit{\alpha }y+{\mathit{\alpha }}^{*}x+x^{*}\mathit{\alpha }+x^{*}x)\Vert \le$$

$$\le \Vert y^{\frac{1}{2}}\mathit{\alpha }*\mathit{\alpha }-\mathit{\alpha }*\mathit{\alpha }{y}^{\frac{1}{2}}\Vert +\Vert y^{\frac{1}{2}}\mathit{\alpha }*x-\mathit{\alpha }*x\Vert +\Vert x*\mathit{\alpha }{y}^{\frac{1}{2}}-x*\mathit{\alpha }\Vert$$

so

$$\underset{y,\mathfrak{F}}{\text{lim}}\Vert (\mathit{\alpha }{y}^{\frac{1}{2}}+x){*}^{}(\mathit{\alpha }{y}^{\frac{1}{2}}+x)-({\mathit{\alpha }}^{*}\mathit{\alpha }y+{\mathit{\alpha }}^{*}x+x^{*}\mathit{\alpha }+x^{*}x)\Vert =0.$$

Since the map $F_{+}⟶F_{+},y⟼y^{\frac{1}{2}}$ maps $\mathfrak{F}$ into itself and

$${\Vert \mathit{\alpha }y+x\Vert }^2=\Vert y{\mathit{\alpha }}^{*}\mathit{\alpha }y+y{\mathit{\alpha }}^{*}x+x^{*}\mathit{\alpha }y+x^{*}x\Vert$$

we have by the above,

$${\Vert (\mathit{\alpha },x)\Vert }^2=\text{sup}\left\{{\Vert \mathit{\alpha }\Vert }^2,\underset{y,\mathfrak{F}}{\text{lim}\text{sup}}{\Vert \mathit{\alpha }{y}^{\frac{1}{2}}+x\Vert }^2\right\}=$$

$$=\text{sup}\left\{{\Vert \mathit{\alpha }\Vert }^2,\underset{y,\mathfrak{F}}{\text{lim}\text{sup}}\Vert (\mathit{\alpha }{y}^{\frac{1}{2}}+x){*}^{}(\mathit{\alpha }{y}^{\frac{1}{2}}+x)\Vert \right\}=$$

$$=\text{sup}\{\Vert \mathit{\alpha }{\Vert }^2,\underset{y,\mathfrak{F}}{\text{lim}\text{sup}}\Vert {\mathit{\alpha }}^{*}\mathit{\alpha }y+{\mathit{\alpha }}^{*}x+x^{*}\mathit{\alpha }+x^{*}x\Vert \}=\Vert (\mathit{\alpha },x{)}^{*}(\mathit{\alpha },x)\Vert .$$

Thus the above norm is a C*-norm and F is adapted.

h) ι is an injective E-C*-homomorphism and its image is equal to Ker π.

COROLLARY 1.2.5 Let F an E-module, G a C*-algebra, and ⊗_σ the spatial tensor product.

• a) F ⊗_σ G is in a natural way an E-module the multiplication being given by

  $$\mathit{\alpha }(x\otimes y)=(\mathit{\alpha }x)\otimes y,(x\otimes y)\mathit{\alpha }=(x\mathit{\alpha })\otimes y$$

  for all α ∈ E, x ∈ F, and y ∈ G.

• b) If F is an E-C*-algebra and G is unital then the map

  $$E⟶F{\otimes }_{\mathit{\sigma }}G,\mathit{\alpha }⟼\mathit{\alpha }\otimes 1_G$$

  is an injective C*-homomorphism. In particular, the E-module F ⊗_σ G is an E-C*-algebra.

• c) If F is an adapted E-module then the E-module F ⊗_σ G is adapted and

  $$\Vert (\mathit{\alpha },z)\Vert =\text{sup}\{\Vert \mathit{\alpha }\Vert ,\Vert \mathit{\alpha }+z\Vert \}$$

  for all (α, z) ∈ E × (F ⊗_σ G).

• d) If F is an adapted E-module and $G≔{\mathcal{C}}_0(\Omega )$ for a locally compact space Ω then ${\mathcal{C}}_0(\Omega ,F)$ is adapted and

  $$\Vert (\mathit{\alpha },x)\Vert =\text{sup}\{\Vert \mathit{\alpha }\Vert ,\Vert \mathit{\alpha }{e}_{\Omega }+x\Vert \}$$

  for all (α, x) ∈ E × C₀(Ω, F).

a) and b) are easy to see.

c) If $\tilde{G}$ denotes the unitization of G then by b), $\check{F}{\otimes }_{\mathit{\sigma }}\tilde{G}$ is an E-C*-algebra and F ⊗_σ G is a closed ideal of it, so the assertion follows from Proposition 1.2.4 d),e).

d) follows from c).

PROPOSITION 1.2.6

• a) If F,G are E-modules and φ : F ⟶ G is an E-linear C*-homomorphism then the map

  $$\check{\mathit{\phi }}:\check{F}⟶\check{G},(\mathit{\alpha },x)⟼(\mathit{\alpha },\mathit{\phi }x)$$

  is an involutive unital algebra homomorphism, injective or surjective if φ is so. If F = G and if φ is the identity map then $\check{\mathit{\phi }}$ is also the identity map.

• b) Let F₁, F₂, F₃ be E-modules and let φ : F₁ ⟶ F₂ and ψ : F₂ ⟶ F₃ be E-linear C*-homomorphisms. Then $\stackrel{ˇ}{\stackrel{}{\overbrace{\mathit{\psi }o\mathit{\phi }}}}=\check{\mathit{\psi }}o\check{\mathit{\phi }}$.

PROPOSITION 1.2.7 Let G be an E-module, F an E-submodule of G which is at the same time an ideal of G, and φ : G ⟶ G/F the quotient map.

• a) G/F has a natural structure of E-module and φ is E-linear.

• b) If G is adapted then G/F is also adapted. Moreover if $\mathit{\psi }:\check{G}⟶\check{G}/F$ denotes the quotient map (where F is identified to {(0, x) |x ∈ F}) then there is an E-C*-isomorphism $\mathit{\theta }:\check{\stackrel{}{\overbrace{G/F}}}⟶\check{G}/F$ such that $\mathit{\psi }=\mathit{\theta }o\check{\mathit{\phi }}$.

a) is easy to see.

b) Let $(\mathit{\alpha },z)\in \check{\stackrel{}{\overbrace{G/F}}}$ and let $x,y\in \stackrel{-1}{\mathit{\phi }}(z)$. Then ψ(α, x) = ψ(α, y) and we put θ(α, z) ≔ ψ(α, x). It is straightforward to show that θ is an isomorphism of involutive algebras. By pulling back the norm of $\check{G}/F$ with respect to θ we see that G/F is adapted.

LEMMA 1.2.8 Let {(F_i)_{i∈ I}, (φ_ij)_{i,j ∈ I})} be an inductive system in the category of C*-algebras, {F,(φ_i)_{i ∈ I}} its inductive limit, G a C*-algebra, for every i ∈ I, ψ_i : F_i ⟶ G a C*-homomorphism such that ψ_j ο φ_ji = ψ_i for all i,j ∈ I, i ≤ j, and ψ : F ⟶ G the resulting C*-homomorphism. If Ker ψ_i ⊂ Ker φ_i for every i ∈ I then ψ is injective.

Let i ∈ I. Since Ker φ_i ⊂ Ker ψ_i is obvious, we have Ker φ_i = Ker ψ_i. Let ρ : F_i ⟶ F_i/Ker ψ_i be the quotient map and

$${\mathit{\phi }}_i^{\prime }:F_i/Ker{\mathit{\psi }}_i⟶F,{\mathit{\psi }}_i^{\prime }:F_i/Ker{\mathit{\psi }}_i⟶G$$

the injective C*-homomorphisms with

$${\mathit{\phi }}_i={\mathit{\phi }}_i^{\prime }o\mathit{\rho },{\mathit{\psi }}_i={\mathit{\psi }}_i^{\prime }o\mathit{\rho }.$$

Then

$${\mathit{\psi }}_i^{\prime }o\mathit{\rho }={\mathit{\psi }}_i=\mathit{\psi }o{\mathit{\phi }}_i=\mathit{\psi }o{\mathit{\phi }}_i^{\prime }o\mathit{\rho }.$$

For x ∈ F_i, since ψ_i′ and φ_i′ are norm-preserving,

$$\Vert \mathit{\rho }x\Vert =\Vert {\mathit{\psi }}_i^{\prime }\mathit{\rho }x\Vert =\Vert \mathit{\psi }{\mathit{\phi }}_i^{\prime }\mathit{\rho }x\Vert \le \Vert {\mathit{\phi }}_i^{\prime }\mathit{\rho }x\Vert =\Vert \mathit{\rho }x\Vert ,$$

$$\Vert \mathit{\psi }{\mathit{\phi }}_{i}x\Vert =\Vert \mathit{\psi }{\mathit{\phi }}_i^{\prime }\mathit{\rho }x\Vert =\Vert {\mathit{\phi }}_i^{\prime }\mathit{\rho }x\Vert =\Vert {\mathit{\phi }}_{i}x\Vert .$$

Thus ψ preserves the norms on ∪_{i∈ I}φ_i(F_i). Since this set is dense in F, ψ is injective.

PROPOSITION 1.2.9 Let {(F_i)_{i ∈ I},(φ_ij)_{i,j ∈ I}} be an inductive system in the category ${\mathfrak{M}}_E$ and let (F,(φ_i)_{i ∈ I}) be its inductive limit in the category of E-modules (Proposition 1.2.4 c)).

• a) F is adapted.

• b) Let (G,(ψ_i)_{i ∈ I}) be the inductive limit in the category ${ℭ}_E^1$ of the inductive system $\{{({\check{F}}_i)}_{i\in I},{({\check{\mathit{\phi }}}_{ij})}_{i,j\in I}\}$ (Proposition 1.2.6 a),b)) and let $\mathit{\psi }:G⟶\check{F}$ be the unital C*-homomorphism such that $\mathit{\psi }o{\mathit{\psi }}_i={\check{\mathit{\phi }}}_i$ for every i ∈ I. Then ψ is an E-C*-isomorphism.

a) Put

$$F_0≔\left\{(\mathit{\alpha },x)\in \check{F}|\mathit{\alpha }\in E,x\in {\displaystyle \underset{i\in I}{\cup }{\mathit{\phi }}_i}(F_i)\right\},$$

$$p:F_0⟶{ℝ}_{+},(\mathit{\alpha },x)⟼\inf \{\Vert (\mathit{\alpha },x_i)\Vert |i\in I,x_i\in F_i,{\mathit{\phi }}_i{x}_i=x\}.$$

F₀ is an involutive unital subalgebra of $\check{F}$. p is a norm and by Proposition 1.2.4 c),

$$q(\mathit{\alpha },x)≔\underset{\begin{array}{l}(\mathit{\alpha },y)\in F_0\\ y⟶x\end{array}}{\text{lim}}p(\mathit{\alpha },y)$$

exists and

$$\Vert \mathit{\alpha }\Vert \le q(\mathit{\alpha },x)\le \Vert \mathit{\alpha }\Vert +\Vert x\Vert ,\Vert x\Vert \le 2q(\mathit{\alpha },x)$$

for every $(\mathit{\alpha },x)\in \check{F}$.

Let (α, x) ∈ F₀. Let further i ∈ I, x_i, y_i ∈ F_i with φ_ix_i = x, φ_iy_i = α*x + x*α + x*x. Then

$$(0,{\mathit{\phi }}_i({\mathit{\alpha }}^{*}{x}_i+x_i^{*}\mathit{\alpha }+x_i^{*}{x}_i-y_i))=\check{{\mathit{\phi }}_i}({(\mathit{\alpha },x_i)}^{*}(\mathit{\alpha },x_i)-({\mathit{\alpha }}^{*}\mathit{\alpha },y_i))=0$$

so

$$\underset{i\le j}{\inf }\Vert {\mathit{\phi }}_{ji}({\mathit{\alpha }}^{*}{x}_i+x_i^{*}\mathit{\alpha }+x_i^{*}{x}_i-y_i)\Vert =0.$$

For ε > 0 there is a j ∈ I, i ≤ j, with

$$\Vert {\mathit{\phi }}_{ji}({\mathit{\alpha }}^{*}{x}_i+x_i^{*}\mathit{\alpha }+x_i^{*}{x}_i-y_i)\Vert <ϵ.$$

We get

$$p{(\mathit{\alpha },x)}^2\le {\Vert (\mathit{\alpha },{\mathit{\phi }}_{ji}{x}_i)\Vert }^2=\Vert {(\mathit{\alpha },{\mathit{\phi }}_{ji}{x}_i)}^{*}(\mathit{\alpha },{\mathit{\phi }}_{ji}{x}_i)\Vert =$$

$$=\Vert ({\mathit{\alpha }}^{*}\mathit{\alpha },{\mathit{\alpha }}^{*}{\mathit{\phi }}_{ji}{x}_i+({\mathit{\phi }}_{ji}{x}_i^{*})\mathit{\alpha }+{\mathit{\phi }}_{ji}(x_i^{*}{x}_i))\Vert =$$

$$=\Vert ({\mathit{\alpha }}^{*}\mathit{\alpha },{\mathit{\phi }}_{ji}({\mathit{\alpha }}^{*}{x}_i+x_i^{*}\mathit{\alpha }+x_i^{*}{x}_i))\Vert \le$$

$$\le \Vert ({\mathit{\alpha }}^{*}\mathit{\alpha },{\mathit{\phi }}_{ji}{y}_i)\Vert +\Vert (0,{\mathit{\phi }}_{ji}({\mathit{\alpha }}^{*}{x}_i+x_i^{*}\mathit{\alpha }+x_i^{*}{x}_i-y_i))\Vert <\Vert ({\mathit{\alpha }}^{*}\mathit{\alpha },{\mathit{\phi }}_{ji}{y}_i)\Vert +ϵ.$$

By taking the infimum on the right side it follows, since ε is arbitrary,

$$p{(\mathit{\alpha },x)}^2\le p({\mathit{\alpha }}^{*}\mathit{\alpha },{\mathit{\alpha }}^{*}x+x^{*}\mathit{\alpha }+x^{*}x)=p({(\mathit{\alpha },x)}^{*}(\mathit{\alpha },x))$$

and this shows that p is a C*-norm. It is easy to see that q is a C*-norms. By the above inequalities, $\check{F}$ endowed with the norm q is complete, i.e. $\check{F}$ is a C*-algebra and F is adapted.

b) Let i ∈ I and let $(\mathit{\alpha },x)\in Ker{\check{\mathit{\phi }}}_i$. Then

$$0={\check{\mathit{\phi }}}_i(\mathit{\alpha },x)=(\mathit{\alpha },{\mathit{\phi }}_{i}x)$$

so

$$\mathit{\alpha }=0,{\mathit{\phi }}_{i}x=0,\underset{j\in I,j\ge i}{\inf }\Vert {\mathit{\phi }}_{ji}x\Vert =0,$$

$$\Vert {\check{\mathit{\phi }}}_{ji}(0,x)\Vert =\Vert (0,{\mathit{\phi }}_{ji}x)\Vert =\Vert {\mathit{\phi }}_{ji}x\Vert ,$$

$$\Vert {\mathit{\psi }}_i(\mathit{\alpha },x)\Vert =\underset{j\in I,j\ge i}{\inf }\Vert {\check{\mathit{\phi }}}_{ji}(0,x)\Vert =0,(\mathit{\alpha },x)\in Ker{\mathit{\psi }}_i.$$

By Lemma 1.2.8, ψ is injective.

Let $(\mathit{\beta },y)\in \check{F}$ and let ε > 0. There are i ∈ I and x ∈ F_i with ‖φ_ix − y‖ < ε. Then

$$\mathit{\psi }{\mathit{\psi }}_i(\mathit{\beta },x)={\check{\mathit{\phi }}}_i(\mathit{\beta },x)=(\mathit{\beta },{\mathit{\phi }}_{i}x),$$

$$\Vert \mathit{\psi }{\mathit{\psi }}_i(\mathit{\beta },x)-(\mathit{\beta },y)\Vert =\Vert {\check{\mathit{\phi }}}_i(\mathit{\beta },x)-(\mathit{\beta },y)\Vert =\Vert {\mathit{\phi }}_{i}x-y\Vert <\mathit{\epsilon }.$$

Thus ψ(G) is dense in $\check{F}$ and ψ is surjective. Hence ψ is a C*-isomorphism.

COROLLARY 1.2.10 We put ${\Phi }_E(F)≔\check{F}$ for every E-module F and similarly ${\Phi }_E(\mathit{\phi })≔\check{\mathit{\phi }}$ for every E-linear C*-homomorphism φ.

• a) Φ_E is a covariant functor from the category ${\mathfrak{M}}_E$ in the category ${ℭ}_E^1$.

• b) The categories ${ℭ}_E^1$ and ${\mathfrak{M}}_E$ possess inductive limits and the functor Φ_E is continuous with respect to the inductive limits.

a) follows from Proposition 1.2.6.

b) follows from Proposition 1.2.9.

Remark. The category ℭ_E does not possess inductive limits in general. This happens for instance if φ_ij = 0 for all i, j ∈ I.

1.3. Some topologies

T is only a set in this section

The purely algebraic projective representation of a group produces only an involutive algebra. In order to obtain a C*-algebra we need to take the closure with respect to a certain topology. For this purpose we shall define some different topologies, but it will be shown that all these topologies conduct to the same construction. The use of more topologies simplifies the manipulations.

We introduce the following notation in order to unify the cases of C*-algebras and (resp. W*-algebras).

DEFINITION 1.3.1

$$\widetilde{⦶}:⦶(\text{resp}.\widetilde{⦶}≔\stackrel{W}{⦶}),$$

$$\tilde{\otimes }≔\otimes (\text{resp}.\tilde{\otimes }≔\bar{\otimes }),$$

$$\tilde{\sum }≔{\displaystyle \sum }(\text{resp}.\tilde{\sum }≔{\displaystyle \sum }^{\ddot{E}}).$$

If $\mathfrak{T}$ is a Hausdorff topology on ${\mathcal{L}}_E(H)$ then for every $\mathcal{G}\subset {\mathcal{L}}_E(H)$, ${\mathcal{G}}_{\mathfrak{T}}$ denotes the set $\mathcal{G}$ endowed with the relative topology $\mathfrak{T}$ and $\stackrel{\mathfrak{T}}{\overline{\mathcal{G}}}$ denotes the closure of $\mathcal{G}$ in ${\mathcal{L}}_E{(H)}_{\mathfrak{T}}$. Moreover $\stackrel{\mathfrak{T}}{\Sigma }$ denotes the sum with respect to $\mathfrak{T}$.

LEMMA 1.3.2 For x ∈ E, by the above identification of E with ${\mathcal{L}}_E(\mathit{Ĕ})$,

$$x\tilde{\otimes }{1}_K:H⟶H,\mathit{\xi }⟼{(x{\mathit{\xi }}_t)}_{t\in T}$$

is well-defined and belongs to ${\mathcal{L}}_E(H)$.

• a) The map

  $$\mathit{\phi }:E⟶{\mathcal{L}}_E(H),x⟼x\tilde{\otimes }{1}_K$$

  is an injective unital C*-homomorphism.

• b) Assume E is a W*-algebra. Then for every $(a,\mathit{\xi },\mathit{\eta })\in \ddot{E}\times H\times H$, the family ${({\mathit{\xi }}_{t}a{\mathit{\eta }}_t^{*})}_{t\in T}$ is summable in ${\ddot{E}}_E$ and for every x ∈ E,

  $$\langle \mathit{\phi }x,(\widetilde{a,\mathit{\xi },n})\rangle =\langle x,{\displaystyle \sum _{t\in T}^E{\mathit{\xi }}_{t}a{\mathit{\eta }}_t^{*}}\rangle .$$

  Thus φ is a W*-homomorphism ([1] Theorem 5.6.3.5 d)) with

  $$\ddot{\mathit{\phi }}(\widetilde{a,\mathit{\xi },\mathit{\eta }})={\displaystyle \sum _{t\in T}^E{\mathit{\xi }}_t}a{\mathit{\eta }}_t^{*},$$

  where $\ddot{\mathit{\phi }}$ denotes the pretranspose of φ.

• c) If we consider E as a canonical unital C**-subalgebra of ${\mathcal{L}}_E(H)$ by using the embedding of a) then ${\mathcal{L}}_E(H)$ is an E-C**-algebra.

a) follows from [6] page 37 (resp. [3] Proposition 1.4).

b) We have

$$\langle x\bar{\otimes }{1}_K,(\widetilde{a,\mathit{\xi },\mathit{\eta }})\rangle =\langle \langle (x\bar{\otimes }{1}_K)\mathit{\xi }|\mathit{\eta }\rangle ,a\rangle =\langle {\displaystyle \sum _{t\in T}^{\ddot{E}}{\mathit{\eta }}_t^{*}}x{\mathit{\xi }}_t,a\rangle =$$

$$={\displaystyle \sum _{t\in T}\langle {\mathit{\eta }}_t^{*}x\mathit{\xi }t,a\rangle }={\displaystyle \sum _{t\in T}\langle x,\mathit{\xi }t,a{\mathit{\eta }}_t^{*}\rangle .}$$

Thus the family ${({\mathit{\xi }}_{t}a{\mathit{\eta }}_t^{*})}_{t\in T}$ is summable in ${\ddot{E}}_E$ and

$$\langle \mathit{\phi }x,(\widetilde{a,\mathit{\xi },\mathit{\eta }})\rangle =\langle x,{\displaystyle \sum _{t\in T}^E{\mathit{\xi }}_t}a{\mathit{\eta }}_t^{*}\rangle .$$

If $\mathit{\phi }\prime :{\mathcal{L}}_E(H)⟶E\prime$ denotes the transpose of φ then

$$\mathit{\phi }\prime (\widetilde{a,\mathit{\xi },\mathit{\eta }})={\displaystyle \sum _{t\in T}^E{\mathit{\xi }}_t}a{\mathit{\eta }}_t^{*}\in \ddot{E}.$$

By continuity $\mathit{\phi }\prime \left(\stackrel{\mathrm{..}}{\overbrace{{\mathcal{L}}_E(H)}}\right)\subset \ddot{E}$ and φ is a unital W*-homomorphism.

c) Let x ∈ E^c and $\mathit{\xi },\mathit{\eta }\in {\mathcal{L}}_E(H)$. By [1] Proposition 5.6.3.17 d),

$$\langle (x\tilde{\otimes }{1}_K)\mathit{\xi }|\mathit{\eta }\rangle =\widetilde{{\displaystyle \sum }_{t\in T}^{}}{\mathit{\eta }}_t^{*}{((x\tilde{\otimes }{1}_K)\mathit{\xi })}_t=\widetilde{{\displaystyle \sum }_{t\in T}^{}}{\mathit{\eta }}_t^{*}x{\mathit{\xi }}_t=$$

$$=\widetilde{{\displaystyle \sum }_{t\in T}^{}}x{\mathit{\eta }}_t^{*}{\mathit{\xi }}_t=x\widetilde{{\displaystyle \sum }_{t\in T}^{}}{\mathit{\eta }}_t^{*}{\mathit{\xi }}_t=x\langle \mathit{\xi }|\mathit{\eta }\rangle .$$

Thus for $u\in {\mathcal{L}}_E(H)$,

$$\langle u(x\tilde{\otimes }{1}_K)\mathit{\xi }|\mathit{\eta }\rangle =\langle (x\tilde{\otimes }{1}_K)\mathit{\xi }|u^{*}\mathit{\eta }\rangle =x\langle \mathit{\xi }|u^{*}\mathit{\eta }\rangle =x\langle u\mathit{\xi }|\mathit{\eta }\rangle ,$$

$$u(x\tilde{\otimes }{1}_K)=(x\tilde{\otimes }{1}_K)u,$$

and so $x\tilde{\otimes }{1}_K\in {\mathcal{L}}_E{(H)}^c$.

DEFINITION 1.3.3 We put for all ξ, η ∈ H (resp. and $a\in {\ddot{E}}_{+}$)

$$P_{\mathit{\xi },\mathit{\eta }}:{\mathcal{L}}_E(H)⟶{ℝ}_{+},X⟼\Vert \langle X\mathit{\xi }|\mathit{\eta }\rangle \Vert ,$$

$$(\text{resp}.P_{\mathit{\xi },\mathit{\eta },a}:{\mathcal{L}}_E(H)⟶{ℝ}_{+},X⟼|\langle \langle X\mathit{\xi }|\mathit{\eta }\rangle ,a\rangle |),$$

$$P_{\mathit{\xi }}:{\mathcal{L}}_E(H)⟶{ℝ}_{+},X⟼\Vert X\mathit{\xi }\Vert =\Vert \langle X\mathit{\xi }|X\mathit{\xi }\rangle {\Vert }^{1/2},$$

$$(\text{resp}.P_{\mathit{\xi },a}:{\mathcal{L}}_E(H)⟶{ℝ}_{+},X⟼{\langle \langle X\mathit{\xi }|X\mathit{\xi }\rangle ,a\rangle }^{1/2}),$$

$$q_{\mathit{\xi }}:{\mathcal{L}}_E(H)⟶{ℝ}_{+},X⟼p_{\mathit{\xi }}(X^{*}),$$

$$(\text{resp}.q_{\mathit{\xi },a}:{\mathcal{L}}_E(H)⟶{ℝ}_{+},X⟼p_{\mathit{\xi },a}(X^{*})).$$

and denote, respectively, by ${\mathfrak{T}}_1,{\mathfrak{T}}_2,{\mathfrak{T}}_3$ the topologies on ${\mathcal{L}}_E(H)$ generated by the set of seminorms

$$\left\{p_{\mathit{\xi },\mathit{\eta }}|\mathit{\xi },\mathit{\eta }\in H\right\},\left(\text{resp}.\left\{p_{\mathit{\xi },\mathit{\eta },a}|\mathit{\xi },\mathit{\eta }\in H,a\in {\ddot{E}}_{+}\right\}\right),$$

$$\left\{p_{\mathit{\xi }}|\mathit{\xi }\in H\right\},\left(\text{resp}.\left\{p_{\mathit{\xi },a}|\mathit{\xi }\in H,a\in {\ddot{E}}_{+}\right\}\right),$$

$$\left\{p_{\mathit{\xi }}|\mathit{\xi }\in H\right\}\cup \left\{q_{\mathit{\xi }}|\mathit{\xi }\in H\right\},$$

$$\left(\text{resp}.\left\{p_{\mathit{\xi },a}|\mathit{\xi }\in H,a\in {\ddot{E}}_{+}\right\}\cup \left\{q_{\mathit{\xi },a}|\mathit{\xi }\in H,a\in {\ddot{E}}_{+}\right\}\right).$$

Moreover ‖⋅‖ denotes the norm topology on ${\mathcal{L}}_E(H)$.

Of course ${\mathfrak{T}}_2\subset {\mathfrak{T}}_3$. In the C*-case, ${\mathfrak{T}}_2$ is the topology of pointwise convergence. If E is finite-dimensional then the C*-case and the W*-case coincide.

PROPOSITION 1.3.4 Let $X\in {\mathcal{L}}_E(H)$ and ξ, η ∈ H (resp. and $a\in \ddot{E}$).

• a) p_ξ,η(X) = p_η,ξ(X*)  (resp. p_ξ,η,|a|(X) = p_η,ξ,|a|(X*)).

• b) p_ξ,η(X) ≤ p_ξ(X) ‖η‖.

• c) If E is a W*-algebra and a = x|a| is the polar representation of a then

  $$p_{\mathit{\xi }x,\mathit{\eta },\left|a\right|}(X)=|\langle X,(\widetilde{a,\mathit{\xi },\mathit{\eta }})\rangle |\le p_{\mathit{\xi }x,\left|a\right|}(X){\langle \langle \mathit{\eta }|\mathit{\eta }\rangle ,\left|a\right|\rangle }^{1/2}.$$

• d) If $Y,Z\in {\mathcal{L}}_E(H)$ then

  $$p_{\mathit{\xi },\mathit{\eta }}(YXZ)=p_{Z\mathit{\xi },Y^{*}\mathit{\eta }}(X)(\text{resp}{p}_{\mathit{\xi },\mathit{\eta },\left|a\right|}(YXZ)=p_{Z\mathit{\xi },Y^{*}\mathit{\eta },\left|a\right|}(X)),$$

  $$p_{\mathit{\xi }}(YXZ)\le \Vert Y\Vert p_{Z\mathit{\xi }}(X)(\text{resp}.p_{\mathit{\xi },\left|a\right|}(YXZ)\le \Vert Y\Vert p_{Z\mathit{\xi },\left|a\right|}(X)).$$

a) From

$$\langle X\mathit{\xi }|\mathit{\eta }\rangle =\langle \mathit{\xi }|X^{*}\mathit{\eta }\rangle ={\langle X^{*}\mathit{\eta }|\mathit{\xi }\rangle }^{*}$$

it follows

$$p_{\mathit{\xi },\mathit{\eta }}(X)=\Vert \langle X\mathit{\xi }|\mathit{\eta }\rangle \Vert =\Vert \langle X^{*}\mathit{\eta }|\mathit{\xi }\rangle \Vert =p_{\mathit{\eta },\mathit{\xi }}(X^{*}),$$

$$(\text{resp}\text{.}{p}_{\mathit{\xi },\mathit{\eta },\left|a\right|}(X)=|\langle \langle X^{*}\mathit{\eta }|\mathit{\xi }\rangle ,\left|a\right|\rangle |=p_{\mathit{\eta },\mathit{\xi },\left|a\right|}(X^{*})).$$

b) $p_{\mathit{\xi },\mathit{\eta }}(X)=\Vert \langle X\mathit{\xi }|\mathit{\eta }\rangle \Vert \le p_{\mathit{\xi }}(X)\Vert \mathit{\eta }\Vert$.

c) We have

$$p_{\mathit{\xi }x,\mathit{\eta },|a|}(X)=|\langle \langle X(\mathit{\xi }x)|\mathit{\eta }\rangle ,|a|\rangle |=|\langle \langle X\mathit{\xi }|\mathit{\eta }\rangle x,|a|\rangle |=$$

$$=|\langle \langle X\mathit{\xi }|\mathit{\eta }\rangle ,x|a|\rangle |=|\langle \langle X\mathit{\xi }|\mathit{\eta }\rangle ,a\rangle |=\left|\langle X,(\widetilde{a,\mathit{\xi },\mathit{\eta }})\rangle \right|.$$

By Schwarz’ inequality ([1] Proposition 2.3.3.9),

$$|\langle \langle X(\mathit{\xi }x)|\mathit{\eta }\rangle ,\left|a\right|\rangle {|}^2\le \langle \langle X(\mathit{\xi }x)|X(\mathit{\xi }x)\rangle ,\left|a\right|\rangle \langle \langle \mathit{\eta }|\mathit{\eta }\rangle ,\left|a\right|\rangle ,$$

so

$$p_{\mathit{\xi }x,\mathit{\eta },\left|a\right|}(X)\le p_{\mathit{\xi }x,\left|a\right|}(X){\langle \langle \mathit{\eta }|\mathit{\eta }\rangle ,\left|a\right|\rangle }^{1/2}.$$

d) The first equation follows from

$$p_{\mathit{\xi },\mathit{\eta }}(YXZ)=\Vert \langle YXZ\mathit{\xi }|\mathit{\eta }\rangle \Vert =\Vert \langle XZ\mathit{\xi }|Y^{*}\mathit{\eta }\rangle \Vert =p_{Z\mathit{\xi },Y^{*}\mathit{\eta }}(X)$$

$$(\text{resp}\text{.}{p}_{\mathit{\xi },\mathit{\eta },|a|}(YXZ)=|\langle \langle YXZ\mathit{\xi }|\mathit{\eta }\rangle ,|a|\rangle |=$$

$$=|\langle \langle XZ\mathit{\xi }|Y^{*}\mathit{\eta }\rangle ,|a|\rangle |=p_{Z\mathit{\xi },Y^{*}\mathit{\eta },|a|}(X))$$

and the second from

$$p_{\mathit{\xi }}(YXZ)=\Vert YXZ\mathit{\xi }\Vert \le \Vert Y\Vert \Vert XZ\mathit{\xi }\Vert =\Vert Y\Vert p_{Z\mathit{\xi }}(X)$$

$$(\text{resp}\text{.}{p}_{\mathit{\xi },\left|a\right|}(YXZ)={\langle \langle YXZ\mathit{\xi }|YXZ\mathit{\xi }\rangle ,\left|a\right|\rangle }^{1/2}\le$$

$$\le \Vert Y\Vert {\langle \langle XZ\mathit{\xi }|XZ\mathit{\xi }\rangle ,\left|a\right|\rangle }^{1/2}=\Vert Y\Vert p_{Z\mathit{\xi },\left|a\right|}(X)).$$

LEMMA 1.3.5 Let n ∈ ℕ and $(x_i{)}_{i\in {\mathbb{N}}_{\text{n}}}$ a family in E. Then

$${\left({\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{x}_i}\right)}^{*}\left({\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{x}_i}\right)\le n{\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{x}_i^{*}{x}_i.}$$

We prove the relation by induction with respect to n. By [1] Corollary 4.2.2.4 and by the hypothesis of the induction,

$$\begin{array}{l}{\left({\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{x}_i}\right)}^{*}\left({\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{x}_i}\right)=\left(x_n^{*}+{\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}-1}}{x}_i^{*}}\right)\left(x_n+{\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}-1}}{x}_i}\right)=\\ =x_n^{*}{x}_n+{\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}-1}}(x_n^{*}{x}_i+x_i^{*}{x}_n)+{\left({\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}-1}}{x}_i}\right)}^{*}\left({\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}-1}}{x}_i}\right)}\le \\ \le x_n^{*}{x}_n+{\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}-1}}(x_n^{*}{x}_n+x_i^{*}{x}_i)+(n-1){\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}-1}}{x}_i^{*}{x}_i=n{\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{x}_i^{*}{x}_i}}}.\end{array}$$

LEMMA 1.3.6 Let n ∈ ℕ, x ∈ E_n,n, and for every j ∈ ℕ_n put

$${\mathit{\eta }}_j:{({\mathit{\delta }}_{ji}{1}_E)}_{i\in {\mathbb{N}}_{\text{n}}}\in \underset{i\in {\mathbb{N}}_{\text{n}}}{⦶}\mathit{Ĕ}.$$

Then

$$\Vert x\Vert \le \sqrt{n}\underset{j\in {\mathbb{N}}_{\text{n}}}{\text{sup}}\Vert x{\mathit{\eta }}_j\Vert .$$

For $\mathit{\xi }\in {\left(\underset{i\in {\mathbb{N}}_{\text{n}}}{⦶}\mathit{Ĕ}\right)}^{\#}$, by Lemma 1.3.5,

$$\begin{array}{l}\langle x\mathit{\xi }|x\mathit{\xi }\rangle ={\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}\langle {(x\mathit{\xi })}_i|{(x\mathit{\xi })}_i\rangle =}{\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{\left({\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{x}_{ij}{\mathit{\xi }}_j}\right)}^{*}\left({\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{x}_{ij{\mathit{\xi }}_j}}\right)\le }\\ \le n{\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{(x_{ij}{\mathit{\xi }}_j)}^{*}(x_{ij}{\mathit{\xi }}_j)=n{\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{\mathit{\xi }}_j^{*}{x}_{ij}^{*}{x}_{ij}{\mathit{\xi }}_j=}}}}\\ =n{\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{\mathit{\xi }}_j^{*}\left({\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{x}_{ij}^{*}{x}_{ij}}\right){\mathit{\xi }}_j}.\end{array}$$

For i, j ∈ ℕ_n,

$$\begin{array}{l}(x{\mathit{\eta }}_j{)}_i={\displaystyle \sum _{k\in {\mathbb{N}}_{\text{n}}}{x}_{ik}{\mathit{\eta }}_{jk}}=x_{ij},\\ \langle x{\mathit{\eta }}_j|x{\mathit{\eta }}_j\rangle ={\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{(x{\mathit{\eta }}_j)}_i^{*}{(x{\mathit{\eta }}_j)}_i={\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{x}_{ij}^{*}{x}_{ij},}}\end{array}$$

so

$$\begin{array}{l}\langle x\mathit{\xi }|x\mathit{\xi }\rangle \le n{\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{\mathit{\xi }}_j^{*}\langle x{\mathit{\eta }}_j|x{\mathit{\eta }}_j\rangle {\mathit{\xi }}_j\le n{\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}\Vert x{n}_j{\Vert }^2{\mathit{\xi }}_j^{*}{\mathit{\xi }}_j}\le }\\ \le n\underset{j\in {\mathbb{N}}_{\text{n}}}{\text{sup}}\Vert x{\mathit{\eta }}_j{\Vert }^2{\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{\mathit{\xi }}_j^{*}{\mathit{\xi }}_j\le n\underset{j\in {\mathbb{N}}_{\text{n}}}{\text{sup}}\Vert x{\mathit{\eta }}_j{\Vert }^2{1}_E,}\\ \Vert x{\Vert }^2\le n\underset{j\in {\mathbb{N}}_{\text{n}}}{\text{sup}}\Vert x{\mathit{\eta }}_j{\Vert }^2.\end{array}$$

COROLLARY 1.3.7

• a) The map

  $${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_1}⟶{\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_1},X⟼X^{*}$$

  is continuous. In particular $Re{\mathcal{L}}_E(H)$ is a closed set of ${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_1}$.

• b) ${\mathfrak{T}}_1\subset {\mathfrak{T}}_2\subset {\mathfrak{T}}_3\subset$ norm topology.

• c) If E is a W*-algebra then the identity map

  $${\mathcal{L}}_E{(H)}_{\stackrel{\dots }{H}}⟶{\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_1}$$

  is continuous so

  $${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_1}^{\#}={\mathcal{L}}_E{(H)}_{\stackrel{\dots }{H}}{\#}^{}$$

  is compact.

• d) For $Y,Z\in {\mathcal{L}}_E(H)$ and k ∈ {1,2}, the map

  $${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_k}⟶{\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_k},X⟼YXZ$$

  is continuous.

• e) ${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_3}$ is complete in the C*-case.

• f) If T is finite then ${\mathfrak{T}}_2$ is the norm topology in the C*-case.

• g) ${\mathcal{K}}_E(H)$ is dense in ${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_3}$.

a) follows from Proposition 1.3.4 a).

b) ${\mathfrak{T}}_1\subset {\mathfrak{T}}_2$ follows from Proposition 1.3.4 b),c). ${\mathfrak{T}}_2\subset {\mathfrak{T}}_3\subset$ norm topology is trivial.

c) follows from Proposition 1.3.4 c) (and [1] Theorem 5.6.3.5 a)).

d) follows from Proposition 1.3.4 d).

e) Let $\mathfrak{F}$ be a Cauchy filter on ${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_3}$. Put

$$Y:H⟶H,\mathit{\xi }⟼\underset{X,\mathfrak{F}}{\text{lim}}(X\mathit{\xi }),$$

$$Z:H⟶H,\mathit{\xi }⟼\underset{X,\mathfrak{F}}{\text{lim}}(X^{*}\mathit{\xi }),$$

where the limits are considered in the norm topology of H. For ξ, η ∈ H,

$$\langle Y\mathit{\xi }|\mathit{\eta }\rangle =\underset{X,\mathfrak{F}}{\text{lim}}\langle X\mathit{\xi }|\mathit{\eta }\rangle =\underset{X,\mathfrak{F}}{\text{lim}}\langle \mathit{\xi }|X^{*}\mathit{\eta }\rangle =\langle \mathit{\xi }|Z\mathit{\eta }\rangle ,$$

so $Y,Z\in {\mathcal{L}}_E(H)$ and Z = Y *. Thus $\mathfrak{F}$ converges to Y in ${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_3}$ and ${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_3}$ is complete.

f) follows from b) and Lemma 1.3.6.

g) Let $X\in {\mathcal{L}}_E(H)$ and ξ ∈ H. For every $S\in {\mathfrak{P}}_f(T)$ put

$$P_S≔{\displaystyle \sum }_{s\in S}^{}{e}_s\langle ·|e_s\rangle \in Pr{\mathcal{K}}_E(H)$$

and let ${\mathfrak{F}}_T$ be the upper section filter or ${\mathfrak{P}}_f(T)$. Then $P_{S}X\in {\mathcal{K}}_E(H)$ for every $S\in {\mathfrak{P}}_f(T)$ and

$$\underset{S,{\mathfrak{F}}_T}{\text{lim}}{P}_{S}X\mathit{\xi }=X\mathit{\xi }$$

in H (resp. in $H_{\ddot{H}}$) ([1] Proposition 5.6.4.1 e) (resp. [1] Proposition 5.6.4.6 c))). Thus

$$\underset{S,{\mathfrak{F}}_T}{\text{lim}}{P}_{S}X=X$$

with respect to the topology ${\mathfrak{T}}_2$. Since the same holds for X*, it follows that X belongs to the closure of ${\mathcal{K}}_E(H)$ in ${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_3}$.

Remark. The inclusions in b) can be strict as it is known from the case E ≔ 𝕂.

LEMMA 1.3.8 Let G be a W*-algebra and F a C*-subalgebra of G. Then the following are equivalent.

• a) F generates G as a W*-algebra.

• b) F^# is dense in $G_{\ddot{G}}^{\#}$.

• c) F is dense in $G_{\ddot{G}}$.

a ⇒ b follows from [1] Corollary 6.3.8.7.

b ⇒ c is trivial.

c ⇒ a follows from [1] Corollary 4.4.4.12 a).

PROPOSITION 1.3.9 Let G be a W*-algebra, F a C*-subalgebra of G generating it as W*-algebra, I a set, and

$$L≔\underset{i\in I}{⦶}\check{F},M≔\underset{i\in I}{\overset{W}{⦶}}\check{G}.$$

• a) M is the extension of L to a selfdual Hilbert right G-module ([2] Proposition 1.3 f)) and L^# is dense in $M_{\ddot{M}}^{\#}$.

• b) If we denote for every $X\in {\mathcal{L}}_F(L)$ by $\bar{X}\in {\mathcal{L}}_G(M)$ its unique extension ([3] Proposition 1.4 a)) then the map

  $${\mathcal{L}}_F(L)⟶{\mathcal{L}}_G(M),X⟼\bar{X}$$

  is an injective C*-homomorphism and its image is dense in ${\mathcal{L}}_G{(M)}_{\stackrel{\dots }{M}}$.

• c) The map

  $${\mathcal{L}}_F{(L)}_{{\mathfrak{T}}_2}^{\#}⟶{\mathcal{L}}_G{(M)}_{{\mathfrak{T}}_1}^{\#},X⟼\bar{X}$$

  is continuous.

a) By Lemma 1.3.8 a ⇒ b, F^# is dense in $G_{\ddot{G}}^{\#}$ so ${\check{F}}^{\#}$ is dense in ${\check{G}}_{\ddot{\check{G}}}^{\#}$ and $\check{G}$ is the extension of $\check{F}$ to a selfdual Hilbert right G-module ([3] Corollary 1.5 a₂ ⇒ a₁). By [3] Proposition 1.8, M is the extension of L to a selfdual Hilbert right G-module. By [3] Corollary 1.5 a₁ ⇒ a₂, L^# is dense in $M_{\ddot{M}}^{\#}$.

b) By a) and [3] Proposition 1.4 e), the map

$${\mathcal{L}}_F(L)⟶{\mathcal{L}}_G(M),X⟼\bar{X}$$

is an injective C*-homomorphism. By [3] Proposition 1.9 b), its image is dense in ${\mathcal{L}}_G{(M)}_{\stackrel{\dots }{M}}$.

c) Denote by N the vector subspace of $\stackrel{⃛}{M}$ generated by

$$\left\{(\widetilde{a,\mathit{\xi },\mathit{\eta }})|(a,\mathit{\xi },\mathit{\eta })\in \ddot{G}\times L\times L\right\}.$$

By a) and [3] Proposition 1.9 a), N is dense in $\stackrel{⃛}{M}$ so by Corollary 1.3.7 c),

$${\mathcal{L}}_G{(M)}_{{\mathfrak{T}}_1}^{\#}={\mathcal{L}}_G{(M)}_N^{\#}.$$

For $(a,\mathit{\xi },\mathit{\eta })\in {\ddot{G}}_{+}\times L\times L$ and $X\in {\mathcal{L}}_F(L)$, by Proposition 1.3.4 c),

$$p_{\mathit{\xi },\mathit{\eta },a}(\bar{X})=|\langle \langle \bar{X}\mathit{\xi }|\mathit{\eta }\rangle ,a\rangle |=$$

$$=|\langle \langle X\mathit{\xi }|\mathit{\eta }\rangle ,a\rangle |\le p_{\mathit{\xi }x,|a|}(X){\langle \langle \mathit{\eta }|\mathit{\eta }\rangle ,|a|\rangle }^{\frac{1}{2}},$$

where a = x|a| is the polar representation of a, so the map

$${\mathcal{L}}_F{(L)}_{{\mathfrak{T}}_2}^{\#}⟶{\mathcal{L}}_G{(M)}_{{\mathfrak{T}}_1}^{\#},X⟼\bar{X}$$

is continuous.

LEMMA 1.3.10 Let $n\in \mathbb{N},\mathit{\xi }\in \underset{i\in {\mathbb{N}}_{\text{n}}}{⦶}\mathit{Ĕ}$, and

$$x≔{[{\mathit{\xi }}_i{\mathit{\delta }}_{j,1}]}_{i,j\in {\mathbb{N}}_{\text{n}}}\in E_{n,n}.$$

Then ‖x‖ = ‖ξ‖.

For $\mathit{\eta }\in \underset{i\in {\mathbb{N}}_{\text{n}}}{⦶}\mathit{Ĕ}$ and i ∈ ℕ_n,

$$\begin{array}{l}(x\mathit{\eta }{)}_i={\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{x}_{ij}{\mathit{\eta }}_j=}{\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{\mathit{\xi }}_i{\mathit{\delta }}_{j,1}{\mathit{\eta }}_j={\mathit{\xi }}_i{\mathit{\eta }}_1,}\\ \langle x\mathit{\eta }|x\mathit{\eta }\rangle ={\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}\langle {(x\mathit{\eta })}_i|{(x\mathit{\eta })}_i\rangle ={\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}\langle {\mathit{\xi }}_i{\mathit{\eta }}_1|{\mathit{\xi }}_i{\mathit{\eta }}_1\rangle ={\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{\mathit{\eta }}_1^{*}{\mathit{\xi }}_1^{*}{\mathit{\xi }}_i{\mathit{\eta }}_1=}}}\\ ={\mathit{\eta }}_1^{*}\left({\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{\mathit{\xi }}_i^{*}{\mathit{\xi }}_i}\right){\mathit{\eta }}_1={\mathit{\eta }}_1^{*}\langle \mathit{\xi }|\mathit{\xi }\rangle {\mathit{\eta }}_1\le \Vert \mathit{\xi }{\Vert }^2{\mathit{\eta }}_1^{*}{\mathit{\eta }}_1,\\ \Vert x\mathit{\eta }{\Vert }^2\le \Vert \mathit{\xi }{\Vert }^2\Vert {\mathit{\eta }}_1{\Vert }^2\le \Vert \mathit{\xi }{\Vert }^2\Vert \mathit{\eta }{\Vert }^2,\Vert x\Vert \le \Vert \mathit{\xi }\Vert .\end{array}$$

On the other hand if we put $\mathit{\zeta }≔{({\mathit{\delta }}_{i,1}{1}_E)}_{i\in {\mathbb{N}}_{\text{n}}}$ then for i ∈ ℕ_n,

$$\begin{array}{l}(x\mathit{\zeta }{)}_i={\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{x}_{ij}{\mathit{\zeta }}_j={\displaystyle \sum _{j\in {\mathbb{N}}_{\text{n}}}{\mathit{\xi }}_i{\mathit{\delta }}_{j,1}{1}_E={\mathit{\xi }}_i,}}\\ \langle x\mathit{\zeta }|x\mathit{\zeta }\rangle ={\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{(x\mathit{\zeta })}_i^{*}{(x\mathit{\zeta })}_i={\displaystyle \sum _{i\in {\mathbb{N}}_{\text{n}}}{\mathit{\xi }}_i^{*}{\mathit{\xi }}_i=\langle \mathit{\xi }|\mathit{\xi }\rangle ,}}\\ \Vert x\Vert \ge \Vert x\mathit{\zeta }\Vert =\Vert \mathit{\xi }\Vert ,\Vert x\Vert =\Vert \mathit{\xi }\Vert .\end{array}$$

LEMMA 1.3.11 Let F,G be unital C**-algebras, φ : F ⟶ G a surjective C**-homomorphism, I a set,

$$L≔\widetilde{\underset{i\in I}{⦶}}\check{F}\approx \check{F}\tilde{\otimes }{l}^2(I),M≔\widetilde{\underset{i\in I}{⦶}}\check{G}\approx \check{G}\tilde{\otimes }{l}^2(I),$$

and for every ξ ∈ L put $\tilde{\mathit{\xi }}≔{(\mathit{\phi }{\mathit{\xi }}_i)}_{i\in I}$.

• a) If ξ, η ∈ L and x ∈ F then

  $$\tilde{\mathit{\xi }}\in M,\Vert \tilde{\mathit{\xi }}\Vert \le \Vert \mathit{\xi }\Vert ,\widetilde{(\mathit{\xi }x)}=(\tilde{\mathit{\xi }})\mathit{\phi }x,\langle \tilde{\mathit{\xi }}|\tilde{\mathit{\eta }}\rangle =\mathit{\phi }\langle \mathit{\xi }|\mathit{\eta }\rangle .$$

• b) For every η ∈ M there is a ξ ∈ L with $\tilde{\mathit{\xi }}=\mathit{\eta },\Vert \mathit{\xi }\Vert =\Vert \mathit{\eta }\Vert$.

• c) In the W*-case the map

  $$L_{\ddot{L}}⟶M_{\ddot{M}},\mathit{\xi }⟼\tilde{\mathit{\xi }}$$

  is continuous.

a) For $J\in {\mathfrak{P}}_f(I)$,

$${\displaystyle \sum }_{i\in J}^{}\langle \mathit{\phi }{\mathit{\xi }}_i|\mathit{\phi }{\mathit{\eta }}_i\rangle ={\displaystyle \sum }_{i\in J}^{}{(\mathit{\phi }{\mathit{\eta }}_i)}^{*}(\mathit{\phi }{\mathit{\xi }}_i)=\mathit{\phi }{\displaystyle \sum }_{i\in J}^{}{\mathit{\eta }}_i^{*}{\mathit{\xi }}_i.$$

It follows $\tilde{\mathit{\xi }}\in M,\Vert \tilde{\mathit{\xi }}\Vert \le \Vert \mathit{\xi }\Vert ,\langle \tilde{\mathit{\xi }}|\tilde{\mathit{\eta }}\rangle =\mathit{\phi }\langle \mathit{\xi }|\mathit{\eta }\rangle$. Moreover for i ∈ I,

$${(\widetilde{\mathit{\xi }x})}_i=\mathit{\phi }{(\mathit{\xi }x)}_i=\mathit{\phi }({\mathit{\xi }}_{i}x)=(\mathit{\phi }{\mathit{\xi }}_i)(\mathit{\phi }x)={\tilde{\mathit{\xi }}}_i(\mathit{\phi }x),\widetilde{\mathit{\xi }x}=\tilde{\mathit{\xi }}(\mathit{\phi }x).$$

b)

Case 1 {i ∈ I | η_i ≠ 0} is finite

For simplicity we assume $\{i\in I|{\mathit{\eta }}_i\ne 0\}={\mathbb{N}}_{\text{n}}$ for some n ∈ ℕ. We put

$$\mathit{\theta }:F_{n,n}⟶G_{n,n},{[x_{ij}]}_{i,j\in {\mathbb{N}}_{\text{n}}}⟼{[\mathit{\phi }{x}_{ij}]}_{i,j\in {\mathbb{N}}_{\text{n}}}.$$

θ is obviously a surjective C*-homomorphism. So if we put

$$y≔{[{\mathit{\eta }}_i{\mathit{\delta }}_{j,1}]}_{i,j\in {\mathbb{N}}_{\text{n}}}\in G_{n,n},$$

then there is an x ∈ F_n,n with θ x = y, ‖x‖ = ‖y‖ ([5] Theorem 10.1.7). If we put

$$\mathit{\xi }:I⟶\check{F},i⟼\{\begin{array}{l}{x}_{i1}\text{if}i\in {\mathbb{N}}_{\text{n}}\\ 0\text{if}i\in I\backslash {\mathbb{N}}_{\text{n}}\end{array}$$

and $x≔{[x_{ij}{\mathit{\delta }}_{j1}]}_{i,j\in {\mathbb{N}}_{\text{n}}}\in F_{n,n}$ then

$$\mathit{\theta }z={[\mathit{\phi }(x_{ij}{\mathit{\delta }}_{j1})]}_{i,j\in {\mathbb{N}}_{\text{n}}}={[y_{ij}{\mathit{\delta }}_{j1}]}_{i,j\in {\mathbb{N}}_{\text{n}}}=y$$

and by [1] Theorem 5.6.6.1 a), ‖z‖ ≤ ‖x‖. We get for i ∈ ℕ_n,

$${\tilde{\mathit{\xi }}}_i=\mathit{\phi }{\mathit{\xi }}_i=\mathit{\phi }{x}_{i1}=y_{i1}={\mathit{\eta }}_i.$$

By a) and Lemma 1.3.10,

$$\Vert \mathit{\xi }\Vert =\Vert z\Vert \le \Vert x\Vert =\Vert y\Vert =\Vert \mathit{\eta }\Vert =\Vert \tilde{\mathit{\xi }}\Vert \le \Vert \mathit{\xi }\Vert ,\Vert \mathit{\xi }\Vert =\Vert \mathit{\eta }\Vert .$$

Case 2 η arbitrary in the W*-case

We may assume ‖η‖ = 1. We put for every $J\in {\mathfrak{P}}_f(I)$,

$${\mathit{\eta }}_J:I⟶G,i⟼\{\begin{array}{ccc}{\mathit{\eta }}_i& \text{if}& i\in J\\ 0& \text{if}& i\in I\setminus J\end{array}.$$

By Case 1, for every $J\in {\mathfrak{P}}_f(I)$ there is a ξ_J ∈ L with ${\tilde{\mathit{\xi }}}_J={\mathit{\eta }}_J$ and ‖ξ_J‖ = ‖η_J‖ ≤ 1. Let $\mathfrak{F}$ be an ultrafilter on ${\mathfrak{P}}_f(I)$ finer than the upper section filter of ${\mathfrak{P}}_f(I)$. By [1] Proposition 5.6.3.3 a ⇒ b,

$$\mathit{\xi }≔\underset{J,\mathfrak{F}}{\text{lim}}{\mathit{\xi }}_J$$

exists in $L_{\ddot{L}}^{\#}$. For i ∈ I,

$${\tilde{\mathit{\xi }}}_i=\mathit{\phi }{\mathit{\xi }}_i=\mathit{\phi }\underset{J,\mathfrak{F}}{\text{lim}}{({\mathit{\xi }}_J)}_i=\underset{J,\mathfrak{F}}{\text{lim}}\mathit{\phi }{({\mathit{\xi }}_J)}_i={\mathit{\eta }}_i$$

so $\tilde{\mathit{\xi }}=\mathit{\eta }$. By a), $1=\Vert \mathit{\eta }\Vert =\Vert \tilde{\mathit{\xi }}\Vert \le \Vert \mathit{\xi }\Vert \le 1$, so ‖ξ‖ = ‖η‖.

Case 3 η arbitrary in the C*-case

We put for every $J\in {\mathfrak{P}}_f(I)$ and every ζ ∈ M,

$${\mathit{\zeta }}_J:I⟼G,i⟼\{\begin{array}{ccc}{\mathit{\zeta }}_i& \text{if}& i\in J\\ 0& \text{if}& i\in I\setminus J\end{array}.$$

Moreover we denote by ${\mathfrak{F}}_I$ the upper section filter of ${\mathfrak{P}}_f(I)$, set

$$M_0≔\left\{\mathit{\zeta }\in M|\left\{i\in I|{\mathit{\zeta }}_i\ne 0\right\}\text{is finite}\right\},$$

and denote by $\mathcal{M}$ the vector subspace of ${\mathcal{K}}_G(M)$ generated by the set

$$\left\{{\mathit{\zeta }}_1\langle ·|{\mathit{\zeta }}_2\rangle |{\mathit{\zeta }}_1,{\mathit{\zeta }}_2\in M_0\right\}.$$

Let $\mathcal{G}$ be the vector subspace of ${\mathcal{K}}_F(L)$ generated by the set

$$\left\{\mathit{\alpha }\langle ·|\mathit{\beta }\rangle |\mathit{\alpha },\mathit{\beta }\in L\right\}.$$

$\mathcal{G}$ is an involutive subalgebra of ${\mathcal{K}}_F(L)$. Let (α_q)_{q ∈ Q}, (β_q)_{q ∈ Q} be finite families in L such that

$${\displaystyle \sum }_{q\in Q}^{}{\mathit{\alpha }}_q\langle ·|{\mathit{\beta }}_q\rangle =0.$$

Let further α ′, β ′ ∈ M₀. By Case 1, there are α, β ∈ L with $\tilde{\mathit{\alpha }}=\mathit{\alpha }\prime ,\tilde{\mathit{\beta }}=\mathit{\beta }\prime$ and we get by a),

$$\langle {\displaystyle \sum }_{q\in Q}^{}{\tilde{\mathit{\alpha }}}_q\langle \mathit{\beta }\prime |{\tilde{\mathit{\beta }}}_q\rangle |\mathit{\alpha }\prime \rangle ={\displaystyle \sum }_{q\in Q}^{}\langle {\tilde{\mathit{\alpha }}}_q|\mathit{\alpha }\prime \rangle \langle \mathit{\beta }\prime |{\tilde{\mathit{\beta }}}_q\rangle =$$

$$={\displaystyle \sum _{q\in Q}\langle {\tilde{\mathit{\alpha }}}_q|\tilde{\mathit{\alpha }}\rangle \langle \tilde{\mathit{\beta }}|{\tilde{\mathit{\beta }}}_q\rangle =\mathit{\phi }\left({\displaystyle \sum _{q\in Q}\langle {\mathit{\alpha }}_q|\mathit{\alpha }\rangle \langle \mathit{\beta }|{\mathit{\beta }}_q\rangle }\right)}=$$

$$=\mathit{\phi }\left(\langle \left({\displaystyle \sum }_{q\in Q}^{}{\mathit{\alpha }}_q\langle ·|{\mathit{\beta }}_q\rangle \right)\mathit{\beta }|\mathit{\alpha }\rangle \right)=0.$$

It follows ([1] Proposition 5.6.4.1 e))

$${\displaystyle \sum }_{q\in Q}^{}{\tilde{\mathit{\alpha }}}_q\langle ·|{\tilde{\mathit{\beta }}}_q\rangle =0.$$

Thus the linear map

$$\mathit{\psi }:\mathcal{G}⟶{\mathcal{K}}_G(M),{\displaystyle \sum }_{q\in Q}^{}{\mathit{\alpha }}_q\langle ·|{\mathit{\beta }}_q\rangle ⟼{\displaystyle \sum }_{q\in Q}^{}{\tilde{\mathit{\alpha }}}_q\langle ·|{\tilde{\mathit{\beta }}}_q\rangle$$

is well-defined and it is easy to see (by a)) that ψ is an involutive algebra homomorphism.

Step 1 ‖ψ‖ ≤ 1; we extend ψ by continuity to a map $\mathit{\psi }:{\mathcal{K}}_F(L)⟶{\mathcal{K}}_G(M)$

Let

$$u≔{\displaystyle \sum }_{q\in Q}^{}{\mathit{\alpha }}_q\langle ·|{\mathit{\beta }}_q\rangle \in \mathcal{G}$$

and let $\mathit{\zeta }\in M_0^{\#}$. By Case 1, there is an α ∈ L^# with $\tilde{\mathit{\alpha }}=\mathit{\zeta }$. By a),

$$(\mathit{\psi }u)\mathit{\zeta }={\displaystyle \sum }_{q\in Q}^{}{\tilde{\mathit{\alpha }}}_q\langle \tilde{\mathit{\alpha }}|{\tilde{\mathit{\beta }}}_q\rangle ={\displaystyle \sum }_{q\in Q}^{}{\tilde{\mathit{\alpha }}}_q\mathit{\phi }\langle \mathit{\alpha }|{\mathit{\beta }}_q\rangle ={\displaystyle \sum }_{q\in Q}^{}\tilde{\stackrel{}{\overbrace{{\mathit{\alpha }}_q\langle \mathit{\alpha }|{\mathit{\beta }}_q\rangle }}}=\widetilde{u\mathit{\alpha }},$$

$$\Vert (\mathit{\psi }u)\mathit{\zeta }\Vert =\Vert \widetilde{u\mathit{\alpha }}\Vert \le \Vert u\mathit{\alpha }\Vert \le \Vert u\Vert .$$

Since M₀ is dense in M ([1] Proposition 5.6.4.1 e)), it follows

$$\Vert \mathit{\psi }u\Vert \le \Vert u\Vert ,\Vert \mathit{\psi }\Vert \le 1.$$

Step 2 $\mathcal{M}$ is dense in ${\mathcal{K}}_G(M)$

Let α, β ∈ M. By [1] Proposition 5.6.4.1 e),

$$\mathit{\alpha }=\underset{J,{\mathfrak{F}}_I}{\text{lim}}{\mathit{\alpha }}_J,\mathit{\beta }=\underset{J,{\mathfrak{F}}_I}{\text{lim}}{\mathit{\beta }}_J$$

so by [1] Proposition 5.6.5.2 a),

$$\mathit{\alpha }\langle ·|\mathit{\beta }\rangle =\underset{J,{\mathfrak{F}}_I}{\text{lim}}{\mathit{\alpha }}_J\langle ·|{\mathit{\beta }}_J\rangle ,$$

which proves the assertion.

Step 3 ψ is a surjective C*-homomorphism

By Step 1, ψ is a C*-homomorphism. Since its image contains $\mathcal{M}$ (by Case 1) it is surjective by Step 2.

Step 4 The assertion

Let j ∈ I. By Step 3 and [5] Theorem 10.1.7 (and [1] Proposition 5.6.5.2 a)), there is a $u\in {\mathcal{K}}_F(L)$ with

$$\mathit{\psi }u=\mathit{\eta }\langle ·|1_G\otimes e_j\rangle ,\Vert u\Vert =\Vert \mathit{\eta }\langle ·|1_G\otimes e_j\rangle \Vert =\Vert \mathit{\eta }\Vert .$$

From

$$\mathit{\psi }(u((1_F\otimes e_j)\langle ·|1_F\otimes e_j\rangle ))=(\mathit{\eta }\langle ·|1_G\otimes e_j\rangle )((1_G\otimes e_j)\langle ·|1_G\otimes e_j\rangle )=$$

$$=\mathit{\eta }\langle ·|1_G\otimes e_j\rangle ,$$

$$\Vert \mathit{\eta }\Vert =\Vert \mathit{\eta }\langle ·|1_G\otimes e_j\rangle \Vert \le \Vert u((1_F\otimes e_j)\langle ·|1_F\otimes e_j\rangle )\Vert \le$$

$$\le \Vert u\Vert \Vert (1_F\otimes e_j)\langle ·|1_F\otimes e_j\rangle \Vert =\Vert u\Vert =\Vert \mathit{\eta }\Vert ,$$

$$\Vert u((1_F\otimes e_j)\langle ·|1_F\otimes e_j\rangle )\Vert =\Vert \mathit{\eta }\Vert$$

we see that we may assume

$$u=u((1_F\otimes e_j)\langle ·|1_F\otimes e_j\rangle ).$$

Then

$$u=(u(1_F\otimes e_j))\langle ·|1_F\otimes e_j\rangle .$$

If we put ξ ≔ u(1_F ⊗ e_j) ∈ L then u = ξ 〈 ⋅ |1_F ⊗ e_j,〉 ‖η‖ = ‖u‖ = ‖ξ‖,

$$\mathit{\eta }\langle ·|1_G\otimes e_j\rangle =\mathit{\psi }u=\tilde{\mathit{\xi }}\langle ·|1_G\otimes e_j\rangle ),$$

$$\mathit{\eta }=\mathit{\eta }\langle 1_G\otimes e_j|1_G\otimes e_j\rangle =\tilde{\mathit{\xi }}\langle 1_G\otimes e_j|1_G\otimes e_j\rangle =\tilde{\mathit{\xi }}.$$

c) Let $(a,{\mathit{\eta }}_0)\in \ddot{G}\times M$. By b), there is a ξ₀ ∈ L with ${\tilde{\mathit{\xi }}}_0={\mathit{\eta }}_0$. By a), for ξ ∈ L,

$$\langle \tilde{\mathit{\xi }},\widetilde{(a,{\mathit{\eta }}_0)}\rangle =\langle \langle \tilde{\mathit{\xi }}|{\mathit{\eta }}_0\rangle ,a\rangle =\langle \langle \tilde{\mathit{\xi }}|{\tilde{\mathit{\xi }}}_0\rangle ,a\rangle =$$

$$=\langle \mathit{\phi }\langle \mathit{\xi }|{\mathit{\xi }}_0\rangle ,a\rangle =\langle \langle \mathit{\xi }|{\mathit{\xi }}_0\rangle ,\ddot{\mathit{\phi }}a\rangle =\langle \mathit{\xi },(\widetilde{\ddot{\mathit{\phi }}a,{\mathit{\xi }}_0})\rangle .$$

We put

$$\mathit{\theta }:L⟶M,\mathit{\xi }⟼\tilde{\mathit{\xi }}$$

and denote by θ ′ : M′ ⟶ L′ its transpose. By the above, $\mathit{\theta }\prime \widetilde{(a,{\mathit{\eta }}_0)}\in \ddot{L}$. Since θ ′ is continuous, $\mathit{\theta }\prime (\ddot{M})\subset \ddot{L}$ and this proves the assertion.

PROPOSITION 1.3.12 We use the notation of Lemma 1.3.11.

• a) If $X\in {\mathcal{L}}_F(L)$ and ξ ∈ L with $\tilde{\mathit{\xi }}=0$ then $\widetilde{X\mathit{\xi }}=0$; we define

  $$\tilde{X}:M⟶M,\mathit{\eta }⟼\widetilde{X\mathit{\xi }},$$

  where ξ ∈ L with $\tilde{\mathit{\xi }}=\mathit{\eta }$ (Lemma 1.3.11 b)).

• b) For every $X\in {\mathcal{L}}_F(L)$, $\tilde{X}$ belongs to ${\mathcal{L}}_G(M)$ and the map

  $${\mathcal{L}}_F(L)⟶{\mathcal{L}}_G(M),X⟼\tilde{X}$$

  is a surjective C**-homomorphism continuous with respect to the topologies ${\mathfrak{T}}_k$ with k ∈ {1,2,3}.

• c) For ξ, η ∈ L,

  $$\tilde{\stackrel{}{\overbrace{\mathit{\eta }\langle ·|\mathit{\xi }\rangle }}}=\tilde{\mathit{\eta }}\langle ·|\tilde{\mathit{\xi }}\rangle$$

  and

  $${\mathcal{K}}_G(M)=\left\{\tilde{X}|X\in {\mathcal{K}}_F(L)\right\}.$$

a) For i ∈ I, $\mathit{\phi }{\mathit{\xi }}_i={\tilde{\mathit{\xi }}}_i=0$ so by Lemma 1.3.11 a),

$$\widetilde{X(e_i{\mathit{\xi }}_i)}=\widetilde{(X{e}_i){\mathit{\xi }}_i}=\widetilde{(X{e}_i)}\mathit{\phi }{\mathit{\xi }}_i=0.$$

By [1] Proposition 5.6.4.1 e) (resp. [1] Proposition 5.6.4.6 c) and [1] Proposition 5.6.3.4 c)),

$$X\mathit{\xi }=X\left({\displaystyle \sum }_{i\in I}^{}{e}_i{\mathit{\xi }}_i\right)={\displaystyle \sum }_{i\in I}^{}X(e_i{\mathit{\xi }}_i)$$

$$\left(\text{resp}\text{.}X\mathit{\xi }=X\left({\displaystyle \sum _{i\in I}^{\ddot{L}}{e}_i}{\mathit{\xi }}_i\right)={\displaystyle \sum _{i\in I}^{\ddot{L}}X}(e_i{\mathit{\xi }}_i)\right)$$

so by Lemma 1.3.11 a) (resp. c)),

$$\widetilde{X\mathit{\xi }}=\tilde{\stackrel{}{\overbrace{{\displaystyle \sum }_{i\in I}^{}X(e_i{\mathit{\xi }}_i)}}}={\displaystyle \sum }_{i\in I}^{}\widetilde{X(e_i{\mathit{\xi }}_i)}=0$$

$$\left(\text{resp}\text{.}\widetilde{X\mathit{\xi }}=\tilde{\stackrel{}{\overbrace{{\displaystyle \sum _{i\in I}^{\ddot{L}}X}(e_i{\mathit{\xi }}_i)}}}={\displaystyle \sum _{i\in I}^{\ddot{M}}\widetilde{X(e_i{\mathit{\xi }}_i)}}=0\right).$$

b) For $X,Y\in {\mathcal{L}}_F(L)$ and ξ, η ∈ L, by Lemma 1.3.11 a),

$$\langle \tilde{X}\tilde{\mathit{\xi }}|\tilde{\mathit{\eta }}\rangle =\langle \widetilde{X\mathit{\xi }}|\tilde{\mathit{\eta }}\rangle =\mathit{\phi }\langle X\mathit{\xi }|\mathit{\eta }\rangle =$$

$$=\mathit{\phi }\langle \mathit{\xi }|X^{*}\mathit{\eta }\rangle =\langle \tilde{\mathit{\xi }}|\widetilde{X^{*}\mathit{\eta }}\rangle =\langle \tilde{\mathit{\xi }}|\widetilde{X^{*}}\tilde{\mathit{\eta }}\rangle ,$$

$$\tilde{X}\tilde{Y}\tilde{\mathit{\xi }}=\tilde{X}\widetilde{Y\mathit{\xi }}=\widetilde{X(Y\mathit{\xi })}=\widetilde{(XY)\mathit{\xi }}=\widetilde{XY}\tilde{\mathit{\xi }}.$$

By Lemma 1.3.11 b), $\tilde{X}\in {\mathcal{L}}_G(M)$, ${(\tilde{X})}^{*}={\tilde{X}}^{*}$, and $\tilde{X}\tilde{Y}=\widetilde{XY}$, i.e. the map is a C*-homomorphism.

For $X\in {\mathcal{L}}_F(L)$ and ξ, η ∈ L (resp. and $a\in {\ddot{M}}_{+}$), by Lemma 1.3.11 a),

$$p_{\tilde{\mathit{\xi }},\tilde{\mathit{\eta }}}(\tilde{X})=\Vert \langle \tilde{X}\tilde{\mathit{\xi }}|\tilde{\mathit{\eta }}\rangle \Vert =\Vert \langle \widetilde{X\mathit{\xi }}|\tilde{\mathit{\eta }}\rangle \Vert =\Vert \mathit{\phi }\langle X\mathit{\xi }|\mathit{\eta }\rangle \Vert \le p_{\mathit{\xi },\mathit{\eta }}(X)$$

$$(\text{resp}\text{.}{p}_{\tilde{\mathit{\xi }},\tilde{\mathit{\eta }},a}(X)=\left|\langle \langle \tilde{X}\tilde{\mathit{\xi }}|\tilde{\mathit{\eta }}\rangle ,a\rangle \right|=|\langle \mathit{\phi }\langle X\mathit{\xi }|\mathit{\eta }\rangle ,a\rangle |=$$

$$=|\langle \langle X\mathit{\xi }|\mathit{\eta }\rangle ,\ddot{\mathit{\phi }}a\rangle |=p_{\mathit{\xi },\mathit{\eta },\ddot{\mathit{\phi }}a}(X)),$$

so by Lemma 1.3.11 b), the map is continuous with respect to the topology ${\mathfrak{T}}_1$. The proof for the other topologies is similar.

c) For ζ ∈ L, by Lemma 1.3.11 a),

$$\tilde{\stackrel{}{\overbrace{\mathit{\eta }\langle ·|\mathit{\xi }\rangle }}}\tilde{\mathit{\zeta }}=\tilde{\stackrel{}{\overbrace{(\mathit{\eta }\langle ·|\mathit{\xi }\rangle )\mathit{\zeta }}}}=\tilde{\stackrel{}{\overbrace{\mathit{\eta }\langle \mathit{\zeta }|\mathit{\xi }\rangle }}}=$$

$$=\tilde{\mathit{\eta }}\mathit{\phi }\langle \mathit{\zeta }|\mathit{\xi }\rangle =\tilde{\mathit{\eta }}\langle \tilde{\mathit{\zeta }}|\tilde{\mathit{\xi }}\rangle =\left(\tilde{\mathit{\eta }}\langle ·|\tilde{\mathit{\xi }}\rangle \right)\tilde{\mathit{\zeta }}$$

so by Lemma 1.3.11 b),

$$\tilde{\stackrel{}{\overbrace{\mathit{\eta }\langle ·|\mathit{\xi }\rangle }}}=\tilde{\mathit{\eta }}\langle ·|\tilde{\mathit{\xi }}\rangle .$$

The last assertion follows now from b).

SCHUR PRODUCTS

Throughout Schur Products we fix $f\in \mathit{ℱ}(T,E)$

2.1. The representations

We present here the projective representation of the groups and its main properties.

DEFINITION 2.1.1 We put for every t ∈ T and ξ ∈ H,

$$\begin{array}{c}{u}_t:\mathit{Ĕ}⟶H,\mathit{\zeta }⟼\mathit{\zeta }\otimes e_t,\\ V_t\mathit{\xi }:T⟶\mathit{Ĕ},s⟼f(t,t^{-1}s)\mathit{\xi }(t^{-1}s).\end{array}$$

If we want to emphasize the role of f then we put $V_t^f$ instead of V_t. For x ∈ E,

$$(x\tilde{\otimes }{1}_K)V_t\mathit{\xi }:T⟶\mathit{Ĕ},s⟼f(t,t^{-1}s)x\mathit{\xi }(t^{-1}s).$$

PROPOSITION 2.1.2 Let s, t ∈ T, x ∈ E, $\mathit{\zeta }\in \mathit{Ĕ}$, and ξ ∈ H.

• a) V_tξ ∈ H.

• b) $V_s{V}_t=(f\left(s,t\right)\tilde{\otimes }{1}_k)V_{st}$.

• c) V_t(ζ ⊗ e_s) = (f(t, s)ζ) ⊗ e_ts.

• d) $V_t(x\tilde{\otimes }{1}_K)=(x\tilde{\otimes }{1}_K)V_t.$

• e) $V_t\in Un{\mathcal{L}}_E(H),V_t^{*}=(\tilde{f}(t)\tilde{\otimes }{1}_K)V_{t^{-1}}.$

• f) $(x\tilde{\otimes }{1}_K)V_t(\mathit{\zeta }\otimes e_s)=(f(t,s)x\mathit{\zeta })\otimes e_{ts}.$

• g) If T is infinite and $\mathfrak{F}$ denotes the filter on T of cofinite subsets, i.e.

$$\mathfrak{F}≔\{S|S\in \mathfrak{P}(T),T\setminus S\in {\mathfrak{P}}_f(T)\},$$

then

$$\underset{t,\mathfrak{F}}{\text{lim}}{V}_t=0$$

in ${\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_1}$.

a) For $R\in {\mathfrak{P}}_f(T)$,

$$\begin{array}{l}{\displaystyle \sum _{r\in R}\langle {(V_t\mathit{\xi })}_r|{(V_t\mathit{\xi })}_r\rangle }={\displaystyle \sum _{r\in R}\langle f(t,t^{-1}r){\mathit{\xi }}_{t^{-1}r}|f(t,t^{-1}r){\mathit{\xi }}_{t^{-1}r}\rangle }=\\ ={\displaystyle \sum _{r\in R}\langle {\mathit{\xi }}_{t^{-1}r}|{\mathit{\xi }}_{t^{-1}r}\rangle }={\displaystyle \sum _{r\in R}\langle {\mathit{\xi }}_r|{\mathit{\xi }}_r\rangle }\le \langle \mathit{\xi }|\mathit{\xi }\rangle \end{array}$$

so V_tξ ∈ H.

b) For r ∈ T,

$$\begin{array}{c}{(V_s{V}_t\mathit{\xi })}_r=f(s,s^{-1}r){\mathit{\xi }}_{s^{-1}r}=f(s,s^{-1}r)f(t,t^{-1}{s}^{-1}r){\mathit{\xi }}_{t^{-1}{s}^{-1}r}=\\ =f(s,t)f(st,t^{-1}{s}^{-1}r){\mathit{\xi }}_{t^{-1}{s}^{-1}r}=f(s,t){(V_{st}\mathit{\xi })}_r=((f(s,t)\tilde{\otimes }{1}_K)V_{st}\mathit{\xi })r\end{array}$$

so

$$V_s{V}_t=(f(s,t)\tilde{\otimes }{1}_K)V_{st}.$$

c) For r ∈ T,

$$\begin{array}{c}(V_t(\mathit{\zeta }\otimes e_s))r=f(t,t^{-1}r){(\mathit{\zeta }\otimes e_s)}_{t^{-1}r}=\\ ={\mathit{\delta }}_{s,t^{-1}r}f(t,t^{-1}r)\mathit{\zeta }={\mathit{\delta }}_{r,ts}f(t,s)\mathit{\zeta }=((f(t,s)\mathit{\zeta })\otimes e_{ts})r\end{array}$$

so

$$V_t(\mathit{\zeta }\otimes e_s)=(f(t,s)\mathit{\zeta })\otimes e_{ts}.$$

d) We have

$${(V_t(x\tilde{\otimes }{1}_K)\mathit{\xi })}_s=f(t,t^{-1}s){((x\tilde{\otimes }{1}_K)\mathit{\xi })}_{t^{-1}s}=f(t,t^{-1}s)x{\mathit{\xi }}_{t^{-1}s}={((x\tilde{\otimes }{1}_K)V_t\mathit{\xi })}_s$$

so

$$V_t(x\tilde{\otimes }{1}_K)=(x\tilde{\otimes }{1}_K)V_t.$$

e) For η ∈ H, by Proposition 1.1.2 a),b),

$$\begin{array}{l}\langle V_t\mathit{\xi }|\mathit{\eta }\rangle =\widetilde{{\displaystyle \sum _{s\in T}}}\langle {(V_t\mathit{\xi })}_s|{\mathit{\eta }}_s\rangle =\widetilde{{\displaystyle \sum _{s\in T}}}\langle f(t,t^{-1}s){\mathit{\xi }}_{t^{-1}s}|{\mathit{\eta }}_s\rangle =\\ =\widetilde{{\displaystyle \sum _{r\in T}}}\langle f(t,r){\mathit{\xi }}_r|{\mathit{\eta }}_{tr}\rangle \widetilde{{\displaystyle \sum _{r\in T}}}\langle {\mathit{\xi }}_r|\tilde{f}(t)f(t^{-1},tr){\mathit{\eta }}_{tr}\rangle =\\ =\widetilde{{\displaystyle \sum _{r\in T}}}\langle {\mathit{\xi }}_r|{(((\tilde{f}(t)\tilde{\otimes }{1}_K)V_{t^{-1}})\mathit{\eta })}_r\rangle =\langle \mathit{\xi }|((\tilde{f}(t)\tilde{\otimes }{1}_K))V_{t^{-1}})\mathit{\eta }\rangle \end{array}$$

so $V_t\in {\mathcal{L}}_E(H)$ with $V_t^{*}=(\tilde{f}(t)\tilde{\otimes }{1}_K)V_{t^{-1}}$. By b) and d),

$$\begin{array}{c}{V}_t^{*}{V}_t=(\tilde{f}(t)\tilde{\otimes }{1}_K)V_{t^{-1}}{V}_t=(\tilde{f}(t)\tilde{\otimes }{1}_K)(f(t^{-1},t)\tilde{\otimes }{1}_K)V_{t^{-1}t}=i{d}_H,\\ V_t{V}_t^{*}=V_t(\tilde{f}(t)\tilde{\otimes }{1}_K)V_{t^{-1}}=(\tilde{f}(t)\tilde{\otimes }{1}_K)V_t{V}_{t^{-1}}=\\ =(\tilde{f}(t)\tilde{\otimes }{1}_K)(f(t,t^{-1})\tilde{\otimes }{1}_K)V_{t{t}^{-1}}=i{d}_H.\end{array}$$

f) follows from c).

g) Let us consider first the C*-case. Let ξ, η ∈ H, t ∈ T, and ε > 0. There is an $S\in {\mathfrak{P}}_f(T)$ such that ║ηe_T\S║ < ε. By e),

$$|\langle V_t\mathit{\xi }|\mathit{\eta }{e}_{T\setminus S}\rangle |\le \Vert V_t\mathit{\xi }\Vert \Vert \mathit{\eta }{e}_{T\setminus S}\Vert \le \mathit{\epsilon }\Vert \mathit{\xi }\Vert$$

so

$$p_{\mathit{\xi },\mathit{\eta }}(V_t)=|\langle V_t\mathit{\xi }|\mathit{\eta }\rangle |\le |\langle V_t\mathit{\xi }|\mathit{\eta }{e}_S\rangle |+|\langle V_t\mathit{\xi }|\mathit{\eta }{e}_{T\setminus S}\rangle |<|\langle V_t\mathit{\xi }|\mathit{\eta }{e}_S\rangle |+\mathit{\epsilon }.$$

From

$$\langle V_t\mathit{\xi }|\mathit{\eta }{e}_S\rangle ={\displaystyle \sum _{s\in S}{\mathit{\eta }}_s^{*}f(t,t^{-1}s){\mathit{\xi }}_{t^{-1}s}}$$

it follows

$$\underset{t,\mathfrak{F}}{\text{lim}}\langle V_t\mathit{\xi }|\mathit{\eta }{e}_S\rangle =0,\underset{t,\mathfrak{F}}{\text{lim}}{p}_{\mathit{\xi },\mathit{\eta }}(V_t)=0.$$

The W*-case can be proved similarly.

Remark. By e), ${\mathfrak{T}}_1$ cannot be replaced by ${\mathfrak{T}}_2$ in g).

PROPOSITION 2.1.3 Let s, t ∈ T.

• a) $u_t\in {\mathcal{L}}_E(\mathit{Ĕ},H),u_t^{*}=\langle ·|1_E\otimes e_t\rangle .$

• b) $u_s^{*}{u}_t={\mathit{\delta }}_{s,t}{1}_E.$

• c) $u_s{u}_t^{*}=1_E\tilde{\otimes }(\langle ·|e_t\rangle e_s).$

• d) ${\displaystyle \sum _{r\in T}^{{\mathfrak{T}}_2}{u}_r}{u}_r^{*}=i{d}_H.$

a) For $\mathit{\zeta }\in \mathit{Ĕ}$ and ξ ∈ H,

$$\langle u_t\mathit{\zeta }|\mathit{\xi }\rangle =\langle \mathit{\zeta }\otimes e_t|\mathit{\xi }\rangle ={\displaystyle \widetilde{\sum _{s\in T}}{\mathit{\xi }}_s^{*}}{(\mathit{\zeta }\otimes e_t)}_s={\mathit{\xi }}_t^{*}\mathit{\zeta }=\langle \mathit{\zeta }|{\mathit{\xi }}_t\rangle$$

so

$$u_t\in {\mathcal{L}}_E(\mathit{Ĕ},H),u_t^{*}\mathit{\xi }={\mathit{\xi }}_t=\langle \mathit{\xi }|1_E\otimes e_t\rangle .$$

b) For $\mathit{\zeta }\in \mathit{Ĕ}$, by a),

$$u_s^{*}{u}_t\mathit{\zeta }=u_s^{*}(\mathit{\zeta }\otimes e_t)=\langle \mathit{\zeta }\otimes e_t|1_E\otimes e_s\rangle ={\mathit{\delta }}_{s,t}\mathit{\zeta }$$

so

$$u_s^{*}{u}_t={\mathit{\delta }}_{s,t}{1}_E.$$

c) For $\mathit{\zeta }\in \mathit{Ĕ}$ and r ∈ T, by a),

$$\begin{array}{c}{u}_s{u}_t^{*}(\mathit{\zeta }\otimes e_r)=u_s{\mathit{\delta }}_{r,t}\mathit{\zeta }={\mathit{\delta }}_{r,t}(\mathit{\zeta }\otimes e_s)=\\ =\mathit{\zeta }\otimes \langle e_r|e_t\rangle e_s=(1_E\tilde{\otimes }(\langle ·|e_t\rangle e_s))(\mathit{\zeta }\otimes e_r),\end{array}$$

so (by a) and [1] Proposition 5.6.4.1 e) (resp. and [1] Proposition 5.6.4.6 c), [1] Proposition 5.6.3.4 c))) $u_s{u}_t^{*}=1_E\tilde{\otimes }(\langle ·|e_t\rangle e_s)$.

d) For ξ ∈ H (resp. and $a\in {\ddot{E}}_{+}$) and $S\in {\mathfrak{P}}_f(T)$, by c),

$$\begin{array}{c}{p}_{\mathit{\xi }}\left({\displaystyle \sum }_{t\in S}^{}{u}_t{u}_t^{*}-i{d}_H\right)=\Vert {\displaystyle \sum }_{t\in T\setminus S}^{}\langle \mathit{\xi }|\mathit{\xi }\rangle {\Vert }^{1/2}\\ \left(\text{resp}\text{.}{p}_{\mathit{\xi },a}({\displaystyle \sum }_{t\in S}^{}{u}_t{u}_t^{*}-i{d}_H\right)=\\ ={\langle \langle {\displaystyle \sum _{t\in S}(u_t{u}_t^{*}-i{d}_H)\mathit{\xi }}|{\displaystyle \sum _{t\in S}(u_t{u}_t^{*}-i{d}_H)\mathit{\xi }}\rangle ,a\rangle }^{1/2}=\\ ={({\displaystyle \sum }_{t\in T\setminus S}^{}\langle \langle \mathit{\xi }|\mathit{\xi }\rangle ,a\rangle )}^{1/2})\end{array}$$

and the assertion follows.

PROPOSITION 2.1.4 Let s, t ∈ T and x ∈ E.

• a) V_s u_t = u_st f(s, t).

• b) $u_s^{*}{V}_t=f(t,t^{-1}s)u_{t^{-1}s}^{*}.$

• c) $(x\tilde{\otimes }{1}_K)u_t=u_{t}x.$

• d) $x{u}_t^{*}=u_t^{*}(x\tilde{\otimes }{1}_K).$

a) For $\mathit{\zeta }\in \mathit{Ĕ}$, by Proposition 2.1.2 c),

$$V_s{u}_t\mathit{\zeta }=V_s(\mathit{\zeta }\otimes e_t)=(f(s,t)\mathit{\zeta })\otimes e_{st}=u_{st}f(s,t)\mathit{\zeta }$$

so V_s u_t = u_st f(s, t).

b) For $\mathit{\zeta }\in \mathit{Ĕ}$ and r ∈ T, by Proposition 2.1.2 c) and Proposition 2.1.3 a),

$$u_s^{*}{V}_t(\mathit{\zeta }\otimes e_r)=u_s^{*}((f(t,r)\mathit{\zeta })\otimes e_{tr})={\mathit{\delta }}_{s,tr}f(t,r)\mathit{\zeta }=$$

$$={\mathit{\delta }}_{t^{-1}s,r}f(t,t^{-1}s)\mathit{\zeta }=f(t,t^{-1}s)u_{t^{-1}s}^{*}(\mathit{\zeta }\otimes e_r)$$

so

$$u_s^{*}{V}_t=f(t,t^{-1}s)u_{t^{-1}s}^{*}.$$

c) For $\mathit{\zeta }\in \mathit{Ĕ}$,

$$(x\tilde{\otimes }{1}_K)u_t\mathit{\zeta }=(x\tilde{\otimes }{1}_K)(\mathit{\zeta }\otimes e_t)=(x\mathit{\zeta })\otimes e_t=u_{t}x\mathit{\zeta }$$

so $(x\tilde{\otimes }{1}_K)u_t=u_{t}x$.

d) follows from c).

DEFINITION 2.1.5 We put for all s, t ∈ T (Proposition 2.1.3 a))

$${\mathit{\phi }}_{s,t}:{\mathcal{L}}_E(H)⟶{\mathcal{L}}_E(\mathit{Ĕ})\approx E,X⟼u_s^{*}X{u}_t$$

and set X_t ≔ φ_{t, 1} X for every $X\in {\mathcal{L}}_E(H)$.

PROPOSITION 2.1.6 Let s, t ∈ T.

• a) φ_{s, t} is linear with ║φ_{s, t}║ = 1.

• b) For $X\in {\mathcal{L}}_E(H)$ and $x,y\in \mathit{Ĕ}$,

  $$\langle ({\mathit{\phi }}_{s,t}X)x|y\rangle =\langle X(x\otimes e_t)|y\otimes e_s\rangle$$

• c) The map

  $${\mathit{\phi }}_{s,t}:{\mathcal{L}}_E{(H)}_{{\mathfrak{T}}_1}⟶E(\text{resp}.E_{\ddot{E}})$$

  is continuous.

• d) φ_{t, t} is involutive and completely positive.

• e) For r ∈ T and x ∈ E,

  $${\mathit{\phi }}_{s,t}((x\tilde{\otimes }{1}_K)V_r)={\mathit{\delta }}_{s,rt}f(r,t)x.$$

• f) If (x_r)_{r ∈ T} ∈ E^(T) and

  $$X≔{\displaystyle \sum _{r\in T}(x_r\tilde{\otimes }{1}_K)V_r}$$

  then

  $${\mathit{\phi }}_{s,t}X=f(s{t}^{-1},t)x_{s{t}^{-1}},X_t=x_t.$$

• g) For $X\in {\mathcal{L}}_E(H)$ and x, y ∈ E,

  $${\mathit{\phi }}_{s,t}((x\tilde{\otimes }{1}_K)X(y\tilde{\otimes }{1}_K))=x({\mathit{\phi }}_{s,t}X)y,$$

  $${((x\tilde{\otimes }{1}_K)X(y\tilde{\otimes }{1}_K))}_t=x{X}_{t}y.$$

a) follows from Proposition 2.1.3 a),b).

b) We have

$$\langle ({\mathit{\phi }}_{s,t}X)x|y\rangle =\langle u_s^{*}X{u}_{t}x|y\rangle =\langle X{u}_{t}x|u_{s}y\rangle =\langle X(x\otimes e_t)|y\otimes e_s\rangle .$$

c)

The C*-case

By b), for $X\in {\mathcal{L}}_E(H)$,

$$\Vert {\mathit{\phi }}_{s,t}X\Vert =\Vert \langle ({\mathit{\phi }}_{s,t}X)1_E|1_E\rangle \Vert =$$

$$=\Vert X(1_E\otimes e_t)|1_E\otimes e_s\rangle \Vert =p{1}_{E\otimes e_t},1_{E\otimes e_s}(X).$$

The W*-case

Let $a\in \ddot{E}$ and let a = x|a| be its polar representation. By b), for $X\in {\mathcal{L}}_E(H)$,

$$|\langle {\mathit{\phi }}_{s,t}X,a\rangle |=|\langle \langle ({\mathit{\phi }}_{s,t}X)1_E|1_E\rangle ,x|a|\rangle |=|\langle \langle ({\mathit{\phi }}_{s,t}X)x|1_E\rangle ,|a|\rangle |=$$

$$=|\langle \langle X(x\otimes e_t)|1_E\otimes e_s\rangle ,|a|\rangle |=p_{x\otimes e_t,1_E\otimes e_s,|a|}(X).$$

d) For $X\in {\mathcal{L}}_E(H)$,

$${({\mathit{\phi }}_{t,t}X)}^{*}={(u_t^{*}X{u}_t)}^{*}=u_t^{*}{X}^{*}{u}_t={\mathit{\phi }}_{t,t}(X^{*})$$

so φ_{t, t} is involutive. For n ∈ ℕ, $X\in {({({\mathcal{L}}_E(H))}_{n,n})}_{+}$, and $\mathit{\zeta }\in {\mathit{Ĕ}}^n$,

$$\begin{array}{c}{\displaystyle \sum _{i\in {\text{ℕ}}_n}\langle {\displaystyle \sum _{j\in {\text{ℕ}}_n}(({\mathit{\phi }}_{t,t}{X}_{ij}){\mathit{\zeta }}_j)|}}{\mathit{\zeta }}_j\rangle ={\displaystyle \sum _{i,j\in {\text{ℕ}}_n}\langle u_t^{*}{X}_{ij}{u}_t{\mathit{\zeta }}_j|}{\mathit{\zeta }}_i\rangle =\\ ={\displaystyle \sum _{i,j\in {\text{ℕ}}_{\text{n}}}\langle X_{ij}{u}_t{\mathit{\zeta }}_j|u_t{\mathit{\zeta }}_i\rangle \ge 0}\end{array}$$

([1] Theorem 5.6.6.1 f) and [1] Theorem 5.6.1.11 c₁ ⇒ c₂) so φ_{t, t} is completely positive ([1] Theorem 5.6.6.1 f) and [1] Theorem 5.6.1.11 c₂ ⇒ c₁).

e) By Proposition 2.1.4 a),d) and Proposition 2.1.3 b),

$${\mathit{\phi }}_{s,t}((x\tilde{\otimes }{1}_K)V_r)=u_s^{*}(x\tilde{\otimes }{1}_K)V_r{u}_t=x{u}_s^{*}{V}_r{u}_t=x{u}_s^{*}{u}_{rt}f(r,t)={\mathit{\delta }}_{s,rt}f(r,t)x.$$

f) By e) (and Proposition 1.1.2 a)),

$${\mathit{\phi }}_{s,t}X={\displaystyle \sum }_{r\in T}^{}{\mathit{\phi }}_{s,t}((x_r\tilde{\otimes }{1}_K)V_r)={\displaystyle \sum }_{r\in T}^{}{\mathit{\delta }}_{s,rt}f(r,t)x_r=f(s{t}^{-1},t)x_{s{t}^{-1}},$$

$$X_t={\mathit{\phi }}_{t,1}X=f(t,1)x_t=x_t.$$

g) By Proposition 2.1.4 c),d),

$${\mathit{\phi }}_{s,t}((x\tilde{\otimes }{1}_K)X(y\tilde{\otimes }{1}_K))=u_s^{*}(x\tilde{\otimes }{1}_K)X(y\tilde{\otimes }{1}_K)u_t=$$

$$=x{u}_s^{*}X{u}_{t}y=x({\mathit{\phi }}_{s,t}X)y.$$

DEFINITION 2.1.7 We put

$$\mathcal{R}(f)≔\left\{{\displaystyle \sum }_{t\in T}^{}(x_t\tilde{\otimes }{1}_K)V_t|{(x_t)}_{t\in T}\in E^{(T)}\right\},$$

$$\mathcal{S}(f)≔\stackrel{{\mathfrak{T}}_3}{\overline{\mathcal{R}(f)}},{\mathcal{S}}_{\Vert ·\Vert }(f)≔\stackrel{\Vert ·\Vert }{\overline{\mathcal{R}(f)}}$$

and call $\mathcal{S}(f)$ the Schur product associated to f. Moreover we put ${\mathcal{S}}_C(f)≔\mathcal{S}(f)$ in the C*-case and ${\mathcal{S}}_W(f)≔\mathcal{S}(f)$ in the W*-case. If F is a subset of E then we put

$$\mathcal{S}(f,F)≔\left\{X\in \mathcal{S}(f)|t\in T\Rightarrow X_t\in F\right\}$$

and use similar notation for the other $\mathcal{S}$.

By Proposition 2.1.2 b),d),e), $\mathcal{R}(f)$ is an involutive unital E-subalgebra of ${\mathcal{L}}_E(H)$ (with V₁ as unit). In particular ${\mathcal{S}}_{\Vert ·\Vert }(f)$ is an E-C*-subalgebra of ${\mathcal{L}}_E(H)$. If T is finite then $\mathcal{R}(f)=\mathcal{S}(f)$. By Corollary 1.3.7 e), ${\mathcal{S}}_C{(f)}_{{\mathfrak{T}}_3}$ is complete.

PROPOSITION 2.1.8 For $X\in \stackrel{{\mathfrak{T}}_1}{\overline{\mathcal{R}(f)}}$ and s, t ∈ T,

$${\mathit{\phi }}_{s,t}X=f(s{t}^{-1},t)X_{s{t}^{-1}}.$$

Let $\mathfrak{F}$ be a filter on $\mathcal{R}(f)$ converging to X in the ${\mathfrak{T}}_1$-topology. By Proposition 2.1.6 c),f) (and Corollary 1.3.7 d)),

$${\mathit{\phi }}_{s,t}X=\underset{Y,\mathfrak{F}}{\text{lim}}{\mathit{\phi }}_{s,t}Y=\underset{Y,\mathfrak{F}}{\text{lim}}f(s{t}^{-1},t)Y_{s{t}^{-1}}=f(s{t}^{-1},t)\underset{Y,\mathfrak{F}}{\text{lim}}{Y}_{s{t}^{-1}}=$$

$$=f(s{t}^{-1},t)\underset{Y,\mathfrak{F}}{\text{lim}}{\mathit{\phi }}_{s{t}^{-1},1}Y=f(s{t}^{-1},t){\mathit{\phi }}_{s{t}^{-1},1}X=f(s{t}^{-1},t)X_{s{t}^{-1}}.$$

THEOREM 2.1.9 Let $X\in \stackrel{{\mathfrak{T}}_1}{\overline{\mathcal{R}(f)}}$.

• a) If (x_t)_{t ∈ T} is a family in E such that

  $$X={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_1}(x_t\tilde{\otimes }{1}_K)}{V}_t$$

  then X_t = x_t for every t ∈ T. In particular, if T is finite then the map

  $$E^T⟼\mathcal{S}(f),x⟼{\displaystyle \sum }_{t\in T}^{}(x_t\otimes 1_K)V_t$$

  is bijective and E-linear (Proposition 2.1.2 d)).

• b) We have

  $$X={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_3}(X_t\tilde{\otimes }{1}_K)}{V}_t\in \mathcal{S}(f).$$

• c) ${(X^{*})}_t=\tilde{f}(t){(X_{t^{-1}})}^{*}$ for every t ∈ T and

  $$X^{*}={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_3}((}{X}_t)*\tilde{\otimes }{1}_K)V_t^{*}\in \stackrel{{\mathfrak{T}}_3}{\overline{\mathcal{R}(f)}}.$$

• d) $\mathcal{S}(f)=\stackrel{{\mathfrak{T}}_1}{\overline{\mathcal{R}(f)}}=\stackrel{{\mathfrak{T}}_2}{\overline{\mathcal{R}(f)}}.$

• e) For ξ ∈ H and t ∈ T,

  $${(X\mathit{\xi })}_t=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f(s,s^{-1}t)X_s{\mathit{\xi }}_{s^{-1}t}.$$

• f) If T is finite and if we identify ${\mathcal{L}}_E(H)$ with E_{T, T} then X is identified with the matrix

  $${[f(s{t}^{-1},t)X_{s{t}^{-1}}]}_{s,t\in T},$$

  and for every r ∈ T, V_r is identified with the matrix

  $${[f(s{t}^{-1},t){\mathit{\delta }}_{s,rt}]}_{s,t\in T}.$$

• g) If $X,Y\in \mathcal{S}(f)$ and t ∈ T then $XY\in \mathcal{S}(f)$ and

  $${(XY)}_t=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f(s,s^{-1}t)X_s{Y}_{s^{-1}t},$$

  $${(X^{*}Y)}_t=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f{(s,t)}^{*}{X}_s^{*}{Y}_{st},{(X{Y}^{*})}_t=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f{(t,s)}^{*}{X}_{ts}{Y}_s^{*},$$

  $${(X^{*}Y)}_1=\widetilde{{\displaystyle \sum }_{s\in T}^{}}{X}_s^{*}{Y}_s,{(X{Y}^{*})}_1=\widetilde{{\displaystyle \sum }_{s\in T}^{}}{X}_s{Y}_s^{*}.$$

• h) The map

  $$E⟶\mathcal{S}(f),x⟼x\tilde{\otimes }{1}_K$$

  is an injective unital C**-homomorphism and so $\mathcal{S}(f)$ is an E-C**-subalgebra of ${\mathcal{L}}_E(H)$ and $Re\mathcal{S}(f)$ is closed in $\mathcal{S}{(f)}_{{\mathfrak{T}}_1}$. In the W*-case, ${\mathcal{S}}_W(f)$ is the W*-subalgebra of ${\mathcal{L}}_E(H)$ generated by $\mathcal{R}(f)$ and $\mathcal{R}{(f)}^{\#}$ is dense in ${\mathcal{S}}_W{(f)}_{{\mathfrak{T}}_1}^{\#}={\mathcal{S}}_W{(f)}_{\stackrel{\dots }{H}}^{\#}$, which is compact.

• i) If E is a W*-algebra then ${\mathcal{S}}_C(f)$ may be identified canonically with a unital C*-subalgebra of ${\mathcal{S}}_W(f)$ by using the map of Proposition 1.3.9. b). By this identification ${\mathcal{S}}_C(f)$ generates ${\mathcal{S}}_W(f)$ as W*-algebra.

• j) If F is a closed ideal of E (resp. of $E_{\ddot{E}}$) then $\mathcal{S}(f,F)$ is a closed ideal of $\mathcal{S}(f)$ (resp. of $\mathcal{S}{(f)}_{\stackrel{\cdot \cdot }{\stackrel{}{\overbrace{\mathcal{S}(f)}}}}$).

• k) If F is a unital C**-subalgebra of E such that f(s, t) ∈ F for all s, t ∈ T then $\mathcal{S}(f,F)$ is a unital C**-subalgebra of $\mathcal{S}(f)$ and the map

  $$\mathcal{S}(f,F)⟶\mathcal{S}(g),X⟼{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_3}(X_t\tilde{\otimes }{1}_K)}{V}_t^g$$

  is an injective C**-homomorphism, where

  $$g:T\times T⟶Un{F}^c,(s,t)⟼f(s,t).$$

  This map induces a C*-isomorphism ${\mathcal{S}}_{\Vert ·\Vert }(f,F)⟶{\mathcal{S}}_{\Vert ·\Vert }(g)$.

• l) $(X,Y)\in \stackrel{}{{\stackrel{°}{\overbrace{\mathcal{S}(f)}}}_{+}}\Rightarrow (X_1,Y_1)\in \stackrel{°}{E_{+}}$.

a) By Proposition 2.1.6 c),e),

$$X_t={\mathit{\phi }}_{t,1}X=\widetilde{{\displaystyle \sum }_{s\in T}^{}}{\mathit{\phi }}_{t,1}((x_s\tilde{\otimes }{1}_K)V_s)=\widetilde{{\displaystyle \sum }_{s\in T}^{}}{\mathit{\delta }}_{t,s}f(s,1)x_s=x_t.$$

b&c&d

$$\text{Step}1X={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}(X_t\tilde{\otimes }{1}_K)}{V}_t$$

By Proposition 2.1.3 d), Corollary 1.3.7 d), Proposition 2.1.8, and Proposition 2.1.4 b),d),

$$X=\left({\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{u}_s}{u}_s^{*}\right)X\left({\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}{u}_t}{u}_t^{*}\right)={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}{u}_s}}{u}_s^{*}X{u}_t{u}_t^{*}=$$

$$={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}{u}_s}}({\mathit{\phi }}_{s,t}X)u_t^{*}={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}{u}_s}}f(s{t}^{-1},t)X_{s{t}^{-1}}{u}_t^{*}=$$

$$={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{\displaystyle \sum _{r\in T}^{{\mathfrak{T}}_2}{u}_s}}{X}_{r}f(r,r^{-1}s)u_{r^{-1}s}^{*}={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{\displaystyle \sum _{r\in T}^{{\mathfrak{T}}_2}{u}_s}}{X}_r{u}_s^{*}{V}_r=$$

$$={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{\displaystyle \sum _{r\in T}^{{\mathfrak{T}}_2}{u}_s}}{u}_s^{*}(X_r\tilde{\otimes }{1}_K)V_r={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{u}_s}{u}_s^{*}\left({\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}(X_t\tilde{\otimes }{1}_K)}{V}_t\right)={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}(X_t\tilde{\otimes }{1}_K)}{V}_t.$$

Step 2 b&c&d

By Step 1, Corollary 1.3.7 a), and Proposition 2.1.2 d),e) (and Proposition 1.1.2. a)),

$$X^{*}={\left({\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_1}(X_s\tilde{\otimes }{1}_K)}{V}_s\right)}^{*}={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_1}(X_s^{*}\tilde{\otimes }{1}_K)}{V}_s^{*}=$$

$$={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_1}(X_s^{*}\tilde{\otimes }{1}_K)(\tilde{f}(s)\tilde{\otimes }{1}_K)}{V}_{s^{-1}}={\displaystyle \sum _{r\in T}^{{\mathfrak{T}}_1}((}\tilde{f}(r)X_{r^{-1}}^{*})\tilde{\otimes }{1}_K)V_r\in \stackrel{{\mathfrak{T}}_1}{\overline{\mathcal{R}(f)}}.$$

By a),

$${\left(X^{*}\right)}_t=\tilde{f}\left(t\right)\left(X_{t-1}\right){*}^{}\text{.}$$

By Step 1 and Proposition 2.1.2 e) (and Proposition 1.1.2 a)),

$$X^{*}={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}((}{X}^{*}{)}_t\tilde{\otimes }{1}_K)V_t={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}((}{X}_{t^{-1}})*\tilde{\otimes }{1}_K)(\tilde{f}(t)\tilde{\otimes }{1}_K)V_t=$$

$$={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}((}{X}_{t^{-1}}{)}^{*}\tilde{\otimes }{1}_K)V_{t^{-1}}^{*}={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}((}{X}_t{)}^{*}\tilde{\otimes }{1}_K)V_t^{*}.$$

Together with Step 1 this proves

$$X={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_3}(X_t\tilde{\otimes }{1}_K)}{V}_t\in \mathcal{S}(f),X^{*}={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_3}((}{X}_t{)}^{*}\tilde{\otimes }{1}_K)V_t^{*}\in \mathcal{S}(f).$$

In particular $\mathcal{S}\left(f\right)=\frac{{\mathfrak{T}}_1}{\mathcal{R}\left(f\right)}=\frac{{\mathfrak{T}}_2}{\mathcal{R}\left(f\right)\text{.}}$

e) By b) and Corollary 1.3.7 b), in the C*-case,

$${(X\mathit{\xi })}_t=\langle \left({\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_1}(X_s\tilde{\otimes }{1}_K)}{V}_s\right)\mathit{\xi }|1_E\otimes e_t\rangle ={\displaystyle \sum }_{s\in T}^{}\langle (X_s\tilde{\otimes }{1}_K)V_s\mathit{\xi }|1_E\otimes e_t\rangle =$$

$$={\displaystyle \sum }_{s\in T}^{}{X}_{s}f(s,s^{-1}t){\mathit{\xi }}_{s^{-1}t}={\displaystyle \sum }_{s\in T}^{}f(s,s^{-1}t)X_s{\mathit{\xi }}_{s^{-1}t}.$$

The proof is similar in the W*-case.

f) For ξ ∈ H and s ∈ T, by e),

$${(X\mathit{\xi })}_s={\displaystyle \sum }_{t\in T}^{}f(t,t^{-1}s)X_t{\mathit{\xi }}_{t^{-1}s}={\displaystyle \sum }_{r\in T}^{}f(s{r}^{-1},r)X_{s{r}^{-1}}{\mathit{\xi }}_r.$$

g) By b), Corollary 1.3.7 b),d), and Proposition 2.1.2 b),d),

$$XY=\left({\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}(X_s\tilde{\otimes }{1}_K)}{V}_s\right)\left({\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}(X_t\tilde{\otimes }{1}_K)}{V}_t\right)=$$

$$={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}(X_s\tilde{\otimes }{1}_K)V_s(Y_t\tilde{\otimes }{1}_K)}}{V}_t={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}(X_s\tilde{\otimes }{1}_K)(Y_t\tilde{\otimes }{1}_K)}}{V}_s{V}_t=$$

$$={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}(X_s\tilde{\otimes }{1}_K)(Y_t\tilde{\otimes }{1}_K)(f(}}s,t)\tilde{\otimes }{1}_K)V_{st}=$$

$$={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_2}{\displaystyle \sum _{r\in T}^{{\mathfrak{T}}_2}((}}f(s,s^{-1}r)X_s{Y}_{s^{-1}r})\tilde{\otimes }{1}_K)V_r.$$

Since by d),

$${\displaystyle \sum _{r\in T}^{{\mathfrak{T}}_2}((}f(s,s^{-1}r)X_s{Y}_{s^{-1}r})\tilde{\otimes }{1}_K)V_r\in \mathcal{S}(f)$$

for every s ∈ T we get $XY\in \mathcal{S}(f)$, again by d). By Corollary 1.3.7 b) and Proposition 2.1.6 c),e),

$${\left(XY\right)}_t={\mathit{\phi }}_{t,1}\left(XY\right)=\widetilde{\sum _{s\in T}}\widetilde{\sum _{r\in T}}{\mathit{\phi }}_{t,1}((f(s,s^{-1}r)X_s{Y}_{s^{-1}r})\tilde{\otimes }{1}_K)V_r=$$

$$=\widetilde{{\displaystyle \sum }_{s\in T}^{}}\widetilde{{\displaystyle \sum }_{r\in T}^{}}{\mathit{\delta }}_{t,r}f(r,1)f(s,s^{-1}r)X_s{Y}_{s^{-1}r}=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f(s,s^{-1}t)X_s{Y}_{s^{-1}t}.$$

By the above, c), and Proposition 1.1.2 b),

$${(X^{*}Y)}_t=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f(s,s^{-1}t){(X^{*})}_s{Y}_{s^{-1}t}=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f(s,s^{-1}t)\tilde{f}(s){(X_{s^{-1}})}^{*}{Y}_{s^{-1}t}=$$

$$=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f{(s^{-1},t)}^{*}{(X_{s^{-1}})}^{*}{Y}_{s^{-1}t}=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f{(s,t)}^{*}{X}_s^{*}{Y}_{st},$$

$${(X{Y}^{*})}_t=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f(s,s^{-1}t)X_s{(Y^{*})}_{s^{-1}t}=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f(s,s^{-1}t)X_s\tilde{f}(s^{-1}t){(Y_{t^{-1}s})}^{*}=$$

$$=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f{(t,t^{-1}s)}^{*}{X}_s{(Y_{t^{-1}s})}^{*}=\widetilde{{\displaystyle \sum }_{s\in T}^{}}f{(t,s)}^{*}{X}_{ts}{Y}_s^{*}.$$

It follows by Proposition 1.1.2 a),

$${(X^{*}Y)}_1=\widetilde{{\displaystyle \sum }_{s\in T}^{}}{X}_s^{*}{Y}_s,{(X{Y}^{*})}_1=\widetilde{{\displaystyle \sum }_{s\in T}^{}}{X}_s{Y}_s^{*}.$$

h) By c) and g), $\mathcal{S}(f)$ is an involutive unital subalgebra of ${\mathcal{L}}_E(H)$. Being closed (resp. closed in ${\mathcal{L}}_E{(H)}_{\stackrel{\dots }{H}}$ (d) and Corollary 1.3.7 c))) it is a C**-subalgebra of ${\mathcal{L}}_E(H)$ (resp. generated by $\mathcal{R}(f)$ [1] Theorem 5.6.3.5 b) and [1] Corollary 4.4.4.12 a) and by [1] Corollary 6.3.8.7 $\mathcal{R}{(f)}^{\#}$ is dense in ${\mathcal{S}}_W{(f)}_{{\mathfrak{T}}_1}^{\#}$, which is compact by Corollary 1.3.7 c)). The assertion concerning E follows from Proposition 2.1.2 d) and Lemma 1.3.2 c). By Corollary 1.3.7 a), $Re\mathcal{S}(f)$ is a closed set of $\mathcal{S}{(f)}_{{\mathfrak{T}}_1}$.

i) The assertion follows from h), Proposition 1.3.9 b), and Lemma 1.3.8 c) ⇒ a).

j) For $X\in \mathcal{S}(f,F)$, $Y\in \mathcal{S}(f)$, and t ∈ T, by g), ${(XY)}_t,{(YX)}_t\in \mathcal{S}(f,F)$ so $\mathcal{S}(f,F)$ is an ideal of $\mathcal{S}(f)$. The closure properties follow from Proposition 2.1.6 c).

k) By c) and g), $\mathcal{S}(f,F)$ is a unital involutive subalgebra of $\mathcal{S}(f)$ and by Proposition 2.1.6 c), $\mathcal{S}(f,F)$ is a C**-subalgebra of $\mathcal{S}(f)$. The last assertion follows from the fact that the image of the map contains $\mathcal{R}(g)$.

l) There are $U,V\in \mathcal{S}(f)$ with

$$(X,Y)={(U,V)}^{*}(U,V)=(U^{*},-V^{*})(U,V)=(U^{*}U+V^{*}V,U^{*}V-V^{*}U).$$

For t ∈ T,

$$0\le {(U_t,V_t)}^{*}(U_t,V_t)=(U_t^{*},-V_t^{*})(U_t,V_t)=(U_t^{*}{U}_t+V_t^{*}{V}_t,U_t^{*}{V}_t-V_t^{*}{U}_t).$$

By g),

$$X_1={(U^{*}U+V^{*}V)}_1=\widetilde{{\displaystyle \sum }_{t\in T}^{}}(U_t^{*}{U}_t+V_t^{*}{V}_t),$$

$$Y_1={(U^{*}V-V^{*}U)}_1=\widetilde{{\displaystyle \sum }_{t\in T}^{}}(U_t^{*}{V}_t-V_t^{*}{U}_t)$$

so

$$(X_1,Y_1)=\widetilde{{\displaystyle \sum }_{t\in T}^{}}(U_t^{*}{U}_t+V_t^{*}{V}_t,U_t^{*}{V}_t-V_t^{*}{U}_t)\in {\stackrel{\circ }{E}}_{+}.$$

Remark. It may happen that by the identification of i), ${\mathcal{S}}_C(f)\ne {\mathcal{S}}_W(f)$ (Remark of Proposition 2.1.23).

COROLLARY 2.1.10

• a) If (x_t)_t∈_T is a family in E such that (║x_t║)_{t ∈ T} is summable then

  $${((x_t\tilde{\otimes }{1}_K)V_t)}_{t\in T}$$

• b) is norm summable in ${\mathcal{L}}_E(H)$ and

  $$\Vert {\displaystyle \sum }_{t\in T}^{}(x_t\tilde{\otimes }{1}_K)V_t\Vert \le {\displaystyle \sum }_{t\in T}^{}\Vert x_t\Vert .$$

• c) The set

  $$\mathcal{A}≔\left\{X\in \mathcal{S}(f)|{\displaystyle \sum }_{t\in T}^{}\Vert X_t\Vert <\infty \right\}$$

  is a dense involutive unital subalgebra of ${\mathcal{S}}_{\Vert ·\Vert }(f)$ with

  $${\displaystyle \sum }_{t\in T}^{}\Vert {(X^{*})}_t\Vert ={\displaystyle \sum }_{t\in T}^{}\Vert X_t\Vert ,$$

  $${\displaystyle \sum }_{t\in T}^{}\Vert {(XY)}_t\Vert \le \left({\displaystyle \sum }_{t\in T}^{}\Vert X_t\Vert \right)\left({\displaystyle \sum }_{t\in T}^{}\Vert Y_t\Vert \right)$$

  for all $X,Y\in \mathcal{A}$.

• d) $\mathcal{A}$ endowed with the norm

  $$\begin{array}{ccc}A⟶{\text{ℝ}}_{+},& & X⟶{\displaystyle \sum _{t\in T}\Vert X_t\Vert }\end{array}$$

  is an involutive Banach algebra and ${\mathcal{S}}_{\Vert ·\Vert }(f)$ is its C*-hull.

a) For $S\in {\mathfrak{P}}_f(T)$, by Proposition 2.1.2 e),

$$\Vert {\displaystyle \sum }_{t\in S}^{}(x_t\tilde{\otimes }{1}_K)V_t\Vert \le {\displaystyle \sum }_{t\in S}^{}\Vert x_t\tilde{\otimes }{1}_K\Vert \Vert V_t\Vert ={\displaystyle \sum }_{t\in S}^{}\Vert x_t\Vert$$

and the assertion follows.

b) By Theorem 2.1.9 c), $X^{*}\in \mathcal{S}(f)$ and

$$\Vert {\left(X^{*}\right)}_t\Vert =\Vert {\left(X_{t^{-1}}\right)}^{*}\Vert =\Vert X_{t^{-1}}\Vert$$

for all t ∈ T so

$${\displaystyle \sum }_{t\in T}^{}\Vert {(X^{*})}_t\Vert ={\displaystyle \sum }_{t\in T}^{}\Vert X_{t^{-1}}\Vert ={\displaystyle \sum }_{t\in T}^{}\Vert X_t\Vert .$$

By Theorem 2.1.9 g), $XY\in \mathcal{S}(f)$ and

$$\Vert {(XY)}_t\Vert =\Vert \widetilde{{\displaystyle \sum }_{s\in T}^{}}f(s,s^{-1}t)X_s{Y}_{s^{-1}t}\Vert \le {\displaystyle \sum }_{s\in T}^{}\Vert X_s\Vert \Vert Y_{s^{-1}t}\Vert$$

for every t ∈ T so

$${\displaystyle \sum }_{t\in T}^{}\Vert {(XY)}_t\Vert \le {\displaystyle \sum }_{t\in T}^{}{\displaystyle \sum }_{s\in T}^{}\Vert X_s\Vert \Vert Y_{s^{-1}t}\Vert ={\displaystyle \sum }_{s\in T}^{}\Vert X_s\Vert \left({\displaystyle \sum }_{t\in T}^{}\Vert Y_{s^{-1}t}\Vert \right)=$$

$$={\displaystyle \sum }_{s\in T}^{}\Vert X_s\Vert \left({\displaystyle \sum }_{t\in T}^{}\Vert Y_t\Vert \right)=\left({\displaystyle \sum }_{t\in T}^{}\Vert X_t\Vert \right)\left({\displaystyle \sum }_{t\in T}^{}\Vert Y_t\Vert \right).$$

c) is easy to see.

Remark. There may exist $X\in {\mathcal{S}}_{\Vert ·\Vert }(f)$ for which ${((X_t\tilde{\otimes }{1}_K)V_t)}_{t\in T}$ is not norm summable, as it is known from the theory of trigonometric series (see Proposition 819). In particular the inclusion $\mathcal{A}\subset {\mathcal{S}}_{\Vert ·\Vert }(f)$ may be strict.

COROLLARY 2.1.11 Let F be a unital C**-algebra and τ : E ⟶ F a positive continuous (resp. W*-continuous) unital trace.

• a) τ ο φ_1,1 is a positive continuous (resp. W*-continuous) unital trace.

• b) If τ is faithful then τ ο φ_1,1 is faithful and V₁ is finite.

• c) In the W*-case, S_W(f) is finite iff E is finite.

a) Let $X,Y\in \mathcal{S}(f)$. By Theorem 2.1.9 g) (and Proposition 1.1.2 a)),

$$\mathit{\tau }{\mathit{\phi }}_{1,1}(XY)=\mathit{\tau }\left(\widetilde{{\displaystyle \sum }_{t\in T}^{}}f(t,t^{-1})X_t{Y}_{t^{-1}}\right)=\mathit{\tau }\left(\widetilde{{\displaystyle \sum }_{t\in T}^{}}f(t,t^{-1})X_{t^{-1}}{Y}_t\right)=$$

$$=\widetilde{{\displaystyle \sum }_{t\in T}^{}}\mathit{\tau }(f(t,t^{-1})X_{t^{-1}}{Y}_t)=\widetilde{{\displaystyle \sum }_{t\in T}^{}}\mathit{\tau }(f(t,t^{-1})Y_t{X}_{t^{-1}})=\mathit{\tau }\left(\widetilde{{\displaystyle \sum }_{t\in T}^{}}f(t,t^{-1})Y_t{X}_{t^{-1}}\right)=$$

$$=\mathit{\tau }{\mathit{\phi }}_{1,1}(YX).$$

Thus τ ο φ_1,1 is a trace which is obviously positive, continuous (resp. W*-continuous), and unital (Proposition 2.1.6 c),d)).

b) By Theorem 2.1.9 g), φ_1,1 is faithful, so τ ο φ is also faithful. Let $X\in \mathcal{S}(f)$ with X*X = V₁. By a),

$$\mathit{\tau }{\mathit{\phi }}_{1,1}(X{X}^{*})=\mathit{\tau }o{\mathit{\phi }}_{1,1}(X^{*}X)=\mathit{\tau }{\mathit{\phi }}_{1,1}{V}_1=1_F$$

so

$$\mathit{\tau }{\mathit{\phi }}_{1,1}(V_1-X{X}^{*})=1_F-1_F=0,V_1=X{X}^{*},$$

and V₁ is finite.

c) By b), if E is finite then ${\mathcal{S}}_W(f)$ is also finite. The reverse implication follows from the fact that E ⊗̅ 1_K is a unital W*-subalgebra of ${\mathcal{S}}_W(f)$ (Theorem 2.1.9 h)).

COROLLARY 2.1.12 Assume T finite and for every x′ ∈ (E′)^T put

$$\begin{array}{ccc}\tilde{x}\text{'}:S\left(f\right)⟶𝕂,& & X⟶{\displaystyle \sum _{t\in T}\langle X_t,x{\text{'}}_t\rangle }.\end{array}$$

• a) $\tilde{x}\prime \in \mathcal{S}(f)\prime$ and

  $$\underset{t\in T}{\text{sup}}\Vert x_t^{\prime }\Vert \le \Vert \tilde{x}\prime \Vert \le {\displaystyle \sum }_{t\in T}^{}\Vert x_t^{\prime }\Vert$$

  for every x′ ∈ (E′)^T and the map

  $$\mathit{\phi }:{(E\prime )}^T⟶\mathcal{S}(f)\prime ,x\prime ⟼\tilde{x}\prime$$

  is an isomorphism of involutive vector spaces such that

  $$\mathit{\phi }(xx\prime )=(x\otimes 1_K)(\mathit{\phi }x\prime ),\mathit{\phi }(x\prime x)=(\mathit{\phi }x\prime )(x\otimes 1_K)$$

  ([1] Proposition 2.2.7.2) for every x ∈ E and x′ ∈ (E′)^T.

• b) If E is a W*-algebra then the map

$$\begin{array}{ccc}\mathit{\psi }:{(\ddot{E})}^T⟶\stackrel{\mathrm{..}}{\widehat{S\left(f\right)}},& & {(a_t)}_{t\in T}⟼{({\tilde{a}}_t)}_{t\in T}\end{array}$$

is an isomorphism of involutive vector spaces such that

$$\mathit{\psi }(xa)=(x\otimes 1_K)(\mathit{\psi }a),\mathit{\psi }(ax)=(\mathit{\psi }a)(x\otimes 1_K)$$

for every x ∈ E and $a\in {(\ddot{E})}^T$.

COROLLARY 2.1.13 Assume T finite and let M be a Hilbert right $\mathcal{S}(f)$ –module. M endowed with the right multiplication

$$M\times E⟶M,(\mathit{\xi },x)⟼\mathit{\xi }(x\tilde{\otimes }{1}_K)$$

and with the inner-product

$$M\times M⟶E,(\mathit{\xi },\mathit{\eta })⟼{\langle \mathit{\xi }|\mathit{\eta }\rangle }_1$$

is a Hilbert right E-module denoted by $\tilde{M}$, ${\mathcal{L}}_{\mathcal{S}(f)}(M)$ is a unital C*-subalgebra of ${\mathcal{L}}_E(\tilde{M})$, and M is selfdual if $\tilde{M}$ is so.

By Proposition 2.1.6 d),g) and Theorem 2.1.9 g),l), for $X,Y\in \mathcal{S}(f)$ and x ∈ E,

$${\mathit{\phi }}_{1,1}(X(x\tilde{\otimes }{1}_K))=({\mathit{\phi }}_{1,1}X)x,X\ge 0\Rightarrow {\mathit{\phi }}_{1,1}X\ge 0,$$

$$\begin{array}{l}(X,Y)\in \stackrel{}{{\stackrel{°}{\overbrace{\mathcal{S}(f)}}}_{+}}\Rightarrow ({\mathit{\phi }}_{1,1}X,{\mathit{\phi }}_{1,1}Y)\in {\stackrel{\circ }{E}}_{+},\\ \inf \left\{\Vert {\mathit{\phi }}_{1,1}X\Vert |X\in \mathcal{S}{(f)}_{+},\Vert X\Vert =1\right\}>0\end{array}$$

and the assertion follows from Proposition 2.1.6 a),c),d) and [1] Proposition 5.6.2.5 a),c),d).

COROLLARY 2.1.14 Let n ∈ ℕ and let $\mathit{\phi }:\mathcal{S}(f)⟶E_{n,n}$ be an E-C*-homomorphism. Then (φ V_t)_{i, j} ∈ E^c for all t ∈ T and all i, j ∈ ℕ_n.

For x ∈ E, by Proposition 2.1.2 d) and Theorem 2.1.9 h),

$$x(\mathit{\phi }{V}_t)=\mathit{\phi }(x\tilde{\otimes }{1}_K)(\mathit{\phi }{V}_t)=\mathit{\phi }((x\tilde{\otimes }{1}_K)V_t)=$$

$$=\mathit{\phi }(V_t(x\tilde{\otimes }{1}_K))=(\mathit{\phi }{V}_t)\mathit{\phi }(x\tilde{\otimes }{1}_K)=(\mathit{\phi }{V}_t)x$$

so (φ V_t)_{i, j} ∈ E^c.

COROLLARY 2.1.15 Let S be a group and $g\in \mathit{ℱ}(S,\mathcal{S}(f))$. If we put

$$h:(T\times S)\times (T\times S)⟶Un\mathcal{S}{(f)}^c,((t_1,s_1),(t_2,s_2))⟼(f(t_1,t_2)\tilde{\otimes }{1}_K)g(s_1,s_2)$$

then $h\in F\left(T\times S,S\left(f\right)\right)$.

The assertion follows from Theorem 2.1.9 h).

COROLLARY 2.1.16 Let $X\in \mathcal{S}(f)(resp.X\in {\mathcal{S}}_{\Vert ·\Vert }(f))$.

• a) For every S ⊂ T,

  $${\displaystyle \sum _{s\in S}^{{\mathfrak{T}}_3}(X_s\tilde{\otimes }{1}_K)}{V}_s\in \mathcal{S}(f)(\text{resp}.{\displaystyle \sum _{s\in S}^{\Vert ·\Vert }(X_s\tilde{\otimes }{1}_K)}{V}_s\in {\mathcal{S}}_{\Vert ·\Vert }(f))$$

  and

  $$\mathit{\gamma }≔\text{sup}\left\{\Vert {\displaystyle \sum }_{t\in S}^{}(X_t\tilde{\otimes }{1}_K)V_t\Vert |S\in {\mathfrak{P}}_f(T)\right\}<\infty .$$

• b) We put for every α ∈ l^∞(T)

  $$\mathit{\alpha }X:T⟶E,t⟼{\mathit{\alpha }}_t{X}_t.$$

  Then $\mathit{\alpha }X\in \mathcal{S}(f)(resp\text{.}\mathit{\alpha }X\in {\mathcal{S}}_{\Vert ·\Vert }(f))$ for every α ∈ l^∞(T) and the map

  $$l^{\infty }(T)⟶\mathcal{S}(f)(\text{resp}.{\mathcal{S}}_{\Vert ·\Vert }(f)),\mathit{\alpha }⟼\mathit{\alpha }X$$

  is norm-continuous.

• c) Assume E is a W*-algebra and let l^∞(T, E) be the C*-direct product of the family (E)_{t ∈ T}, which is a W*-algebra ([1] Proposition 4.4.4.21 a)). We put for every α ∈ l^∞(T, E),

  $$\mathit{\alpha }X:T⟶E,t⟼{\mathit{\alpha }}_t{X}_t.$$

  Then $\mathit{\alpha }X\in {\mathcal{S}}_W(f)$ for every α ∈ l^∞(T, E) and the map

  $$l^{\infty }(T,E)⟶{\mathcal{S}}_W(f),\mathit{\alpha }⟼\mathit{\alpha }X$$

  is continuous and W*-continuous.

a) In the C*-case the family ((X_s ⊗ 1_K)V_s)_{s ∈ S} is summable since ${\mathcal{S}}_C{(f)}_{{\mathfrak{T}}_3}$ is complete. By Banach-Steinhaus Theorem, γ is finite. In the W*-case the summability follows now from Corollary 1.3.7 b),c) and Theorem 2.1.9 b).

b) Let G be the vector subspace {α ∈ l^∞(T)|α(T) is finite} of l^∞(T). By a), the map

$$G⟶\mathcal{S}(f)(\text{resp}\text{.}{\mathcal{S}}_{\Vert ·\Vert }(f)),\mathit{\alpha }⟼\mathit{\alpha }X$$

is well-defined, linear, and continuous. The assertion follows by continuity.

c) Let x ∈ E, S ⊂ T, and α ≔ xe_S. For ξ, η ∈ H and $a\in \ddot{E}$, by a) and Lemma 1.3.2 b) (and Theorem 2.1.9 b)),

$$\langle \mathit{\alpha }X,\tilde{\stackrel{}{\overbrace{(a,\mathit{\xi },\mathit{\eta })}}}\rangle =\langle \langle \mathit{\alpha }X\mathit{\xi }|\mathit{\eta }\rangle ,a\rangle =\langle {\displaystyle \sum _{t\in T}^{\ddot{E}}{\mathit{\eta }}_t^{*}}x{((e_{S}X)\mathit{\xi })}_t,a\rangle =$$

$$={\displaystyle \sum }_{t\in T}^{}\langle x,{((e_{S}X)\mathit{\xi })}_{t}a{\mathit{\eta }}_t^{*}\rangle =\langle x,{\displaystyle \sum _{t\in T}^E((}{e}_{S}X)\mathit{\xi }{)}_{t}a{\mathit{\eta }}_t^{*}\rangle .$$

Let G be the involutive subalgebra {α ∈ l^∞(T, E)|α(T) is finite} of l^∞(T, E) and let $\overline{G}$ be its norm-closure in l^∞(T, E), which is a C*-subalgebra of l^∞(T, E). By [1] Proposition 4.4.4.21 a), G is dense in $l^{\infty }{(T,E)}_{\ddot{F}}$, where F ≔ l^∞(T, E).

Let α ∈ l^∞(T, E)^# and let $\mathfrak{F}$ be a filter on G^# converging to α in $l^{\infty }{(T,E)}_{\ddot{F}}$ ([1] Corollary 6.3.8.7). By the above (and by Theorem 2.1.9 h)),

$$\underset{\mathit{\beta },\mathfrak{F}}{\text{lim}}\mathit{\beta }X=\mathit{\alpha }X$$

in ${\mathcal{S}}_W{(f)}_{\stackrel{\mathrm{..}}{\overbrace{{\mathcal{S}}_W(f)}}}$ and so $\mathit{\alpha }X\in {\mathcal{S}}_W(f)$. The assertion follows.

COROLLARY 2.1.17 Let S be a subgroup of T. Put

$$f_S≔f|(S\times S),K_S≔l^2(S),\mathcal{G}≔\left\{X\in \mathcal{S}(f)|t\in T\setminus S\Rightarrow X_t=0\right\}.$$

• a) $f_S\in \mathit{ℱ}(S,E)$.

• b) $\mathcal{G}$ is an E-C**-subalgebra of $\mathcal{S}(f)$.

• c) For every $X\in \mathcal{G}$, the family ${((X_s\tilde{\otimes }{1}_{K_S})V_s^{f_S})}_{s\in S}$ is summable in ${\mathcal{L}}_E{(K_S)}_{{\mathfrak{T}}_3}$ and the map

  $$\mathit{\phi }:\mathcal{G}⟶\mathcal{S}(f_S),X⟼{\displaystyle \sum _{s\in S}^{{\mathfrak{T}}_3}(X_s\tilde{\otimes }{1}_{K_S})}{V}_s^{f_S}$$

  is an injective E-C**-homomorphism.

• d) If $X\in \mathcal{G}\cap {\mathcal{S}}_{\Vert ·\Vert }(f)$ then $\mathit{\phi }X\in {\mathcal{S}}_{\Vert ·\Vert }(f_S)$ and the map

  $$\mathcal{G}\cap {\mathcal{S}}_{\Vert ·\Vert }(f)⟶{\mathcal{S}}_{\Vert ·\Vert }(f_S),X⟼\mathit{\phi }X$$

  is an E-C*-isomorphism.

• e) If S is finite then the map

  $$\mathcal{G}⟶\mathcal{S}(f_S),X⟼{\displaystyle \sum }_{t\in S}^{}(X_t\otimes 1_{K_S})V_t^{f_S}$$

  is an E-C*-isomorphism.

a) is obvious.

b) By Theorem 2.1.9 c),g), $\mathcal{G}$ is an involutive unital subalgebra of $\mathcal{S}(f)$ and by Proposition 2.1.6 a) (resp. Proposition 2.1.6 c) and Corollary 1.3.7 c)) and Theorem 2.1.9 h), it is an E-C**-subalgebra of $\mathcal{S}(f)$.

c) follows from Theorem 2.1.9 b) and Corollary 2.1.16 a).

d) follows from c).

e) is contained in d).

DEFINITION 2.1.18 We denote by 𝔖_T the set of finite subgroups of T and call T locally finite if 𝔖_T is upward directed and

$${\displaystyle \underset{S\in {𝔖}_T}{\cup }S}=T.$$

T is locally finite iff the subgroups of T generated by finite subsets of T are finite.

COROLLARY 2.1.19 Assume T locally finite. We put f_S ≔ f | (S × S) for every $S\in {𝔖}_T$ and identify $\mathcal{S}(f_S)$ with $\left\{X\in \mathcal{S}(f)|t\in T\setminus S\Rightarrow X_t=0\right\}$ (Corollary 2.1.17 e)).

• a) For every $X\in {\mathcal{S}}_{\Vert ·\Vert }(f)$ and ε > 0 there is an $S\in {𝔖}_T$ such that

  $$\Vert {\displaystyle \sum }_{t\in R}^{}(X_t\otimes 1_K)V_t-X\Vert <\mathit{\epsilon }$$

  for every $R\in {𝔖}_T$ with S ⊂ R.

• b) ${\mathcal{S}}_{\Vert ·\Vert }(f)$ is the norm closure of ${\cup }_{s\in {𝔖}_T}\mathcal{S}(f_S)$ and so it is canonically isomorphic to the inductive limit of the inductive system $\left\{\mathcal{S}(f_S)|S\in {𝔖}_T\right\}$ and for every $S\in {𝔖}_T$ the inclusion map $\mathcal{S}(f_S)⟶{\mathcal{S}}_{\Vert ·\Vert }(f)a$ is the associated canonical morphism.

a) There is a $Y\in \mathcal{R}(f)$ with $\Vert X-Y\Vert <\frac{\mathit{\epsilon }}{2}$. Let $S\in {𝔖}_T$ with $Y\in \mathcal{S}(f_S)$. By Corollary 2.1.17 b), for $R\in {𝔖}_T$ with S ⊂ R,

$$\Vert {\displaystyle \sum }_{t\in R}^{}((X_t-Y_t)\tilde{\otimes }{1}_K)V_t\Vert \le \Vert X-Y\Vert <\frac{\mathit{\epsilon }}{2}$$

so

$$\Vert {\displaystyle \sum }_{t\in R}^{}(X_t\tilde{\otimes }{1}_K)V_t-X\Vert \le \Vert {\displaystyle \sum }_{t\in R}^{}((X_t-Y_t)\tilde{\otimes }{1}_K)V_t\Vert +\Vert Y-X\Vert <\frac{\mathit{\epsilon }}{2}+\frac{\mathit{\epsilon }}{2}=\mathit{\epsilon }.$$

b) follows from a).

Remark. The C*-algebras of the form ${\mathcal{S}}_{\Vert ·\Vert }(f)$ with T locally finite can be seen as a kind of AF-E-C*-algebras.

PROPOSITION 2.1.20 The following are equivalent for all t ∈ T with t² = 1 and α ∈ Un E.

• a) $\frac{1}{2}(V_1+(\mathit{\alpha }\tilde{\otimes }{1}_K)V_t)\in Pr\hspace{0.17em}\mathcal{S}(f)$.

• b) ${\mathit{\alpha }}^2=\tilde{f}(t)$.

By Proposition 2.1.2 b),d),e),

$${(V_t)}^{*}=(\tilde{f}(t)\tilde{\otimes }{1}_K)V_t\hspace{0.17em},{(V_t)}^2=(\tilde{f}{(t)}^{*}\tilde{\otimes }{1}_K)V_1$$

so

$$\frac{1}{2}{(V_1+(\mathit{\alpha }\tilde{\otimes }{1}_K)V_t)}^{*}=\frac{1}{2}(V_1+(({\mathit{\alpha }}^{*}\tilde{f}(t))\tilde{\otimes }{1}_K)V_t)\hspace{0.17em},$$

$${\left(\frac{1}{2}(V_1+(\mathit{\alpha }\tilde{\otimes }{1}_K)V_t)\right)}^2=\frac{1}{4}((1_E+{\mathit{\alpha }}^2\tilde{f}{(t)}^{*})\tilde{\otimes }{1}_K)V_1+\frac{1}{2}(\mathit{\alpha }\tilde{\otimes }{1}_K)V_t .$$

Thus a) is equivalent to ${\mathit{\alpha }}^{*}\tilde{f}(t)=\mathit{\alpha }$ and $a^2\tilde{f}(t{)}^{*}=1_E$, which is equivalent to b).

COROLLARY 2.1.21 Let t ∈ T such that t² = 1 and $\tilde{f}(t)=1_E$. Then

$$\frac{1}{2}(V_1\pm V_t)\in Pr \mathcal{S}(f),(V_1+V_t)(V_1-V_t)=0\hspace{0.17em}.$$

The assertion follows from Proposition 2.1.20.

COROLLARY 2.1.22 Let α, β ∈ Un E, s, t ∈ T with s² = t² = 1, st = ts,

$$\gamma ≔\frac{1}{2}({\mathit{\alpha }}^{*}\mathit{\beta }f{(s,st)}^{*}+{\mathit{\beta }}^{*}\mathit{\alpha }f{(t,st)}^{*})\hspace{0.17em},\hspace{1em}\gamma \prime ≔\frac{1}{2}(\mathit{\alpha }{\mathit{\beta }}^{*}f{(st,t)}^{*}+\mathit{\beta }{\mathit{\alpha }}^{*}f{(st,s)}^{*})\hspace{0.17em},$$

and

$$X≔\frac{1}{2}((\mathit{\alpha }\tilde{\otimes }{1}_K)V_s+(\mathit{\beta }\tilde{\otimes }{1}_K)V_t) .$$

• a) $f(s,st)f(t,st)=f(st,t)f(st,s)=\tilde{f}{(st)}^{*}$.

• b) f(st, t) f(s, st) = f(st, s) f(t, st).

• c) $X^{*}X=\frac{1}{2}(V_1+(\gamma \tilde{\otimes }{1}_K)V_{st})\hspace{0.17em},\hspace{1em}X{X}^{*}=\frac{1}{2}(V_1+(\gamma \prime \tilde{\otimes }{1}_K)V_{st})$.

• d) The following are equivalent.

  • d₁)$X^{*}X\in Pr\hspace{0.17em}\mathcal{S}(f)$.

  • d₂)$X{X}^{*}\in Pr\hspace{0.17em}\mathcal{S}(f)$.

  • d₃)α^*βf(t, st) = β^*αf(s, st).

  • d₄)α^*βf(st, t) = β^*αf(st, s).

a) and b) follow from the equation of Schur functions (Definition 703) and Proposition 1.1.2 a).

c) By Proposition 2.1.2 b),e) and Proposition 1.1.2 b),

$$X^{*}=\frac{1}{2}((({\mathit{\alpha }}^{*}\tilde{f}(s))\tilde{\otimes }{1}_K)V_s+(({\mathit{\beta }}^{*}\tilde{f}(t))\tilde{\otimes }{1}_K)V_t)\hspace{0.17em},$$

$$X^{*}X=\frac{1}{2}{V}_1+\frac{1}{4}(({\mathit{\alpha }}^{*}\mathit{\beta }\tilde{f}(s)f(s,t)+{\mathit{\beta }}^{*}\mathit{\alpha }\tilde{f}(t)f(t,s))\tilde{\otimes }{1}_K)V_{st}=$$

$$=\frac{1}{2}{V}_1+\frac{1}{4}(({\mathit{\alpha }}^{*}\mathit{\beta }f{(s,st)}^{*}+{\mathit{\beta }}^{*}\mathit{\alpha }f{(t,st)}^{*})\tilde{\otimes }{1}_K)V_{st}=\frac{1}{2}(V_1+(\gamma \tilde{\otimes }{1}_K)V_{st})\hspace{0.17em},$$

$$X{X}^{*}=\frac{1}{2}{V}_1+\frac{1}{4}((\mathit{\alpha }{\mathit{\beta }}^{*}\tilde{f}(t)f(s,t)+\mathit{\beta }{\mathit{\alpha }}^{*}\tilde{f}(s)f(t,s))\tilde{\otimes }{1}_K)V_{st}=$$

$$=\frac{1}{2}{V}_1+\frac{1}{4}((\mathit{\alpha }{\mathit{\beta }}^{*}f{(st,t)}^{*}+\mathit{\beta }{\mathit{\alpha }}^{*}f{(st,s)}^{*})\tilde{\otimes }{1}_K)V_{st}=\frac{1}{2}(V_1+(\gamma \prime \tilde{\otimes }{1}_K)V_{st}) .$$

d₁ ⇔ d₂ is known.

d₁ ⇔ d₃. By a),

$${\gamma }^2-\tilde{f}(st)=\frac{1}{4}({\mathit{\alpha }}^{*}\mathit{\beta }{\mathit{\alpha }}^{*}\mathit{\beta }f{(s,st)}^{*2}+{\mathit{\beta }}^{*}\mathit{\alpha }{\mathit{\beta }}^{*}\mathit{\alpha }f{(t,st)}^{*2}+2f{(s,st)}^{*}f{(t,st)}^{*})-$$

$$-f{(s,st)}^{*}f{(t,st)}^{*}=\frac{1}{4}{({\mathit{\alpha }}^{*}\mathit{\beta }f{(s,st)}^{*}-{\mathit{\beta }}^{*}\mathit{\alpha }f{(t,st)}^{*})}^2 .$$

By Proposition 2.1.20 d₁) is equivalent to ${\gamma }^2=\tilde{f}(st)$ so, by the above, since α^*βf(s, st)^* − β^*αf(t, st)^* is normal, it is equivalent to

$${\mathit{\alpha }}^{*}\mathit{\beta }f{(s,st)}^{*}={\mathit{\beta }}^{*}\mathit{\alpha }f{(t,st)}^{*}\hspace{1em}\text{or to}\hspace{1em}{\mathit{\beta }}^{*}\mathit{\alpha }f(s,st)={\mathit{\alpha }}^{*}\mathit{\beta }f(t,st) .$$

d₃ ⇔ d₄ follows from b).

PROPOSITION 2.1.23 Let $X\in \mathcal{S}(f)$.

• a) $\widetilde{{\displaystyle \sum }_{t\in T}^{}}{X}_t^{*}{X}_t={(X^{*}X)}_1\hspace{0.17em},\hspace{1em}\widetilde{{\displaystyle \sum }_{t\in T}^{}}(X_t{X}_t^{*})={(X{X}^{*})}_1$.

• b) ${(X_t)}_{t\in T}\hspace{0.17em},{(X_t^{*})}_{t\in T}\in \underset{t\in T}{\widetilde{⦶}}\mathit{Ĕ}$,

  $$\Vert {(X_t)}_{t\in T}\Vert \le \Vert X\Vert \hspace{0.17em},\hspace{1em}\Vert {(X_t^{*})}_{t\in T}\Vert \le \Vert X\Vert .$$

• c) If T is finite and f is constant then there is an $X\in \mathcal{S}(f)$ with

  $$\Vert X\Vert \ge \sqrt{Card T}\Vert {(X_t)}_{t\in T}\Vert \hspace{0.17em},\hspace{1em}\Vert X\Vert \ge \sqrt{Card T}\Vert {(X_t^{*})}_{t\in T}\Vert .$$

• d) If T is infinite and locally finite and f is constant then the map

  $$\mathcal{S}(f)⟶\widetilde{\underset{t\in T}{⦶}}\mathit{Ĕ},\hspace{1em}X⟼{(X_t)}_{t\in T}$$

  is not surjective.

a) follows from Theorem 2.1.9 g).

b) By a),

$${(X_t)}_{t\in T}\hspace{0.17em},{(X_t^{*})}_{t\in T}\in \widetilde{\underset{t\in T}{⦶}}\mathit{Ĕ}$$

and by Proposition 2.1.6 a),

$${\Vert {(X_t)}_{t\in T}\Vert }^2=\Vert {\mathit{\phi }}_{1,1}(X^{*}X)\Vert \le \Vert X^{*}X\Vert ={\Vert X\Vert }^2\hspace{0.17em},$$

$${\Vert {(X_t^{*})}_{t\in T}\Vert }^2=\Vert {\mathit{\phi }}_{1,1}(X{X}^{*})\Vert \le \Vert X{X}^{*}\Vert ={\Vert X\Vert }^2 .$$

c) Let n ≔ Card T and for every t ∈ T put X_t ≔ 1_E, ξ_t ≔ 1_E. Then

$${\Vert {(X_t)}_{t\in T}\Vert }^2={\Vert {(X_t^{*})}_{t\in T}\Vert }^2=n\hspace{0.17em},\hspace{1em}{\Vert {({\xi }_t)}_{t\in T}\Vert }^2=n .$$

For t ∈ T, by Theorem 2.1.9 e),

$${(X\xi )}_t={\displaystyle \sum }_{s\in T}^{}f(s,s^{-1}t)X_s{\xi }_{s^{-1}t}=n{1}_E$$

so

$$\langle \hspace{0.17em}X\xi \hspace{0.17em}|\hspace{0.17em}X\xi \hspace{0.17em}\rangle =n^3{1}_E\hspace{0.17em},\hspace{1em}n{\Vert X\Vert }^2={\Vert X\Vert }^2{\Vert \xi \Vert }^2\ge {\Vert X\xi \Vert }^2=n^3\hspace{0.17em},$$

$${\Vert X\Vert }^2\ge n{\Vert {(X_t)}_{t\in T}\Vert }^2\hspace{0.17em},\hspace{1em}\Vert X\Vert \ge \sqrt{n}\Vert {(X_t)}_{t\in T}\Vert .$$

d) follows from c), Theorem 2.1.9 a), and the Principle of Inverse Operator.

Remark. If E is a W*-algebra then it may exist a family (x_t)_{t ∈ T} in E such that the family ${((x_t\tilde{\otimes }{1}_K)V_t)}_{t\in T}$ is summable in ${ℒ}_E{(H)}_{{\mathfrak{T}}_2}$ in the W*-case but not in the C*-case as the following example shows. Take $T≔\mathbb{Z}$, f constant, $E≔l^{\infty }(\mathbb{Z})$, and x_t ≔ (δ_{t, s})_{s ∈ T}∈E for every t ∈ T. By Proposition 2.1.23 b), ((x_t ⊗ 1_K)V_t)_{t ∈ T} is not summable in ${ℒ}_E{(H)}_{{\mathfrak{T}}_2}$ in the C*-case. In the W*-case for ξ ∈ H and s, t ∈ T,

$$\langle \hspace{0.17em}{((x_t\bar{\otimes }{1}_K)V_t\xi )}_s\hspace{0.17em}|\hspace{0.17em}{((x_t\bar{\otimes }{1}_K)V_t\xi )}_s\hspace{0.17em}\rangle =e_t|{\xi }_{s-t}{|}^2\hspace{0.17em},$$

$$\langle \hspace{0.17em}(x_t\bar{\otimes }{1}_K)V_t\xi \hspace{0.17em}|\hspace{0.17em}(x_t\bar{\otimes }{1}_K)V_t\xi \hspace{0.17em}\rangle =e_t{\Vert \xi \Vert }^2 .$$

Thus

$$X≔{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}(x_t\bar{\otimes }{1}_K)}{V}_t\in {\mathcal{S}}_W(f) .$$

Using the identification of Theorem 2.1.9 i), we get $X\in {\mathcal{S}}_W(f)\setminus {\mathcal{S}}_C(f)$.

COROLLARY 2.1.24 Let $X\in \mathcal{S}(f)$.

• a) $X\in {\{\hspace{0.17em}x\tilde{\otimes }{1}_K | x\in E\hspace{0.17em}\}}^c$ iff X_t ∈ E^c for all t ∈ T.

• b) X∈{V_t|t ∈ T}^c iff

  $$X_{s^{-1}ts}=f{(s,s^{-1}ts)}^{*}f(t,s)X_t=f(s^{-1},ts)f(t,s)\tilde{f}(s)X_t$$

  for all s, t ∈ T.

• c) $X\in \mathcal{S}{(f)}^c$ iff for all s, t ∈ T

  $$X_t\in E^c\hspace{0.17em},\hspace{1em}{X}_{s^{-1}ts}=f{(s,s^{-1}ts)}^{*}f(t,s)X_t=f(s^{-1},ts)f(t,s)\tilde{f}(s)X_t .$$

  In particular if f(s, t) = f(t, s) for all s, t ∈ T then $X\in \mathcal{S}{(f)}^c$ iff X_t ∈ E^c for all t ∈ T.

• d) ${\mathit{\phi }}_{1,1}(\mathcal{S}{(f)}^c)=E^c$.

• e) If the conjugacy class of t ∈ T (i.e. the set {s⁻¹ts|s ∈ T}) is infinite and X ∈{V_t|t ∈ T}^c then X_t = 0.

• f) If the conjugacy class of every t ∈ T \{1} is infinite then

  $${\left\{\hspace{0.17em}{V}_t | t\in T\hspace{0.17em}\right\}}^c=\left\{\hspace{0.17em}x\tilde{\otimes }{1}_K | x\in E\hspace{0.17em}\right\}\hspace{0.17em},\hspace{1em}\mathcal{S}{(f)}^c=\left\{\hspace{0.17em}x\tilde{\otimes }{1}_K | x\in E^c\hspace{0.17em}\right\} .$$

  Thus if E is a factor then $\mathcal{S}(f)$ is also a factor.

• g) The following are equivalent:

  1. g₁)$\mathcal{S}(f)$ is commutative.

  2. g₂)T and E are commutative and f(s, t) = f(t, s) for all s, t ∈ T.

For s, t ∈ T, x ∈ E, and $Y≔(x\tilde{\otimes }{1}_K)V_s$, by Theorem 2.1.9 g),

$${(XY)}_t=\widetilde{{\displaystyle \sum }_{r\in T}^{}}f(r,r^{-1}t)X_r{Y}_{r^{-1}t}=\widetilde{{\displaystyle \sum }_{r\in T}^{}}f(r,r^{-1}t)X_r{\delta }_{s,r^{-1}t}x=f(t{s}^{-1},s)X_{t{s}^{-1}}x\hspace{0.17em},$$

$${(YX)}_t=\widetilde{{\displaystyle \sum }_{r\in T}^{}}f(r,r^{-1}t)Y_r{X}_{r^{-1}t}=\widetilde{{\displaystyle \sum }_{r\in T}^{}}f(r,r^{-1}t){\delta }_{r,s}x{X}_{r^{-1}t}=f(s,s^{-1}t)x{X}_{s^{-1}t} .$$

a) follows from the above by putting s ≔ 1 (Proposition 1.1.2 a)).

b) follows from the above by putting x ≔ 1_E and t ≔ rs (Proposition 1.1.2).

c) follows from a), b), and Corollary 1.3.7 d). The last assertion follows using Proposition 1.1.5 a).

d) follows from c) (and Proposition 1.1.2 a)).

e) follows from b) and Proposition 2.1.23 b).

f) follows from c), e), and Proposition 2.1.2 d).

g₁ ⇒ g₂. By a), E is commutative. By Proposition 2.1.2 b),

$$f(s,t)V_{st}=V_s{V}_t=V_t{V}_s=f(t,s)V_t{V}_s=f(t,s)V_{ts}$$

and so by Theorem 2.1.9 a), st = ts and f(s, t) = f(t, s).

g₂ ⇒ g₁ follows from c).

COROLLARY 2.1.25 If 𝕂 = ℝ then the following are equivalent:

• a) $\mathcal{S}{(f)}^c=\mathcal{S}(f)=Re \mathcal{S}(f)$.

• b) T is commutative, E^c = E = Re E, and

  $$f(s,t)=f(t,s)\hspace{0.17em},\hspace{1em}\tilde{f}(t)=1_E\hspace{0.17em},\hspace{1em}{t}^2=1$$

  for all s, t ∈ T.

a ⇒ b. By Corollary 2.1.24 g₁ ⇒ g₂, T is commutative, E = E^c, and f(s, t) = f(t, s) for all s, t ∈ T. Since E is isomorphic with a C*-subalgebra of $\mathcal{S}(f)$ (Theorem 2.1.9 h)), E = Re E. By Proposition 2.1.2 e),

$$V_t=V_t^{*}=(\tilde{f}(t)\tilde{\otimes }{1}_K)V_{t^{-1}}$$

so by Theorem 2.1.9 a), $t=t^{-1}\hspace{0.17em},\hspace{0.17em}\tilde{f}(t)=1_E$, so t² = 1.

b ⇒ a. By Corollary 2.1.24 g₂ ⇒ g₁, $\mathcal{S}{(f)}^c=\mathcal{S}(f)$. For $X\in \mathcal{S}(f)$ and t ∈ T, by Theorem 2.1.9 c),

$${(X^{*})}_t=\tilde{f}(t){(X_{t^{-1}})}^{*}={(X_t)}^{*}=X_t$$

so X^* = X (Theorem 2.1.9 a)).

PROPOSITION 2.1.26 Let (E_i)_{i ∈ I} be a family of unital C**-algebras such that E is the C*-direct product of this family. For every i ∈ I, we identify E_i with the corresponding closed ideal of E (resp. of $E_{\ddot{E}}$) and put

$$f_i:T\times T⟶Un E_i^c\hspace{0.17em},\hspace{1em}(s,t)⟼f{(s,t)}_i .$$

• a) For every i ∈ I, $f_i\in \mathit{ℱ}(T,E_i)$. We put (by Theorem 2.1.9 b))

  $${\mathit{\phi }}_i:\mathcal{S}(f)⟶\mathcal{S}(f_i)\hspace{0.17em},\hspace{1em}X⟼{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}((}{X}_t{)}_i\tilde{\otimes }{1}_K)V_t^{f_i} .$$

  φ_i is a surjective C**-homomorphism.

• b) In the C*-case, if T is finite then $\mathcal{R}(f)={\mathcal{S}}_{\Vert \cdot \Vert }(f)={\mathcal{S}}_C(f)$ is isomorphic to the C*-direct product of the family

  $${(\mathcal{R}(f_i)={\mathcal{S}}_{\Vert \cdot \Vert }(f_i)={\mathcal{S}}_C(f_i))}_{i\in I} .$$

• c) In the C*-case, if I is finite then ${\mathcal{S}}_C(f)$ (resp. ${\mathcal{S}}_{\Vert \cdot \Vert }(f)$) is isomorphic to ${\displaystyle \prod _{i\in I}{\mathcal{S}}_C(f_i)}$ (resp. ${\displaystyle \prod _{i\in I}{\mathcal{S}}_{\Vert \cdot \Vert }(f_i)}$).

• d) In the W*-case, ${\mathcal{S}}_W(f)$ is isomorphic to the C*-direct product of the family ${({\mathcal{S}}_W(f_i))}_{i\in I}$.

Remark. The C*-isomorphisms of b) and c) cease to be surjective in general if T and I are both infinite. Take T ≔ (ℤ₂)^ℕ, I ≔ ℕ, E_i ≔ 𝕂 for every i ∈ I, and E ≔ l^∞ (i.e. E is the C*-direct product of the family (E_i)_{i ∈ I}). For every n ∈ ℕ put $t_n≔{\left({\delta }_{m,n}\right)}_{m\in \mathbb{N}}\in T$. Assume there is an $X\in {\mathcal{S}}_C(f)$ (resp. $X\in {\mathcal{S}}_{\Vert \cdot \Vert }(f)$) with $\mathit{\psi }X={(V_{t_i}^{f_i})}_{i\in I}$ (resp. $\mathit{\phi }X={(V_{t_i}^{f_i})}_{i\in I}$), where ψ and φ are the maps of b) and c), respectively. Then ${\left(X_{t_n}\right)}_i={\delta }_{i,n}$ for all i, n ∈ ℕ and this implies ${(X_t)}_{t\in T}\notin \hspace{0.17em}\underset{t\in T}{⦶}\mathit{Ĕ}$, which contradicts Proposition 2.1.23 b).

PROPOSITION 2.1.27 Let S be a finite group, K′ ≔ l²(S), K″ ≔ l²(S × T), and $g\in \mathit{ℱ}(S,\mathcal{S}(f))$ such that g(s₁, s₂)∈Un E^c (where Un E^c is identified with $(Un E^c)\tilde{\otimes }{1}_K\subset Un \mathcal{S}{(f)}^c$) for all s₁, s₂∈S and put

$$h:(S\times T)\times (S\times T)⟶Un E^c\hspace{0.17em},\hspace{1em}((s_1,t_1),(s_2,t_2))⟼g(s_1,s_2)f(t_1,t_2) .$$

• a) $h\in \mathit{ℱ}(S\times T,E)$; for every $X\in \mathcal{S}(g)$ put

  $$\mathit{\phi }X≔{\displaystyle \sum }_{s\in S}^{}{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_3}((}{X}_s{)}_t\tilde{\otimes }{1}_{K^{″}})V_{(s,t)}^h\in \mathcal{S}(h) .$$

• b) $\mathit{\phi }:\mathcal{S}(g)⟶\mathcal{S}(h)$ is an E-C*-isomorphism.

a) is obvious.

b) For $X,Y\in \mathcal{S}(g)$ and (s, t)∈S × T, by Theorem 2.1.9 c),g) and Proposition 2.1.6 g),

$${(\mathit{\phi }{X}^{*})}_{(s,t)}={({(X^{*})}_s)}_t=\tilde{g}(s){({(X_{s^{-1}})}^{*})}_t=$$

$$=\tilde{g}(s)\tilde{f}(t){({(X_{s^{-1}})}_{t^{-1}})}^{*}=\tilde{h}(s,t){(X_{{(s,t)}^{-1}})}^{*}={({(\mathit{\phi }X)}^{*})}_{(s,t)}\hspace{0.17em},$$

$${(\mathit{\phi }(XY))}_{(s,t)}={({(XY)}_s)}_t={\displaystyle \sum }_{r\in S}^{}g(r,r^{-1}s){(X_r{Y}_{r^{-1}s})}_t=$$

$$={\displaystyle \sum }_{r\in S}^{}g(r,r^{-1}s)\widetilde{{\displaystyle \sum }_{q\in T}^{}}f(q,q^{-1}t){(X_r)}_q{(Y_{r^{-1}s})}_{q^{-1}t}=$$

$$=\widetilde{{\displaystyle \sum }_{(r,q)\in S\times T}^{}}h((r,q),{(r,q)}^{-1}(s,t))X_{(r,q)}{Y}_{{(r,q)}^{-1}(s,t)}={((\mathit{\phi }X)(\mathit{\phi }Y))}_{(s,t)}\hspace{0.17em},$$

so φ is a C*-homomorphism. If φ X = 0 then X_{(s, t)} = 0 for all (s, t)∈S × T, so X = 0 and φ is injective. Let $Z\in \mathcal{S}(h)$. For every s ∈ S put

$$X_s≔{\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_3}(X_{(s,t)}\tilde{\otimes }{1}_K)V_t^f\in \mathcal{S}(f),}$$

$$X≔{\displaystyle \sum }_{s\in S}^{}(X_s\otimes 1_{K^{\prime }})V_s^g\in \mathcal{S}(g) .$$

Then φ X = Z and φ is surjective.

PROPOSITION 2.1.28 If T is infinite and $X\in \mathcal{S}(f)\setminus \left\{0\right\}$ then X(H^#) is not precompact.

Let t ∈ T with X_t ≠ 0. There is an $x\prime \in E_{+}^{\prime }$ (resp. $x\prime \in {\ddot{E}}_{+}$) with $\langle \hspace{0.17em}{X}_t^{*}{X}_t\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle >0$. We put t₁ ≔ 1 and construct a sequence (t_n)_{n ∈ ℕ} recursively in T such that for all m, n ∈ ℕ, m < n,

$$\left|\langle \hspace{0.17em}f{(t,t_m)}^{*}f(t{t}_m{t}_n^{-1},t_n)X_t^{*}{X}_{t{t}_m{t}_n^{-1}}\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle \right|<\frac{1}{2}\langle \hspace{0.17em}{X}_t^{*}{X}_t\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle .$$

Let n ∈ ℕ \ {1} and assume the sequence was constructed up to n − 1. Since (Proposition 2.1.23 a))

$${\displaystyle \sum }_{s\in T}^{}\langle \hspace{0.17em}{X}_{t{t}_m{s}^{-1}}^{*}{X}_{t{t}_m{s}^{-1}}\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle <\infty$$

for all m ∈ ℕ_{n − 1} there is a t_n ∈ T with

$$\langle \hspace{0.17em}{X}_{t{t}_m{t}_n^{-1}}^{*}{X}_{t{t}_m{t}_n^{-1}}\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle <\frac{1}{4}\langle \hspace{0.17em}{X}_t^{*}{X}_t\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle$$

for all m ∈ ℕ_{n − 1}. By Schwarz’ inequality ([1] Proposition 2.3.4.6 c)) for m ∈ ℕ_{n − 1},

$${\left|\langle \hspace{0.17em}f{(t,t_m)}^{*}f(t{t}_m{t}_n^{-1},t_n)X_t^{*}{X}_{t{t}_m{t}_n^{-1}}\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle \right|}^2\le$$

$$\le \langle \hspace{0.17em}{X}_t^{*}{X}_t\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle \langle \hspace{0.17em}{X}_{t{t}_m{t}_n^{-1}}^{*}{X}_{t{t}_m{t}_n^{-1}}\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle <\frac{1}{4}{\langle \hspace{0.17em}{X}_t^{*}{X}_t\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle }^2 .$$

This finishes the recursive construction.

For r, s ∈ T, by Theorem 2.1.9 e),

$${(X(1_E\otimes e_r))}_s=\widetilde{{\displaystyle \sum }_{q\in T}^{}}f(q,q^{-1}s)X_q{\delta }_{r,q^{-1}s}=f(s{r}^{-1},r)X_{s{r}^{-1}}\hspace{0.17em},$$

$$\langle \hspace{0.17em}X(1_E\otimes e_r)\hspace{0.17em}|\hspace{0.17em}{X}_t\otimes e_s\hspace{0.17em}\rangle =f(s{r}^{-1},r)X_t^{*}{X}_{s{r}^{-1}} .$$

For m, n ∈ ℕ, m < n, it follows

$$\langle \hspace{0.17em}X(1_E\otimes e_{t_m})\hspace{0.17em}|\hspace{0.17em}{X}_t\otimes e_{t{t}_m}\hspace{0.17em}\rangle =f(t,t_m)X_t^{*}{X}_t\hspace{0.17em},$$

$$\langle \hspace{0.17em}\langle \hspace{0.17em}X(1_E\otimes e_{t_m})\hspace{0.17em}|\hspace{0.17em}{X}_t\otimes e_{t{t}_m}\hspace{0.17em}\rangle \hspace{0.17em},\hspace{0.17em}x\prime f{(t,t_m)}^{*}\hspace{0.17em}\rangle =\langle \hspace{0.17em}{X}_t^{*}{X}_t\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle \hspace{0.17em},$$

$$\langle \hspace{0.17em}X(1_E\otimes e_{t_n})\hspace{0.17em}|\hspace{0.17em}{X}_t\otimes e_{t{t}_m}\hspace{0.17em}\rangle =f(t{t}_m{t}_n^{-1},t_n)X_t^{*}{X}_{t{t}_m{t}_n^{-1}}\hspace{0.17em},$$

$$\left|\langle \hspace{0.17em}\langle \hspace{0.17em}X(1_E\otimes e_{t_n})\hspace{0.17em}|\hspace{0.17em}{X}_t\otimes e_{t{t}_m}\hspace{0.17em}\rangle \hspace{0.17em},\hspace{0.17em}x\prime f{(t,t_m)}^{*}\hspace{0.17em}\rangle \right|=$$

$$\left|\langle \hspace{0.17em}f{(t,t_m)}^{*}f(t{t}_m{t}_n^{-1},t_n)X_t^{*}{X}_{t{t}_m{t}_n^{-1}}\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle \right|<\frac{1}{2}\langle \hspace{0.17em}{X}_t^{*}{X}_t\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle \hspace{0.17em},$$

$$\Vert x\prime \Vert \Vert X(1_E\otimes e_{t_m})-X(1_E\otimes e_{t_n})\Vert \Vert X_t\Vert \ge$$

$$\ge \left|\langle \hspace{0.17em}\langle \hspace{0.17em}X(1_E\otimes e_{t_m})-X(1_E\otimes e_{t_n})\hspace{0.17em}|\hspace{0.17em}{X}_t\otimes e_{t{t}_m}\hspace{0.17em}\rangle \hspace{0.17em},\hspace{0.17em}x\prime f{(t,t_m)}^{*}\hspace{0.17em}\rangle \right|\ge$$

$$\ge \left|\langle \hspace{0.17em}\langle \hspace{0.17em}X(1_E\otimes e_{t_m})\hspace{0.17em}|\hspace{0.17em}{X}_t\otimes e_{t{t}_m}\hspace{0.17em}\rangle \hspace{0.17em},\hspace{0.17em}x\prime f{(t,t_m)}^{*}\hspace{0.17em}\rangle \right|-$$

$$-\left|\langle \hspace{0.17em}\langle \hspace{0.17em}X(1_E\otimes e_{t_n})\hspace{0.17em}|\hspace{0.17em}{X}_t\otimes e_{t{t}_m}\hspace{0.17em}\rangle \hspace{0.17em},\hspace{0.17em}x\prime f{(t,t_m)}^{*}\hspace{0.17em}\rangle \right|>$$

$$>\langle \hspace{0.17em}{X}_t^{*}{X}_t\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle -\frac{1}{2}\langle \hspace{0.17em}{X}_t^{*}{X}_t\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle =\frac{1}{2}\langle \hspace{0.17em}{X}_T^{*}{X}_T\hspace{0.17em},\hspace{0.17em}x\prime \hspace{0.17em}\rangle .$$

Thus the sequence ${\left(X\left(1_E\otimes e_{t_n}\right)\right)}_{n\in \mathbb{N}}$ has no Cauchy subsequence and therefore X(H^#) is not precompact.

PROPOSITION 2.1.29 Assume T finite and let Ω be a compact space, ω₀ ∈ Ω,

$$g:T\times T⟶Un \mathcal{C}(\Omega ,E)\hspace{0.17em},\hspace{1em}(s,t)⟼f(s,t)1_{\Omega }\hspace{0.17em},$$

$$A≔\left\{\hspace{0.17em}X\in \mathcal{S}(g) | t\in T,t\ne 1\Rightarrow X_t({\mathit{\omega }}_0)=0\hspace{0.17em}\right\}\hspace{0.17em},$$

$$B≔\left\{\hspace{0.17em}Y\in \mathcal{C}(\Omega ,\mathcal{S}(f)) | t\in T,t\ne 1\Rightarrow Y{({\mathit{\omega }}_0)}_t=0\hspace{0.17em}\right\} .$$

Then $g\in \mathit{ℱ}(T,\mathcal{C}(\Omega ,E))$ and we define for every X ∈ A and Y ∈ B,

$$\mathit{\phi }X:\Omega ⟶\mathcal{S}(f)\hspace{0.17em},\hspace{1em}\mathit{\omega }⟼{\displaystyle \sum }_{t\in T}^{}(X_t(\mathit{\omega })\otimes 1_K)V_t^f\hspace{0.17em},$$

$$\mathit{\psi }Y≔{\displaystyle \sum }_{t\in T}^{}(Y{(o)}_t\otimes 1_K)V_t^g .$$

Then A (resp. B) is a unital C*-subalgebra of $\mathcal{S}(g)$ (resp. of $\mathcal{C}(\Omega ,\mathcal{S}(f))$)

$$\mathit{\phi }:A⟶B\hspace{0.17em},\mathit{\psi }:B⟶A$$

are C*-isomorphisms, and φ = ψ⁻¹.

It is easy to see that A (resp. B) is a unital C*-subalgebra of $\mathcal{S}(g)$ (resp. of $\mathcal{C}(\Omega ,\mathcal{S}(f))$) and that φ and ψ are well-defined. For X, X′ ∈A, t ∈ T, and ω ∈ Ω, by Theorem 2.1.9 c),g) and Proposition 2.1.2 e),

$${(((\mathit{\phi }X)(\mathit{\phi }X\prime ))(\mathit{\omega }))}_t={\displaystyle \sum }_{s\in T}^{}f(s,s^{-1}t){((\mathit{\phi }X)(\mathit{\omega }))}_s{((\mathit{\phi }X\prime )(\mathit{\omega }))}_{s^{-1}t}=$$

$$={\displaystyle \sum }_{s\in T}^{}f(s,s^{-1}t)X_s(\mathit{\omega }){X^{\prime }}_{s^{-1}t}(\mathit{\omega })={\displaystyle \sum }_{s\in T}^{}(f(s,s^{-1}t)X_s{X^{\prime }}_{s^{-1}t})(\mathit{\omega })=$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={(XX\prime )}_t(\mathit{\omega })={(\mathit{\phi }(XX\prime )(\mathit{\omega }))}_t\hspace{0.17em},$$

$$(\mathit{\phi }{X}^{*})(\mathit{\omega })={\displaystyle \sum }_{s\in T}^{}(({(X^{*})}_s(\mathit{\omega }))\otimes 1_K)V_s^f={\displaystyle \sum }_{s\in T}^{}((\tilde{f}(s)({(X_{s^{-1}})}^{*}(\mathit{\omega })))\otimes 1_K)V_s^f=$$

$$={\displaystyle \sum }_{s\in T}^{}((X_{s^{-1}}){(\mathit{\omega })}^{*}\otimes 1_K){(V_{s^{-1}}^f)}^{*}={\displaystyle \sum }_{s\in T}^{}(X_s{(\mathit{\omega })}^{*}\otimes 1_K){(V_s^f)}^{*}={(\mathit{\phi }X)}^{*}(\mathit{\omega })$$

so φ is a C*-homomorphism and we have

$${(\mathit{\psi }\mathit{\phi }X)}_t={(\mathit{\phi }X)}_t=X_t .$$

Moreover for Y ∈ B,

$${(\mathit{\phi }\mathit{\psi }Y)}_t(\mathit{\omega })={((\mathit{\psi }Y)(\mathit{\omega }))}_t=Y_t(\mathit{\omega })$$

which proves the assertion.

2.2. Variation of the parameters

In this section we examine the changes produced by the replacement of the groups and of the Schur functions.

DEFINITION 2.2.1 We put for every λ ∈ Λ (T, E) (Definition 1.1.3)

$$U_{\lambda }:H⟶H,\hspace{1em}\xi ⟼{(\lambda (t){\xi }_t)}_{t\in T} .$$

It is easy to see that U_λ is well-defined, $U_{\lambda }\in Un {ℒ}_E(H)$, and the map

$$\Lambda (T,E)⟶Un {ℒ}_E(H),\hspace{1em}\lambda ⟼U_{\lambda }$$

is an injective group homomorphism with $U_{\lambda }^{*}=U_{\lambda *}$ (Proposition 1.1.4 c)). Moreover

$$\Vert U_{\lambda }-U_{\mu }\Vert \le {\Vert \lambda -\mu \Vert }_{\infty }$$

for all λ, μ ∈ Λ (T, E).

PROPOSITION 2.2.2 Let $f,g\in \mathit{ℱ}(T,E)$ and λ ∈ Λ (T, E).

• a) The following are equivalent:

  • a₁) g = fδλ.

  • a₂) There is a (unique) E-C*-isomorphism

    $$\mathit{\phi }:\mathcal{S}(f)⟶\mathcal{S}(g)$$

    continuous with respect to the ${\mathfrak{T}}_2$-topologies such that for all t ∈ T and x ∈ E,

    $$\mathit{\phi }{V}_t^f=(\lambda {(t)}^{*}\tilde{\otimes }{1}_K)V_t^g$$

    (we call such an isomorphism an $\mathcal{S}$-isomorphism and denote it by ${\approx }_{\mathcal{S}}$)

• b) If the above equivalent assertions are fulfilled then for $X\in \mathcal{S}(f)$ and t ∈ T,

  $$\mathit{\phi }X=U_{\lambda }^{*}X{U}_{\lambda }\hspace{0.17em},\hspace{1em}{(\mathit{\phi }X)}_t=\lambda {(t)}^{*}{X}_t .$$

• c) There is a natural bijection

  $$\left\{\hspace{0.17em}\mathcal{S}(f) | f\in \mathit{ℱ}(T,E)\hspace{0.17em}\right\}/{\approx }_{\mathcal{S}}⟶\mathit{ℱ}(T,E)/\left\{\hspace{0.17em}\delta \lambda | \lambda \in \Lambda (T,E)\hspace{0.17em}\right\} .$$

By Proposition 1.1.4 c), $\delta \lambda \in \mathit{ℱ}(T,E)$ for every λ ∈ Λ (T, E).

a₁ ⇒ a₂& b. For s, t ∈ T and $\zeta \in \mathit{Ĕ}$, by Proposition 2.1.2 c),

$$U_{\lambda }^{*}{V}_t^f{U}_{\lambda }(\zeta \otimes e_s)=U_{\lambda }^{*}{V}_t^f((\lambda (s)\zeta )\otimes e_s)=U_{\lambda }^{*}((f(t,s)\lambda (s)\zeta )\otimes e_{ts})=$$

$$=(\lambda {(ts)}^{*}f(t,s)\lambda (s)\zeta )\otimes e_{ts}=(\lambda {(t)}^{*}g(t,s)\zeta )\otimes e_{ts}=(\lambda {(t)}^{*}\tilde{\otimes }{1}_K)V_t^g(\zeta \otimes e_s)$$

so (by Proposition 2.1.2 e))

$$U_{\lambda }^{*}{V}_t^f{U}_{\lambda }=(\lambda {(t)}^{*}\tilde{\otimes }{1}_K)V_t^g .$$

Thus the map

$$\mathit{\phi }:\mathcal{S}(f)⟶\mathcal{S}(g),\hspace{1em}X⟼U_{\lambda }^{*}X{U}_{\lambda }$$

is well-defined. It is obvious that it has the properties described in a₂). The uniqueness follows from Theorem 2.1.9 b).

We have

$$\mathit{\phi }((X_t\tilde{\otimes }{1}_K)V_t^f)=(X_t\tilde{\otimes }{1}_K)(\lambda {(t)}^{*}\tilde{\otimes }{1}_K)V_t^g=((\lambda {(t)}^{*}{X}_t)\tilde{\otimes }{1}_K)V_t^g$$

so (φX)_t = λ(t)^*X_t.

a₂ ⇒ a₁. Put h ≔ fδλ. By the above, for t ∈ T,

$$(\lambda {(t)}^{*}\tilde{\otimes }{1}_K)V_t^g=\mathit{\phi }{V}_t^f=(\lambda {(t)}^{*}\tilde{\otimes }{1}_K)V_t^h$$

so $V_t^g=V_t^h$ and this implies g = h.

c) follows from a).

Remark. Not every E-C*-isomorphism $\mathcal{S}(f)⟶\mathcal{S}(g)$ is an $\mathcal{S}$ isomorphism (see Remark of Proposition 3.2.3).

COROLLARY 2.2.3 Let

$${\Lambda }_0(T,E)≔\left\{\lambda \in \Lambda (T,E) | \lambda \mathit{\text{is a group homomorphism}}\right\}$$

and for every λ ∈ Λ₀(T, E) put

$${\mathit{\phi }}_{\lambda }:\mathcal{S}(f)⟶\mathcal{S}(f),\hspace{1em}X⟼U_{\lambda }^{*}X{U}_{\lambda } .$$

Then the map λ ⟼ φ_λ is an injective group homomorphism.

By Proposition 1.1.4 c), Λ₀(T, E) is the kernel of the map

$$\Lambda (T,E)⟶\mathit{ℱ}(T,E),\hspace{1em}\lambda ⟼\delta \lambda$$

so by Proposition 2.2.2, φ_λ is well-defined. Thus only the injectivity of the map has to be proved. For t ∈ T and $\zeta \in \mathit{Ĕ}$, by Proposition 2.1.2 c),

$$U_{\lambda }^{*}{V}_t{U}_{\lambda }(\zeta \otimes e_1)=U_{\lambda }^{*}{V}_t(\zeta \otimes e_1)=U_{\lambda }^{*}(\zeta \otimes e_t)=$$

$$=(\lambda {(t)}^{*}\zeta )\otimes e_t=(\lambda {(t)}^{*}\tilde{\otimes }{1}_K)V_t(\zeta \otimes e_1) .$$

So if φ_λ is the identity map then λ(t) = 1_E for every t ∈ T.

PROPOSITION 2.2.4 Let F be a unital C**-algebra, φ : E ⟶ F a surjective C**-homomorphism, $g≔\text{ο}\circ f\in \mathit{ℱ}(T,F)$, and $L≔\widetilde{\underset{t\in T}{⦶}}\check{F}$. We put for all ξ ∈ H, η ∈ L, and $X\in {ℒ}_E(H)$,

$$\tilde{\xi }≔{(\mathit{\phi }{\xi }_i)}_{i\in I}\in L\hspace{0.17em},\tilde{X}\eta ≔\widetilde{X\zeta }\in L\hspace{0.17em},$$

where ζ ∈ H with $\tilde{\zeta }=\eta$ (Lemma 1.3.11 a),b) and Proposition 1.3.12 a)). Then

$$\tilde{X}={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_3}((}\mathit{\phi }{X}_t)\tilde{\otimes }{1}_K)V_t^g\in \mathcal{S}(g)$$

for every $X\in \mathcal{S}(f)$ and the map

$$\tilde{\mathit{\phi }}:\mathcal{S}(f)⟶\mathcal{S}(g),\hspace{1em}X⟼\tilde{X}$$

is a surjective C**-homomorphism, continuous with respect to the topologies ${\mathfrak{T}}_k$, k∈{1, 2, 3} such that

$$Ker\hspace{0.17em}\tilde{\mathit{\phi }}=\left\{X\in \mathcal{S}(f) | t\in T\Rightarrow X_t\in Ker\hspace{0.17em}\mathit{\phi }\right\} .$$

For s, t ∈ T and ξ ∈ H,

$${\left(\tilde{\stackrel{}{\overbrace{(X_t\tilde{\otimes }{1}_K)V_t^f}}}\tilde{\xi }\right)}_s={(\tilde{\stackrel{}{\overbrace{(X_t\tilde{\otimes }{1}_K)V_t^f\xi }}})}_s=\mathit{\phi }{((X_t\tilde{\otimes }{1}_K)V_t^f\xi )}_s=$$

$$=\mathit{\phi }(f(t,t^{-1}s)X_t{\xi }_{t^{-1}s})=g(t,t^{-1}s)(\mathit{\phi }{X}_t){\tilde{\xi }}_{t^{-1}s}={(((\mathit{\phi }{X}_t)\tilde{\otimes }{1}_K)V_t^g\tilde{\xi })}_s$$

so by Lemma 1.3.11 b),

$$\tilde{\stackrel{}{\overbrace{(X_t\tilde{\otimes }{1}_K)V_t^f}}}=((\mathit{\phi }{X}_t)\tilde{\otimes }{1}_K)V_t^g .$$

By Theorem 2.1.9 b),

$$X={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_3}(X_t\tilde{\otimes }{1}_K)}{V}_t^f$$

so by the above and by Proposition 1.3.12 b),

$$\tilde{X}={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_3}((}\mathit{\phi }{X}_t)\tilde{\otimes }{1}_K)V_t^g\in \mathcal{S}(g) .$$

By Proposition 1.3.12 b), $\tilde{\mathit{\phi }}$ is a surjective C**-homomorphism, continuous with respect to the topologies ${\mathfrak{T}}_k (k\in \{1,2,3\})$. The last assertion is easy to see.

COROLLARY 2.2.5 Let F be a unital C*-algebra, φ : E ⟶ F a unital C*-homomorphism such that φ(Un E^c) ⊂ F^c, $g≔\mathit{\phi }\text{ο}f\in \mathit{ℱ}(T,F)$, and $L≔\underset{t\in T}{⦶}\check{F}$. Then the map

$$\tilde{\mathit{\phi }}:{\mathcal{S}}_{\Vert \cdot \Vert }(f)⟶{\mathcal{S}}_{\Vert \cdot \Vert }(g),\hspace{1em}X⟼{\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }((}\mathit{\phi }{X}_t)\otimes 1_L)V_t^g$$

is C*-homomorphism.

Put G ≔ E/Ker φ and denote by φ₁: E ⟶ G the quotient map and by φ₂: G ⟶ F the corresponding injective C*-homomorphism. By Proposition 2.2.4, the corresponding map

$${\tilde{\mathit{\phi }}}_1:{\mathcal{S}}_{\Vert \cdot \Vert }(f)⟶{\mathcal{S}}_{\Vert \cdot \Vert }({\mathit{\phi }}_1\text{ο}f)$$

is a C*-homomorphism and by Theorem 2.1.9 k), the corresponding map

$${\tilde{\mathit{\phi }}}_2:{\mathcal{S}}_{\Vert \cdot \Vert }({\mathit{\phi }}_1\text{ο}f)⟶{\mathcal{S}}_{\Vert \cdot \Vert }(g)$$

is also a C*-homomorphism. The assertion follows from $\tilde{\mathit{\phi }}={\tilde{\mathit{\phi }}}_2\text{ο}{\tilde{\mathit{\phi }}}_1$.

PROPOSITION 2.2.6 Let T′ be a group, K′ ≔ l²(T′), $H\prime ≔\mathit{Ĕ}\tilde{\otimes }K\prime$, ψ : T ⟶ T′ a surjective group homomorphism such that

$$\underset{t\prime \in T\prime }{\text{sup}}Card\hspace{0.17em}\hspace{0.17em}\stackrel{-1}{\mathit{\psi }}(t\prime )\in \mathbb{N},$$

and $f\prime \in \mathit{ℱ}(T\prime ,E)$ such that f′ ο (ψ × ψ) = f. If we put

$$X_{t\text{'}}^{\text{'}}≔{\displaystyle \sum _{t\in \stackrel{-1}{\mathit{\psi }}(t\text{'})}{X}_t}$$

for every $X\in \mathcal{S}(f)$ and t′ ∈ T′ then the family ${((X_{t\prime }^{\prime }\tilde{\otimes }{1}_{K\prime })V_{t\prime }^{f\prime })}_{t\prime \in T\prime }$ is summable in ${ℒ}_E{(H\prime )}_{{\mathfrak{T}}_2}$ for every $X\in \mathcal{S}(f)$ and the map

$$\tilde{\mathit{\psi }}:\mathcal{S}(f)⟶\mathcal{S}(f\prime ),\hspace{1em}X⟼X\prime ≔{\displaystyle \sum _{t\prime \in T\prime }^{{\mathfrak{T}}_1}(X_{t\prime }^{\prime }\tilde{\otimes }{1}_{K\prime })}{V}_{t\prime }^{f\prime }$$

is a surjective E-C**-homomorphism.

We may drop the hypothesis that ψ is surjective if we replace $\mathcal{S}$ by ${\mathcal{S}}_{\Vert \cdot \Vert }$.

Let $X\in \mathcal{S}(f)$. By Corollary 2.1.16 a), since ψ is surjective and

$$\underset{t\prime \in T\prime }{\text{sup}}Card\hspace{0.17em}\hspace{0.17em}\stackrel{-1}{\mathit{\psi }}(t\prime )\in \mathbb{N},$$

it follows that the family ${((X_{t\prime }^{\prime }\tilde{\otimes }{1}_{K\prime })V_{t\prime }^{f\prime })}_{t\prime \in T\prime }$ is summable in ${ℒ}_E{(H\prime )}_{{\mathfrak{T}}_2}$ and therefore $X\prime \in \mathcal{S}(f\prime )$.

Let $X,Y\in \mathcal{S}(f)$. By Theorem 2.1.9 c),g), for t′ ∈ T′,

$$\begin{array}{l}(X^{\prime }*{)}_{t^{\prime }}=\widetilde{f^{\prime }}(t^{\prime })(X_{t^{\prime -1}}{)}^{*}=\widetilde{f^{\prime }}(t^{\prime }){\left({\displaystyle \sum _{t\in \stackrel{-1}{\mathit{\psi }}(t^{\prime -1})}{X}_t}\right)}^{*}=\widetilde{f^{\prime }}(t^{\prime }){\displaystyle \sum _{s\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}{(X_{s^{-1}})}^{*}=}\\ \hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \sum _{s\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}\tilde{f}(s){(X_{s^{-1}})}^{*}}={\displaystyle \sum _{s\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}{(X^{*})}_s}={(X*)}_{t\text{'}}^{\text{'}},\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(X^{\prime }{Y}^{\prime }{)}_{t^{\prime }}=\tilde{{\displaystyle \sum _{s^{\prime }\in T^{\prime }}}}{f}^{\prime }(s^{\prime },s^{\prime -1}{t}^{\prime }){X^{\prime }}_{s^{\prime }}{Y^{\prime }}_{s^{\prime -1}{t}^{\prime }}=\\ =\tilde{{\displaystyle \sum _{s^{\prime }\in T^{\prime }}}}{f}^{\prime }(s^{\prime },s^{\prime -1}{t}^{\prime })\left({\displaystyle \sum _{s\in \stackrel{-1}{\mathit{\psi }}(s^{\prime })}{X}_s}\right)\left({\displaystyle \sum _{r\in \stackrel{-1}{\mathit{\psi }}({s^{\prime }}^{-1}{t}^{\prime })}{Y}_r}\right)=\\ \hspace{1em}\hspace{1em}=\tilde{{\displaystyle \sum _{s^{\prime }\in T^{\prime }}}}{f}^{\prime }(s^{\prime },{s^{\prime }}^{-1}{t}^{\prime })\left({\displaystyle \sum _{s\in \stackrel{-1}{\mathit{\psi }}(s^{\prime })}{\displaystyle \sum _{t\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}{X}_s{Y}_{s^{-1}t}}}\right)=\\ \hspace{1em}\hspace{1em}=\tilde{{\displaystyle \sum _{s^{\prime }\in T^{\prime }}}}\left({\displaystyle \sum _{s\in \stackrel{-1}{\mathit{\psi }}(s^{\prime })}{\displaystyle \sum _{t\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}f(s,s^{-1}t)X_s{Y}_{s^{-1}t}}}\right)=\\ ={\displaystyle \sum _{t\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}\tilde{{\displaystyle \sum _{s^{\prime }\in T}}}f(s,s^{-1}t)X_s{Y}_{s^{-1}t}}={\displaystyle \sum _{t\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}{(XY)}_t={(XY{)}^{\prime }}_{t^{\prime }}.}\end{array}$$

Thus ψ is a C*-homomorphism. The other assertions are easy to see.

The last assertion follows from Corollary 2.1.17 d).

COROLLARY 2.2.7 If we use the notation of Proposition 2.2.6 and Corollary 2.2.5 and define $\widetilde{{\mathit{\phi }}^{\prime }}$ and $\widetilde{{\mathit{\psi }}^{\prime }}$ in an obvious way then $\widetilde{{\mathit{\phi }}^{\prime }}\text{ο}\tilde{\mathit{\psi }}=\widetilde{{\mathit{\psi }}^{\prime }}\text{ο}\tilde{\mathit{\phi }}$.

For $X\in \mathcal{S}(f)$ and t^′∈T^′,

$$\begin{array}{c}({\tilde{\mathit{\phi }}}^{\prime }\tilde{\mathit{\psi }}X{)}_{t^{\prime }}=\mathit{\phi }((\tilde{\mathit{\psi }}X{)}_{t^{\prime }})=\mathit{\phi }{\displaystyle \sum _{t\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}{X}_t}={\displaystyle \sum _{t\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}\mathit{\phi }{X}_t},\\ ({\tilde{\mathit{\psi }}}^{\prime }\tilde{\mathit{\phi }}X{)}_{t^{\prime }}={\displaystyle \sum _{t\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}{(\tilde{\mathit{\phi }}X)}_t}={\displaystyle \sum _{t\in \stackrel{-1}{\mathit{\psi }}(t^{\prime })}\mathit{\phi }{X}_t},\end{array}$$

so

$$\widetilde{{\mathit{\phi }}^{\prime }}\text{ο}\tilde{\mathit{\psi }}=\widetilde{{\mathit{\psi }}^{\prime }}\text{ο}\tilde{\mathit{\phi }} .$$

PROPOSITION 2.2.8 Let F be a unital C*-subalgebra of E such that f(s, t)∈F for all s, t ∈ T. We denote by ψ : F ⟶ E the inclusion map and put

$$f^F:T\times T⟶Un F^c,\hspace{1em}(s,t)⟼f(s,t),$$

$$H^F≔\underset{t\in T}{⦶}\check{F}\approx \check{F}\otimes K,$$

$$\tilde{\mathit{\psi }}:H^F⟶H,\hspace{1em}\xi ⟼{(\mathit{\psi }{\xi }_t)}_{t\in T} .$$

Moreover we denote for all s, t ∈ T by $u_t^F,\hspace{0.17em}{V}_t^F$, and ${\mathit{\phi }}_{s,t}^F$ the corresponding operators associated with F $(f^F\in \mathit{ℱ}(T,F))$. Let $X\in {\mathcal{S}}_C(f)$ such that $X(\tilde{\mathit{\psi }}\xi )\in \tilde{\mathit{\psi }}(H^F)$ for every ξ ∈ H^F and put

$$X^F:H^F⟶H^F,\hspace{1em}\xi ⟼\xi \prime ,$$

where ξ′ ∈ H^F with $\tilde{\mathit{\psi }}\xi \prime =X(\tilde{\mathit{\psi }}\xi )$, and $X_t^F≔{(u_1^F)}^{*}{X}^F{u}_t^F\in F$ (by the canonical identification of F with ${ℒ}_F(\check{F})$) for every t ∈ T.

• a) $\xi ,\eta \in H^F\Rightarrow \langle \hspace{0.17em}\tilde{\mathit{\psi }}\xi \hspace{0.17em}|\hspace{0.17em}\tilde{\mathit{\psi }}\eta \hspace{0.17em}\rangle =\mathit{\psi }\langle \hspace{0.17em}\xi \hspace{0.17em}|\hspace{0.17em}\eta \hspace{0.17em}\rangle$.

• b) $\tilde{\mathit{\psi }}$ is linear and continuous with $\Vert \tilde{\mathit{\psi }}\Vert =1$.

• c) X^F is linear and continuous with ‖X^F‖ = ‖X‖.

• d) For s, t ∈ T,

  $$\mathit{\psi }{\mathit{\phi }}_{s,t}^F{X}^F={\mathit{\phi }}_{s,t}X\hspace{0.17em},\mathit{\psi }{X}_t^F=X_t\hspace{0.17em},{\mathit{\phi }}_{s,t}^F{X}^F=f^F(s{t}^{-1},t)X_{s{t}^{-1}}^F .$$

• e) $X^F\in \mathcal{S}(f^F)$.

• f) $\xi \in H^F\Rightarrow X(\tilde{\mathit{\psi }}\xi )={\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }(X_t\otimes 1_K)V_t(\tilde{\mathit{\psi }}\xi )}$.

a&b&c are easy to see.

d) By a) and Proposition 2.1.6 b),

$${\mathit{\phi }}_{s,t}^F{X}^F=\langle \hspace{0.17em}{X}^F(1_F\otimes e_t)\hspace{0.17em}|\hspace{0.17em}{1}_F\otimes e_s\hspace{0.17em}\rangle \hspace{0.17em},$$

$$\mathit{\psi }{\mathit{\phi }}_{s,t}^F{X}^F=\mathit{\psi }\langle \hspace{0.17em}{X}^F(1_F\otimes e_t)\hspace{0.17em}|\hspace{0.17em}{1}_F\otimes e_s\hspace{0.17em}\rangle =$$

$$=\langle \hspace{0.17em}\tilde{\mathit{\psi }}(X^F(1_F\otimes e_t))\hspace{0.17em}|\hspace{0.17em}\tilde{\mathit{\psi }}(1_F\otimes e_s)\hspace{0.17em}\rangle =\langle \hspace{0.17em}X(1_E\otimes e_t)\hspace{0.17em}|\hspace{0.17em}{1}_E\otimes e_s\hspace{0.17em}\rangle ={\mathit{\phi }}_{s,t}X .$$

In particular

$$\mathit{\psi }{X}_t^F=\mathit{\psi }{\mathit{\phi }}_{1,t}^F{X}^F={\mathit{\phi }}_{1,t}X=X_t$$

and by Proposition 2.1.8,

$$\mathit{\psi }{\mathit{\phi }}_{s,t}^F{X}^F={\mathit{\phi }}_{s,t}X=f(s{t}^{-1},t)X_{s{t}^{-1}}=\mathit{\psi }(f^F(s{t}^{-1},t)X_{s{t}^{-1}}^F)\hspace{0.17em},$$

$${\mathit{\phi }}_{s,t}^F{X}^F=f^F(s{t}^{-1},t)X_{s{t}^{-1}}^F .$$

e) By c) and Proposition 2.1.3 d), for ξ ∈ H^F,

$${\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }{u}_t^F}{(u_t^F)}^{*}\xi =\xi \hspace{0.17em},$$

$$X^F\xi =X^F{\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }{u}_t^F}{(u_t^F)}^{*}\xi ={\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }{X}^F}{u}_t^F{(u_t^F)}^{*}\xi \hspace{0.17em},$$

$$X^F\xi ={\displaystyle \sum _{s\in T}^{\Vert \cdot \Vert }{u}_s^F}{(u_s^F)}^{*}{X}^F\xi ={\displaystyle \sum _{s\in T}^{\Vert \cdot \Vert }{\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }{u}_s^F}}({(u_s^F)}^{*}{X}^F{u}_t^F){(u_t^F)}^{*}\xi .$$

By d) and Proposition 2.1.4 b),d),

$$X^F\xi ={\displaystyle \sum _{s\in T}^{\Vert \cdot \Vert }{\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }{u}_s^F}}{f}^F(s{t}^{-1},t)X_{s{t}^{-1}}^F{(u_t^F)}^{*}\xi ={\displaystyle \sum _{s\in T}^{\Vert \cdot \Vert }{\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }{u}_s^F}}{X}_{s{t}^{-1}}^F{(u_s^F)}^{*}{V}_{s{t}^{-1}}^F\xi =$$

$$={\displaystyle \sum _{s\in T}^{\Vert \cdot \Vert }{\displaystyle \sum _{r\in T}^{\Vert \cdot \Vert }{u}_s^F}}{X}_r^F{(u_s^F)}^{*}{V}_r^F\xi ={\displaystyle \sum _{s\in T}^{\Vert \cdot \Vert }{\displaystyle \sum _{r\in T}^{\Vert \cdot \Vert }{u}_s^F}}{(u_s^F)}^{*}(X_r^F\otimes 1_F)V_r^F\xi =$$

$$={\displaystyle \sum _{s\in T}^{\Vert \cdot \Vert }{u}_s^F}{(u_s^F)}^{*}{\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }(X_t^F\otimes 1_K)}{V}_t^F\xi ={\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }(X_t^F\otimes 1_K)}{V}_t^F\xi$$

by Proposition 2.1.3 d), again. Thus

$$X^F={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}(X_t^F\otimes 1_K)}{V}_t^F\in {\mathcal{S}}_C(f^F) .$$

f) For s, t ∈ T, by d),

$${(\tilde{\mathit{\psi }}((X_t^F\otimes 1_K)V_t^F\xi ))}_s=\mathit{\psi }{((X_t^F\otimes 1_K)V_t^F\xi )}_s=\mathit{\psi }(f^F(t,t^{-1}s)X_t^F{\xi }_{t^{-1}s})=$$

$$=f(t,t^{-1}s)X_t{(\tilde{\mathit{\psi }}\xi )}_{t^{-1}s}={((X_t\otimes 1_K)V_t\tilde{\mathit{\psi }}\xi )}_s\hspace{0.17em},$$

$$\tilde{\mathit{\psi }}((X_t^F\otimes 1_K)V_t^F\xi )=(X_t\otimes 1_K)V_t\tilde{\mathit{\psi }}\xi$$

so by b) and e),

$$X(\tilde{\mathit{\psi }}\xi )=\tilde{\mathit{\psi }}(X^F\xi )=\tilde{\mathit{\psi }}\left({\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }(X_t^F\otimes 1_K)}{V}_t^F\xi \right)=$$

$$={\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }\tilde{\mathit{\psi }}}((X_t^F\otimes 1_K)V_t^F\xi )={\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }(X_t\otimes 1_K)V_t(\tilde{\mathit{\psi }}\xi )} .$$

PROPOSITION 2.2.9 Let F be a W*-algebra such that E is a unital C*-subalgebra of F generating it as W*-algebra, φ : E ⟶ F the inclusion map, and $\tilde{\xi }≔{(\mathit{\phi }{\xi }_t)}_{t\in T}\in L$ for every ξ ∈ H, where

$$L≔\underset{t\in T}{\overset{W}{⦶}}\check{F}\approx \check{F}\bar{\otimes }K .$$

• a) φ(Un E^c) ⊂ Un F^c and $g≔\mathit{\phi }\text{ο}f\in \mathit{ℱ}(T,F)$.

• b) If

  $$\mathit{\psi }:{ℒ}_E(H)⟶{ℒ}_F(L),\hspace{1em}X⟼\bar{X}$$

  is the injective C*-homomorphism defined in Proposition 1.3.9 b), then $\mathit{\psi }({\mathcal{S}}_C(f))\subset {\mathcal{S}}_W(g)$, $\mathit{\psi }({\mathcal{S}}_C(f))$ generates ${\mathcal{S}}_W(g)$ as W*-algebra, and for every $X\in {\mathcal{S}}_C(f)$ and t ∈ T we have ${(\bar{X})}_t=\mathit{\phi }{X}_t$.

• c) The following are equivalent for every $Y\in {\mathcal{S}}_W(g)$:

  • c₁) $Y\in \mathit{\psi }({\mathcal{S}}_C(f))$.

  • c₂) $\xi \in H\Rightarrow Y\tilde{\xi }\in H$.

    If these conditions are fulfilled then

  • c₃) (Y_t)_{t ∈ T} ∈ H.

  • c₄) ${\left(Y_t^{*}\right)}_{t\in T}\in H$.

  • c₅) $\xi \in H\Rightarrow Y\tilde{\xi }={\displaystyle \sum _{t\in T}^{\Vert \cdot \Vert }(Y_t\bar{\otimes }{1}_K)}{V}_t^g\tilde{\xi }\in H$.

a) follows from the density of φ(E) in $F_{\ddot{F}}$ (Lemma 1.3.8 a ⇒ c).

b) For x ∈ E, t ∈ T, and ξ ∈ H,

$${(((\mathit{\phi }x)\bar{\otimes }{1}_K)V_t^g\tilde{\xi })}_s=g(t,t^{-1}s)(\mathit{\phi }x){\tilde{\xi }}_{t^{-1}s}=$$

$$=\mathit{\phi }(f(t,t^{-1}s)x{\xi }_{t^{-1}s})=\mathit{\phi }((x\otimes 1_K)V_t{\xi }_s)$$

so

$$((\mathit{\phi }x)\bar{\otimes }{1}_K)V_t^g=\overline{(x\otimes 1_K)V_t^f} .$$

Let now $X\in \mathcal{S}(f)$. By Theorem 2.1.9 b),

$$X={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_2}(X_t\otimes 1_K)}{V}_t^f$$

so by the above and by Proposition 1.3.9 c) (and Theorem 2.1.9 d)),

$$\bar{X}={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_1}\overline{(X_t\otimes 1_K)V_t^f}}={\displaystyle \sum _{t\in T}^{{\mathfrak{T}}_1}((}\mathit{\phi }{X}_t)\bar{\otimes }{1}_K)V_t^g\in {\mathcal{S}}_W(g)$$

so $\mathit{\psi }({\mathcal{S}}_C(f))\subset {\mathcal{S}}_W(f)$. By Theorem 2.1.9 a), ${(\overline{X})}_t=\mathit{\phi }{X}_t$ for every t ∈ T.

Since φ (E) is dense in $F_{\ddot{F}}$ (Lemma 1.3.8 a) ⇒ c)) it follows that

$$\mathcal{R}(g)\subset \stackrel{{\mathfrak{T}}_1}{\overline{\mathit{\phi }(\mathcal{R}(f))}}$$

so $\mathit{\psi }(\mathcal{S}(f))$ is dense in $\mathcal{S}{(g)}_{\stackrel{\cdot \cdot }{\overbrace{\mathcal{S}(g)}}}$ and therefore generates $\mathcal{S}(g)$ as W*-algebra (Lemma 1.3.8 c ⇒ a).

c₁ ⇒ c₂ follows from the definition of ψ.

c₂ ⇒ c₁ follows from Proposition 2.2.8 e).

c₂ ⇒ c₃&c₄ follows from Proposition 2.1.23 b).

c₂ ⇒ c₅ follows from Proposition 2.2.8 f).

LEMMA 2.2.10 Let E, F be W*-algebras, $G≔E\bar{\otimes }F$, and

$$L≔\underset{t\in T}{\overset{W}{⦶}}\check{G}\approx \check{G}\bar{\otimes }K$$

• a) If z∈ G^# then z ⊗̅ 1_K belongs to the closure of

  $$\left\{\hspace{0.17em}w\bar{\otimes }{1}_K | w\in E\odot F, \Vert w\Vert \le 1\hspace{0.17em}\right\}$$

  in ${ℒ}_G{(L)}_{\stackrel{\dots }{L}}$.

• b) For every y ∈ F, the map

  $$E_{\ddot{E}}^{\#}⟶G_{\ddot{G}},\hspace{1em}x⟼x\otimes y$$

  is continuous.

a) By [1] Corollary 6.3.8.7, there is a filter $\mathfrak{F}$ on $\left\{w\in E\hspace{0.17em}\odot \hspace{0.17em}F|\hspace{0.17em}\Vert w\Vert \le 1\right\}$ converging to z in $G_{\ddot{G}}^{\#}$. By Lemma 1.3.2 b), for $(a,\xi ,\eta )\in \ddot{G}\times L\times L$,

$$\langle \hspace{0.17em}z\bar{\otimes }{1}_K\hspace{0.17em},\hspace{0.17em}(\widetilde{a,\xi ,\eta })\hspace{0.17em}\rangle =\langle \hspace{0.17em}z\hspace{0.17em},\hspace{0.17em}{\displaystyle \sum _{t\in T}^G{\xi }_t}\hspace{0.17em}a\hspace{0.17em}{\eta }_t^{*}\hspace{0.17em}\rangle =$$

$$=\underset{w,\mathfrak{F}}{\text{lim}}\langle \hspace{0.17em}w\hspace{0.17em},\hspace{0.17em}{\displaystyle \sum _{t\in T}^G{\xi }_t}\hspace{0.17em}a\hspace{0.17em}{\eta }_t^{*}\hspace{0.17em}\rangle =\underset{w,\mathfrak{F}}{\text{lim}}\langle \hspace{0.17em}w\bar{\otimes }{1}_K\hspace{0.17em},\hspace{0.17em}(\widetilde{a,\xi ,\eta })\hspace{0.17em}\rangle$$

which proves the assertion.

b) Let (a_i, b_i)_{i ∈ I} be a finite family in $\ddot{E}\times \ddot{F}$. For x ∈ E,

$$\langle x\otimes y,{\displaystyle \sum _{i\in I}{a}_i\otimes b_i}\rangle ={\displaystyle \sum _{i\in I}\langle x,a_i\rangle \langle y,b_i\rangle }=\langle x,{\displaystyle \sum _{i\in I}\langle y,b_i\rangle a_i}\rangle .$$

Since {x ⊗ y | x ∈ E^#} is a bounded set of G, the above identity proves the continuity.

PROPOSITION 2.2.11 Let F be a unital C**-algebra, S a group, and $g\in \mathit{ℱ}(S,F)$. We denote by ⊗_σ the spatial tensor product and put

$$G≔E{\otimes }_{\sigma }F(resp. G≔E\bar{\otimes }F),$$

$$L≔\widetilde{\underset{s\in S}{⦶}}\check{F}\approx \check{F}\tilde{\otimes }{l}^2(S),\hspace{1em}\hspace{1em}M≔\widetilde{\underset{(t,s)\in T\times S}{⦶}}\check{G}\approx \check{G}\tilde{\otimes }{l}^2(T\times S),$$

$$h:(T\times S)\times (T\times S)⟶Un\hspace{0.17em}{G}^c,\hspace{1em}((t_1,s_1),(t_2,s_2))⟼f(t_1,t_2)\otimes g(s_1,s_2) .$$

• a) $h\in \mathit{ℱ}(T\times S,G),M\approx H\tilde{\otimes }L,$

  $${ℒ}_E(H){\otimes }_{\sigma }{ℒ}_F(L)\subset {ℒ}_G(M) in\hspace{0.17em}the\hspace{0.17em}C\text{*-}case,$$

  $${ℒ}_E(H)\bar{\otimes }{ℒ}_F(L)\approx {ℒ}_G(M) in\hspace{0.17em}the\hspace{0.17em}W\text{*-}case .$$

• b) For t ∈ T, s ∈ S, x ∈ E, y ∈ F,

  $$((x\tilde{\otimes }{1}_{l^2(T)})V_t^f)\otimes ((y\tilde{\otimes }{1}_{l^2(S)})V_s^g)=((x\otimes y)\tilde{\otimes }{1}_{l^2(T\times S)})V_{(t,s)}^h .$$

• c) In the C*-case, ${\mathcal{S}}_{\Vert \cdot \Vert }(f){\otimes }_{\sigma }{\mathcal{S}}_{\Vert \cdot \Vert }(g)\approx {\mathcal{S}}_{\Vert \cdot \Vert }(h)$ and ${\mathcal{S}}_C(f){\otimes }_{\sigma }{\mathcal{S}}_C(g)\approx {\mathcal{S}}_C(h)$.

• d) In the W*-case, if z ∈ G^# and (t, s)∈T × S then $(z\bar{\otimes }{1}_{l^2(T\times S)})V_{(t,s)}^h$ belongs to the closure of $\left\{\hspace{0.17em}(w\bar{\otimes }{1}_{l^2(T\times S)})V_{(t,s)}^h | w\in {(E\odot F)}^{\#}\hspace{0.17em}\right\}$ in ${ℒ}_G{(M)}_{\stackrel{\dots }{M}}$

• e) In the W*-case, ${\mathcal{S}}_W(f)\bar{\otimes }{\mathcal{S}}_W(g)\approx {\mathcal{S}}_W(h)$.

a) $h\in \mathit{ℱ}(T\times S,G)$ is obvious.

Let us treat the C*-case first. For ξ, ξ′ ∈ H and η, η′ ∈ L,

$$\begin{array}{c}\langle {\xi }^{\prime }\otimes {\eta }^{\prime }|\xi \otimes \eta \rangle =\langle {\xi }^{\prime }|\xi \rangle \otimes \langle {\eta }^{\prime }|\eta \rangle =\left({\displaystyle \sum _{t\in T}{\xi }_t^{*}{{\xi }^{\prime }}_t}\right)\otimes \left({\displaystyle \sum _{s\in S}{\eta }_s^{*}{{\eta }^{\prime }}_s}\right)=\\ ={\displaystyle \sum _{(t,s)\in T\times S}(({\xi }_t^{*}{{\xi }^{\prime }}_t)\otimes ({\eta }_s^{*}{{\eta }^{\prime }}_s))}={\displaystyle \sum _{(t,s)\in T\times S}({\xi }_t^{*}\otimes {\eta }_s^{*})({{\xi }^{\prime }}_t\otimes {{\eta }^{\prime }}_s)=}\\ ={\displaystyle \sum _{(t,s)\in T\times S}{({\xi }_t\otimes {\eta }_s)}^{*}({{\xi }^{\prime }}_t\otimes {{\eta }^{\prime }}_s),}\end{array}$$

so the linear map

$$H\odot L⟶M,\hspace{1em}\xi \otimes \eta ⟼{({\xi }_t\otimes {\eta }_s)}_{(t,s)\in T\times S}$$

preserves the scalar products and it may be extended to a linear map φ : H ⊗ L ⟶ M preserving the scalar products.

Let z ∈ G, (t, s)∈T × S, and ε > 0. There is a finite family (x_i, y_i)_{i ∈ I} in E × F such that

$$\Vert {\displaystyle \sum _{i\in I}{x}_i\otimes y_i-z}\Vert <\epsilon .$$

Then

$$\Vert {\displaystyle \sum _{i\in I}(x_i\otimes e_t)\otimes (y_i\otimes e_s)-z\otimes e_{(t,s)}}\Vert <\epsilon$$

so $z\otimes e_{(t,s)}\in \overline{\mathit{\phi }(H\otimes L)}=\mathit{\phi }(H\otimes L)$. It follows that φ is surjective and so H ⊗ L ≈ M.

The proof for the inclusion ${ℒ}_E(H){\otimes }_{\sigma }{ℒ}_F(L)\subset {ℒ}_G(M)$ can be found in [6] page 37.

Let us now discus the W*-case. $\mathit{Ĕ}\bar{\otimes }\check{F}\approx \check{G}$ follows from [2] Proposition 1.3 e), $M\approx H\bar{\otimes }L$ follows from [3] Corollary 2.2, and ${ℒ}_E(H)\bar{\otimes }{ℒ}_F(L)\approx {ℒ}_G(M)$ follows from [2] Theorem 2.4 d) or [3] Theorem 2.4.

b) For t₁, t₂∈T, s₁, s₂∈S, $\xi \in \mathit{Ĕ}$, and $\eta \in \check{F}$, by Proposition 2.1.2 f) and [3] Corollary 2.11,

$$(((x\tilde{\otimes }{1}_{l^2(T)})V_{t_1}^f)\tilde{\otimes }((y\tilde{\otimes }{1}_{l^2(S)})V_{s_1}^g))((\xi \otimes e_{t_2})\otimes (\eta \otimes e_{s_2}))=$$

$$=(((x\tilde{\otimes }{1}_{l^2(T)})V_{t_1}^f)(\xi \otimes e_{t_2}))\tilde{\otimes }(((y\tilde{\otimes }{1}_{l^2(S)})V_{s_1}^g)(\eta \otimes e_{s_2}))\hspace{0.17em},$$

$$((((x\otimes y)\tilde{\otimes }{1}_{l^2(T\times S)}))V_{(t_1,s_1)}^h)((\xi \otimes \eta )\otimes e_{(t_2,s_2)})=$$

$$=(h((t_1,s_1),(t_2,s_2))(x\otimes y)(\xi \otimes \eta ))\otimes e_{(t_1{t}_2,s_1{s}_2)}=$$

$$=((f(t_1,t_2)x\xi )\otimes (g(s_1,s_2)y\eta ))\otimes e_{t_1{t}_2}\otimes e_{s_1{s}_2}=$$

$$=(((x\tilde{\otimes }{1}_{l^2(T)})V_{t_1}^f)(\xi \otimes e_{t_2}))\tilde{\otimes }(((y\tilde{\otimes }{1}_{l^2(S)})V_{s_1}^g)(\eta \otimes e_{s_2})) .$$

We put

$$u≔((x\tilde{\otimes }{1}_{l^2(T)})V_t^f)\tilde{\otimes }((y\tilde{\otimes }{1}_{l^2(S)})V_s^g)-((x\otimes y)\tilde{\otimes }{1}_{l^2(T\times S)})V_{t,s}^h\in {ℒ}_G(M) .$$

By the above, u(ζ ⊗ e_r) = 0 for all $\zeta \in \mathit{Ĕ}\odot \check{F}$ and r ∈ T × S.

Let us consider the C*-case first. Since $\mathit{Ĕ}\odot \check{F}$ is dense in $\check{G}$, we get u(z ⊗ e_r) = 0 for all $z\in \check{G}$ and r ∈ T × S. For ζ ∈ M, by [1] Proposition 5.6.4.1 e),

$$u\zeta =u\left({\displaystyle \sum _{r\in T\times S}({\zeta }_r\otimes e_r)}\right)={\displaystyle \sum _{r\in T\times S}u({\zeta }_r\otimes e_r)=0,}$$

which proves the assertion in this case.

Let us consider now the W*-case. Let z ∈ G^# and r ∈ T × S and let $\mathfrak{F}$ be a filter on ${(E\hspace{0.17em}\odot \hspace{0.17em}F)}^{\#}$ converging to z in $G_{\ddot{G}}$ ([1] Corollary 6.3.8.7). For η ∈ M, $a\in \ddot{G}$, and r ∈ T × S,

$$\langle \hspace{0.17em}z\otimes e_r\hspace{0.17em},\hspace{0.17em}\widetilde{(a,\eta )}\hspace{0.17em}\rangle =\langle \hspace{0.17em}\langle \hspace{0.17em}z\otimes e_r\hspace{0.17em}|\hspace{0.17em}\eta \hspace{0.17em}\rangle \hspace{0.17em},\hspace{0.17em}a\hspace{0.17em}\rangle =\langle \hspace{0.17em}{\eta }_r^{*}z\hspace{0.17em},\hspace{0.17em}a\hspace{0.17em}\rangle =\langle \hspace{0.17em}z\hspace{0.17em},\hspace{0.17em}a{\eta }_r^{*}\hspace{0.17em}\rangle =$$

$$=\underset{w,\mathfrak{F}}{\text{lim}}\langle \hspace{0.17em}w\hspace{0.17em},\hspace{0.17em}a{\eta }_r^{*}\hspace{0.17em}\rangle =\underset{w,\mathfrak{F}}{\text{lim}}\langle \hspace{0.17em}w\otimes e_r\hspace{0.17em},\hspace{0.17em}\widetilde{(a,\eta )}\hspace{0.17em}\rangle \hspace{0.17em},$$

$$\underset{w,\mathfrak{F}}{\text{lim}}w\otimes e_r=z\otimes e_r$$

in $M_{\ddot{M}}$. Since $u:M_{\ddot{M}}⟶M_{\ddot{M}}$ is continuous ([1] Proposition 5.6.3.4 c)), we get by the above u(z ⊗ e_r) = 0. For ζ ∈ M it follows by [1] Proposition 5.6.4.6 c),

$$u\zeta =u\left({\displaystyle \sum _{r\in T\times S}^{\ddot{M}}({\zeta }_r\otimes e_r)}\right)={\displaystyle \sum _{r\in T\times S}^{\ddot{M}}u}({\zeta }_r\otimes e_r)=0$$

which proves the assertion in the W*-case.

c) By b), $\mathcal{R}(f)\odot \mathcal{R}(g)\subset \mathcal{R}(h)$ so by a),

$${\mathcal{S}}_{\Vert \cdot \Vert }(f)\odot {\mathcal{S}}_{\Vert \cdot \Vert }(g)\subset {\mathcal{S}}_{\Vert \cdot \Vert }(h)\hspace{0.17em},{\mathcal{S}}_C(f)\odot {\mathcal{S}}_C(g)\subset {\mathcal{S}}_C(h)\hspace{0.17em},$$

$${\mathcal{S}}_{\Vert \cdot \Vert }(f){\otimes }_{\sigma }{\mathcal{S}}_{\Vert \cdot \Vert }(g)\subset {\mathcal{S}}_{\Vert \cdot \Vert }(h)\hspace{0.17em},{\mathcal{S}}_C(f){\otimes }_{\sigma }{\mathcal{S}}_C(g)\subset {\mathcal{S}}_C(h) .$$

Let z ∈ G^#, (t, s)∈T × S, and ε > 0. There is a finite family (x_i, y_i)_{i ∈ I} in E × F such that

$$\Vert {\displaystyle \sum _{i\in I}(x_i\otimes y_i)}\Vert <1,\hspace{1em}\hspace{1em}\Vert {\displaystyle \sum _{i\in I}(x_i\otimes y_i)-z}\Vert <\epsilon .$$

By b),

$$\Vert {\displaystyle \sum _{i\in I}(((x_i\otimes 1_{l^2(T)})V_t^f)\otimes ((y_i\otimes 1_{l^2(S)})V_s^g))-(z\otimes 1_{l^2(T\times S)})V_{(t,s)}^h}\Vert <\epsilon$$

and so by a),

$$\mathcal{R}(h)\subset \stackrel{\Vert \cdot \Vert }{\overline{\mathcal{R}(f)\odot \mathcal{R}(g)}} \subset \stackrel{{\mathfrak{T}}_2}{\overline{\mathcal{R}(f)\odot \mathcal{R}(g)}}\hspace{0.17em},$$

$${\mathcal{S}}_{\Vert \cdot \Vert }(h)\subset {\mathcal{S}}_{\Vert \cdot \Vert }(f){\otimes }_{\sigma }{\mathcal{S}}_{\Vert \cdot \Vert }(g)\hspace{0.17em},{\mathcal{S}}_C(h)\subset {\mathcal{S}}_C(f){\otimes }_{\sigma }{\mathcal{S}}_C(g) .$$

d) By a) and Lemma 2.2.10 a), there is a filter $\mathfrak{F}$ on

$$\left\{\hspace{0.17em}w\bar{\otimes }{1}_{l^2(T\times S)} | w\in {(E\odot F)}^{\#}\hspace{0.17em}\right\}$$

converging to $z\bar{\otimes }{1}_{l^2(T\times S)}$ in ${ℒ}_G{(M)}_{\stackrel{\dots }{M}}$. For ξ, η ∈ M and $a\in \ddot{G}$,

$$\langle \hspace{0.17em}(z\bar{\otimes }{1}_{l^2(T\times S)})V_{(t,s)}^h\hspace{0.17em},\hspace{0.17em}\widetilde{(a,\xi ,\eta )}\hspace{0.17em}\rangle =\langle \hspace{0.17em}z\bar{\otimes }{1}_{l^2(T\times S)}\hspace{0.17em},\hspace{0.17em}{V}_{(t,s)}^h\widetilde{(a,\xi ,\eta )}\hspace{0.17em}\rangle =$$

$$=\underset{w,\mathfrak{F}}{\text{lim}}\langle \hspace{0.17em}w\bar{\otimes }{1}_{l^2(T\times S)}{V}_{(t,s)}^h\hspace{0.17em},\hspace{0.17em}\widetilde{(a,\xi ,\eta )}\hspace{0.17em}\rangle =\underset{w,\mathfrak{F}}{\text{lim}}\langle \hspace{0.17em}(w\bar{\otimes }{1}_{l^2(T\times S)})V_{(t,s)}^h\hspace{0.17em},\hspace{0.17em}\widetilde{(a,\xi ,\eta )}\hspace{0.17em}\rangle \hspace{0.17em},$$

which proves the assertion.

e) By Theorem 2.1.9 h),

$${\left(\stackrel{\stackrel{\dots }{H}}{\overline{\mathcal{R}(f)}}\right)}^{\#}={\mathcal{S}}_W{(f)}^{\#}\subset {ℒ}_E(H),{\left(\stackrel{\stackrel{\dots }{L}}{\overline{\mathcal{R}(g)}}\right)}^{\#}={\mathcal{S}}_W{(g)}^{\#}\subset {ℒ}_F(L) .$$

By b), $\mathcal{R}(f)\odot \mathcal{R}(g)\subset \mathcal{R}(h)$, so by Lemma 2.2.10 b),

$${\mathcal{S}}_W{(f)}^{\#}\odot \mathcal{R}{(g)}^{\#}\subset {\mathcal{S}}_W{(h)}^{\#},{\mathcal{S}}_W{(f)}^{\#}\otimes {\mathcal{S}}_W{(g)}^{\#}\subset {\mathcal{S}}_W{(h)}^{\#} .$$

$${\mathcal{S}}_W(f)\otimes {\mathcal{S}}_W(g)\subset {\mathcal{S}}_W(h)$$

By [3] Proposition 2.5,

$${\mathcal{S}}_W(f)\bar{\otimes }{\mathcal{S}}_W(g)\approx \stackrel{\stackrel{\dots }{M}}{\overline{{\mathcal{S}}_W(f)\otimes {\mathcal{S}}_W(g)}}\subset {\mathcal{S}}_W(h) .$$

For x ∈ E, y ∈ F, and (t, s)∈T × S, by b),

$$((x\otimes y)\bar{\otimes }{1}_{l^2(T\times S)})V_{(t,s)}^h=((x\bar{\otimes }{1}_{l^2(T)})V_t^f)\bar{\otimes }((y\bar{\otimes }{1}_{l^2(S)})V_s^g)\in {\mathcal{S}}_W(f)\bar{\otimes }{\mathcal{S}}_W(g) .$$

Let z ∈ G^#. By d), there is a filter $\mathfrak{F}$ on

$$\left\{\hspace{0.17em}(w\bar{\otimes }{1}_{l^2(T\times S)})V_{(t,s)}^h | w\in {(E\odot F)}^{\#}\hspace{0.17em}\right\}$$

converging to $(z\bar{\otimes }{1}_{l^2(T\times S)})V_{(t,s)}^h$ in ${ℒ}_G{(M)}_{\stackrel{\dots }{M}}$, so by the above

$$(z\bar{\otimes }{1}_{l^2(T\times S)})V_{(t,s)}^h\in {\mathcal{S}}_W(f)\bar{\otimes }{\mathcal{S}}_W(g) .$$

We get

$$\mathcal{R}(h)\subset {\mathcal{S}}_W(f)\bar{\otimes }{\mathcal{S}}_W(g),{\mathcal{S}}_W(h)\subset {\mathcal{S}}_W(f)\bar{\otimes }{\mathcal{S}}_W(g)\hspace{0.17em},$$

$${\mathcal{S}}_W(h)={\mathcal{S}}_W(f)\bar{\otimes }{\mathcal{S}}_W(g) .$$

COROLLARY 2.2.12 Let n ∈ ℕ and

$$g:T\times T⟶Un {(E_{n,n})}^c,\hspace{1em}(s,t)⟼{[{\delta }_{i,j}f(s,t)]}_{i,j\in {\mathbb{N}}_{\text{n}}} .$$

• a) ${(\mathcal{S}(f))}_{n,n}\approx \mathcal{S}(g),\hspace{1em}{({\mathcal{S}}_{\Vert \cdot \Vert }(f))}_{n,n}\approx {\mathcal{S}}_{\Vert \cdot \Vert }(g)$.

• b) Let us denote by $\rho :\mathcal{S}(g)⟶{(\mathcal{S}(f))}_{n,n}$ the isomorphism of a). For $X\in \mathcal{S}(g)$, t ∈ T, and $i,j\in {\mathbb{N}}_{\text{n}}$,

  $${({(\rho X)}_{i,j})}_t={(X_t)}_{i,j} .$$

a) Take F ≔ 𝕂_{n, n} and S ≔ {1} in Proposition 2.2.11. Then G ≈ E_{n, n} and

$$g:T\times T⟶Un\hspace{0.17em}{G}^c,\hspace{1em}(s,t)⟼f(s,t)\otimes 1_F .$$

By Proposition 2.2.11 c),e),

$$\mathcal{S}(g)\approx \mathcal{S}(f)\otimes {𝕂}_{n,n}\approx {(\mathcal{S}(f))}_{n,n}$$

$${\mathcal{S}}_{\Vert \cdot \Vert }(g)\approx {\mathcal{S}}_{\Vert \cdot \Vert }(f)\otimes {𝕂}_{\text{n},\text{n}}\approx {({\mathcal{S}}_{\Vert \cdot \Vert }(f))}_{n,n} .$$

b) By Theorem 2.1.9 b),

$$X={\displaystyle \sum _{s\in T}^{{\mathfrak{T}}_3}(X_s\tilde{\otimes }{1}_K)}{V}_s^g$$

so

$${(\rho X)}_{i,j}={\displaystyle \sum _{s\in t}^{{\mathfrak{T}}_3}((}{X}_s{)}_{i,j}\tilde{\otimes }{1}_K)V_s^f\hspace{0.17em},$$

$${({(\rho X)}_{i,j})}_t={(X_t)}_{i,j}$$

by Theorem 2.1.9 a).

COROLLARY 2.2.13 Let n ∈ ℕ. If 𝕂 = ℂ (resp. if n = 4^m for some m ∈ ℕ) then there is an $f\in \mathit{ℱ}({\mathbb{Z}}_{\text{n}}\times {\mathbb{Z}}_{\text{n}},E)$ (resp. $f\in \mathit{ℱ}( {({\mathbb{Z}}_{\text{2}})}^{\text{2}m},E)$) such that

$$\mathcal{R}(f)=\mathcal{S}(f)\approx E_{n,n} .$$

By [1] Proposition 7.1.4.9 b),d) (resp. [1] Theorem 7.2.2.7 i),k)) there is a $g\in \mathit{ℱ}({\mathbb{Z}}_{\text{n}}\times {\mathbb{Z}}_{\text{n}},\mathcal{C})$ (resp. $g\in F({({\mathbb{Z}}_{\text{n}})}^{2m},𝕂)$) such that

$$\mathcal{S}(g)\approx {\mathcal{C}}_{n,n}(\text{resp}.\mathcal{S}(g)\approx {𝕂}_{n,n}).$$

If we put

$$f:({\mathbb{Z}}_{\text{n}}\times {\mathbb{Z}}_{\text{n}})\times ({\mathbb{Z}}_{\text{n}}\times {\mathbb{Z}}_{\text{n}})⟶Un\hspace{0.17em}{E}^c,\hspace{1em}(s,t)⟼g(s,t)\otimes 1_E$$

$$(\text{resp}\text{.} f:{({\mathbb{Z}}_{\text{2}})}^{2m}\times {({\mathbb{Z}}_{\text{2}})}^{2m}⟶Un\hspace{0.17em}{E}^c,\hspace{1em}(s,t)⟼g(s,t)\otimes 1_E)$$

then by Proposition 2.2.11 a),e), $f\in \mathit{ℱ}({\mathbb{Z}}_{\text{n}}\times {\mathbb{Z}}_{\text{n}},E)$ (resp. $f\in \mathit{ℱ}( {({\mathbb{Z}}_{\text{2}})}^{2m},E)$) and

$$\mathcal{S}(f)\approx \mathcal{S}(g)\otimes E\approx {𝕂}_{n,n}\otimes E\approx E_{n,n}.$$

COROLLARY 2.2.14 Let F be a unital C**-algebra, $G≔E\tilde{\otimes }F$, and

$$h:T\times T⟶Un\hspace{0.17em}{G}^c,\hspace{1em}(s,t)⟼f(s,t)\otimes 1_F .$$

Then $h\in \mathit{ℱ}(T,G)$ and

$${\mathcal{S}}_{\Vert \cdot \Vert }(h)\approx {\mathcal{S}}_{\Vert \cdot \Vert }(f)\otimes F\hspace{0.17em},\mathcal{S}(h)\approx \mathcal{S}(f)\tilde{\otimes }F .$$

COROLLARY 2.2.15 If E is a W*-algebra then the following are equivalent:

• a) E is semifinite.

• b) ${\mathcal{S}}_W(f)$ is semifinite.

a ⇒ b. Assume first that there are a finite W*-algebra F and a Hilbert space L such that $E\approx F\bar{\otimes }ℒ(L)$. Put

$$g:T\times T⟶Un\hspace{0.17em}{F}^c,\hspace{1em}(s,t)⟼f(s,t) .$$

By Corollary 2.2.14,

$${\mathcal{S}}_W(f)\approx {\mathcal{S}}_W(g)\bar{\otimes }ℒ(L) .$$

By Corollary 2.1.11 c), ${\mathcal{S}}_W(g)$ is finite and so ${\mathcal{S}}_W(f)$ is semifinite.

The general case follows from the fact that E is the C*-direct product of W*-algebras of the above form ([8] Proposition V.1.40).

b ⇒ a. E is isomorphic to a W*-subalgebra of ${\mathcal{S}}_W(f)$ (Theorem 2.1.9 h)) and the assertion follows from [8] Theorem V.2.15.

PROPOSITION 2.2.16 Let S, T be finite groups and $g\in \mathit{ℱ}(S,\mathcal{S}(f))$ and put L ≔ l²(S), M ≔ l²(S × T), and

$$h:(S\times T)\times (S\times T)⟶Un \mathcal{S}{(f)}^c,\hspace{1em}((s_1,t_1),(s_2,t_2))⟼f(t_1,t_2)g(s_1,s_2) .$$

Then $h\in \mathit{ℱ}(S\times T,\mathcal{S}(f))$ and the map

$$\mathit{\phi }:\mathcal{S}(g)⟶\mathcal{S}(h),\hspace{1em}X⟼{\displaystyle \sum _{(s,t)\in S\times T}({(X_s)}_t\otimes 1_M)V_{(s,t)}^h}$$

is an $\mathcal{S}(f)$-C*-isomorphism.

For $X,Y\in \mathcal{S}(g)$, $Z\in \mathcal{S}(f)$, and (s, t)∈S × T, by Theorem 2.1.9 c), g),

$${(\mathit{\phi }(X^{*}))}_{(s,t)}={({(X^{*})}_s)}_t={(\tilde{g}(s){(X_{s^{-1}})}^{*})}_t={({(\tilde{g}{(s)}^{*}{X}_{s^{-1}})}^{*})}_t=$$

$$=\tilde{f}(t){({(\tilde{g}{(s)}^{*}{X}_{s^{-1}})}_{t^{-1}})}^{*}=\tilde{f}(t)\tilde{g}(s){({(X_{s^{-1}})}_{t^{-1}})}^{*}=$$

$$=\tilde{h}(s,t){({(\mathit{\phi }X)}_{(s^{-1},t^{-1})})}^{*}=\tilde{h}(s,t){({(\mathit{\phi }X)}_{{(s,t)}^{-1}})}^{*}={({(\mathit{\phi }X)}^{*})}_{(s,t)}\hspace{0.17em},$$

$${((\mathit{\phi }X)(\mathit{\phi }Y))}_{(s,t)}={\displaystyle \sum _{(r,u)\in S\times T}h((r,u),{(r,u)}^{-1}(s,t)){(\mathit{\phi }X)}_{(r,u)}{(\mathit{\phi }Y)}_{{(r,u)}^{-1}(s,t)}=}$$

$$={\displaystyle \sum _{(r,u)\in S\times T}g(r,r^{-1}s)f(u,u^{-1}t){(X_r)}_u{(Y_{r^{-1}s})}_{u^{-1}t}}={\displaystyle \sum _{r\in S}g(r,r^{-1}s){(X_r{Y}_{r^{-1}s})}_t=}$$

$$={\left({\displaystyle \sum _{r\in S}g\left(r,r^{-1}s\right)X_r{Y}_{r^{-1}s}}\right)}_t={({(XY)}_s)}_t={(\mathit{\phi }(XY))}_{(s,t)},$$

$${(\mathit{\phi }(ZX))}_{(s,t)}={({(ZX)}_s)}_t={({(ZX)}_s)}_t={(Z{X}_s)}_t=Z{(X_s)}_t=Z{(\mathit{\phi }X)}_{(s,t)}$$

so

$$\mathit{\phi }(X^{*})={(\mathit{\phi }X)}^{*}\hspace{0.17em},\mathit{\phi }(XY)=(\mathit{\phi }X)(\mathit{\phi }Y),\mathit{\phi }(ZX)=Z\mathit{\phi }(X)$$

and φ is an $\mathcal{S}(f)$-C*-homomorphism.

If $X\in \mathcal{S}(g)$ with φ X = 0 then for (s, t)∈S × T,

$${(X_s)}_t={(\mathit{\phi }X)}_{(s,t)}=0\hspace{0.17em},X_s=0\hspace{0.17em},X=0$$

so φ is injective.

Let x ∈ E and (s, t)∈S × T. Put

$$Z≔(x\otimes 1_K)V_t^f\in \mathcal{S}(f)\hspace{0.17em},X≔(Z\otimes 1_L)V_s^g\in \mathcal{S}(g) .$$

Then for (r, u)∈S × T,

$${(\mathit{\phi }X)}_{(r,u)}={(X_r)}_u={\delta }_{r,s}{Z}_u={\delta }_{r,s}{\delta }_{u,t}x$$

so

$$\mathit{\phi }X=(x\otimes 1_M)V_{(s,t)}^h$$

and φ is surjective.

PROPOSITION 2.2.17 Let S be a finite subgroup of T and g ≔ f|(S × S). We identify $\mathcal{S}(g)$ with the E-C**-subalgebra $\left\{\hspace{0.17em}Z\in \mathcal{S}(f) | t\in T\setminus S\Rightarrow Z_t=0\hspace{0.17em}\right\}$ of $\mathcal{S}(f)$ (Corollary 2.1.17 e)). Let $X\in \mathcal{S}(f)\cap \mathcal{S}{(g)}^c$, P₊ ≔ X^*X, and P₋ ≔ XX^* and assume $P_{\pm }\in Pr \mathcal{S}(f)$.

• a) $P_{\pm }\in \mathcal{S}{(g)}^c$.

• b) The map

  $${\mathit{\phi }}_{\pm }:\mathcal{S}(g)⟶P_{\pm }\mathcal{S}(f)P_{\pm },\hspace{1em}Y⟼P_{\pm }Y{P}_{\pm }$$

  is a unital C**-homomorphism.

• c) For every $Z\in {\mathit{\phi }}_{+}(\mathcal{S}(g))$, $XZ{X}^{*}\in {\mathit{\phi }}_{-}(\mathcal{S}(g))$ and the map

  $$\mathit{\psi }:{\mathit{\phi }}_{+}(\mathcal{S}(g))⟶{\mathit{\phi }}_{-}(\mathcal{S}(g))\hspace{0.17em},\hspace{1em}Z⟼XZ{X}^{*}$$

  is a C*-isomorphism with inverse

  $${\mathit{\phi }}_{-}(\mathcal{S}(g))⟶{\mathit{\phi }}_{+}(\mathcal{S}(g)),\hspace{1em}Z⟼X^{*}ZX$$

  such that φ₋ = ψ ο φ₊.

• d) If $p\in Pr\hspace{0.17em}\mathcal{S}(g)$ then

  $${(X({\mathit{\phi }}_{+}p))}^{*}(X({\mathit{\phi }}_{+}p))={\mathit{\phi }}_{+}p\hspace{0.17em},(X(({\mathit{\phi }}_{+}p)){(X({\mathit{\phi }}_{+}p))}^{*}={\mathit{\phi }}_{-}p .$$

• e) If φ₊ is injective then φ₋ is also injective, the map

  $$E⟶P_{\pm }\mathcal{S}(f)P_{\pm },\hspace{1em}x⟼P_{\pm }(x\tilde{\otimes }{1}_K)P_{\pm }$$

  is an injective unital C**-homomorphism, $P_{\pm }\mathcal{S}(f)P_{\pm }$ is an E-C**-algebra, ${\mathit{\phi }}_{\pm }(\mathcal{S}(g))$ is an E-C**-subalgebra of it, and φ_± and ψ are E-C**-homo-morphisms.

• f) The above results still hold for an arbitrary subgroup S of T if we replace $\mathcal{S}$ by ${\mathcal{S}}_{\Vert \cdot \Vert }$.

a) follows from the hypothesis on X.

b) follows from a).

c) Let $Y\in \mathcal{S}(g)$ with Z = P₊Y P₊. By the hypotheses of the Proposition,

$$XZ{X}^{*}=X{P}_{+}Y{P}_{+}{X}^{*}=X{X}^{*}XY{X}^{*}X{X}^{*}=$$

$$=X{X}^{*}YX{X}^{*}X{X}^{*}=P_{-}Y{P}_{-}\in {\mathit{\phi }}_{-}(\mathcal{S}(g))$$

and ψ is a C*-homomorphism. The other assertions follow from

$$X^{*}(XZ{X}^{*})X=P_{+}Z{P}_{+}=P_{+}Y{P}_{+} .$$

d) By b) and c),

$${(X({\mathit{\phi }}_{+}p))}^{*}(X({\mathit{\phi }}_{+}p))=({\mathit{\phi }}_{+}p)X^{*}X({\mathit{\phi }}_{+}p)=({\mathit{\phi }}_{+}p)P_{+}({\mathit{\phi }}_{+}p)={\mathit{\phi }}_{+}p\hspace{0.17em},$$

$$(X({\mathit{\phi }}_{+}p)){(X({\mathit{\phi }}_{+}p))}^{*}=X({\mathit{\phi }}_{+}p){({\mathit{\phi }}_{+}p)}^{*}{X}^{*}=X({\mathit{\phi }}_{+}p)X^{*}=\mathit{\psi }{\mathit{\phi }}_{+}p={\mathit{\phi }}_{-}p .$$

e) follows from b), c), and Lemma 1.3.2.

f) follows from Corollary 2.1.17 d).

Remark. Even if φ_± is injective $P_{\pm }\mathcal{S}(f)P_{\pm }$ is not an E-C*-subalgebra of $\mathcal{S}(f)$.

THEOREM 2.2.18 Let S be a finite subgroup of T, L ≔ l²(S), g ≔ f|(S × S), $\mathit{\omega }:{\mathbb{Z}}_{\text{2}}\times {\mathbb{Z}}_{\text{2}}⟶T$ an injective group homomorphism such that $S\cap \mathit{\omega }({\mathbb{Z}}_{\text{2}}\times {\mathbb{Z}}_{\text{2}})=\left\{1\right\}$,

$$a≔\mathit{\omega }(1,0)\hspace{0.17em},\hspace{1em}b≔\mathit{\omega }(0,1)\hspace{0.17em},\hspace{1em}c≔\mathit{\omega }(1,1)\hspace{0.17em},\hspace{1em}{\mathit{\alpha }}_1≔f(a,a)\hspace{0.17em},\hspace{1em}{\mathit{\alpha }}_2≔f(b,b)\hspace{0.17em},$$

β₁, β₂∈Un E^c such that ${\mathit{\alpha }}_1{\mathit{\beta }}_1^2+{\mathit{\alpha }}_2{\mathit{\beta }}_2^2=0$,

$$\gamma ≔\frac{1}{2}({\mathit{\alpha }}_1^{*}{\mathit{\beta }}_1^{*}{\mathit{\beta }}_2-{\mathit{\alpha }}_2^{*}{\mathit{\beta }}_1{\mathit{\beta }}_2^{*})={\mathit{\alpha }}_1^{*}{\mathit{\beta }}_1^{*}{\mathit{\beta }}_2=-{\mathit{\alpha }}_2^{*}{\mathit{\beta }}_1{\mathit{\beta }}_2^{*}\hspace{0.17em},$$

$$X≔\frac{1}{2}(({\mathit{\beta }}_1\tilde{\otimes }{1}_K)V_a^f+({\mathit{\beta }}_2\tilde{\otimes }{1}_K)V_b^f)\hspace{0.17em},P_{+}≔X^{*}X\hspace{0.17em},\hspace{1em}{P}_{-}≔X{X}^{*} .$$

We assume f(s, c) = f(c, s) and cs = sc for every s ∈ S, and f(a, b) = −f(b, a) = 1_E. Moreover we consider $\mathcal{S}(g)$ as an E-C**-subalgebra of $\mathcal{S}(f)$ (Corollary 2.1.17 e)).

• a) We have

  $$f(a,c)=-f(c,a)={\mathit{\alpha }}_1\hspace{0.17em},\hspace{1em}f(b,c)=-f(c,b)=-{\mathit{\alpha }}_2\hspace{0.17em},\hspace{1em}f(c,c)=-{\mathit{\alpha }}_1{\mathit{\alpha }}_2\hspace{0.17em},$$

  $${\gamma }^2=-{\mathit{\alpha }}_1^{*}{\mathit{\alpha }}_2^{*}\hspace{0.17em},V_c^f\in \mathcal{S}{(g)}^c .$$

• b) We have

  $$P_{\pm }=\frac{1}{2}(V_1^f\pm (\gamma \tilde{\otimes }{1}_K)V_c^f)\in \mathcal{S}{(g)}^c\cap Pr\hspace{0.17em}\mathcal{S}(f), P_{+}+P_{-}=V_1^f, P_{+}{P}_{-}=0\hspace{0.17em},$$

  $$X^2=0, X{P}_{+}=X,\hspace{0.17em}{P}_{-}X=X, P_{+}X=X{P}_{-}=0, X+X^{*}\in Un \mathcal{S}(f)\hspace{0.17em},$$

  $$Y\in \mathcal{S}(g)\Rightarrow XYX=0 .$$

• c) The map

  $$E⟶P_{\pm }\mathcal{S}(f)P_{\pm },\hspace{1em}x⟼(x\tilde{\otimes }{1}_K)P_{\pm }$$

  is a unital injective C**-homomorphism; we shall consider $P_{\pm }\mathcal{S}(f)P_{\pm }$ as an E-C**-algebra using this map.

• d) The maps

  $${\mathit{\phi }}_{+}:\mathcal{S}(g)⟶P_{+}\mathcal{S}(f)P_{+},\hspace{1em}Y⟼P_{+}Y{P}_{+},$$

  $${\mathit{\phi }}_{-}:\mathcal{S}(g)⟶P_{-}\mathcal{S}(f)P_{-},\hspace{1em}Y⟼XY\hspace{0.17em}{X}^{*}$$

  are orthogonal injective E-C**-homomorphisms and φ₊ + φ₋ is an injective E-C*-homomorphism. If $Y_1,Y_2\in Un \mathcal{S}(g)$ (resp. $Y_1,Y_2\in Pr \mathcal{S}(g)$) then ${\mathit{\phi }}_{+}{Y}_1+{\mathit{\phi }}_{-}{Y}_2\in Un \mathcal{S}(f)$ (resp. ${\mathit{\phi }}_{+}{Y}_1+{\mathit{\phi }}_{-}{Y}_2\in Pr \mathcal{S}(f)$). Moreover the map

  $$\mathit{\psi }:\mathcal{S}(f)⟶\mathcal{S}(f),\hspace{1em}Z⟼(X+X^{*})Z(X+X^{*})$$

  is an E-C**-isomorphism such that

  $${\mathit{\psi }}^{-1}=\mathit{\psi },\mathit{\psi }(P_{+}\mathcal{S}(f)P_{+})=P_{-}\mathcal{S}(f)P_{-},\mathit{\psi }\text{ο}{\mathit{\phi }}_{+}={\mathit{\phi }}_{-}.$$

  If 𝕂 = ℂ then X + X^* is homotopic to $V_1^f$ in $Un\mathcal{S}(f)$ and ψ is homotopic to the identity map of $\mathcal{S}(f)$. Using this homotopy we find that φ₊Y is homotopic in the above sense to φ₋Y for every $Y\in \mathcal{S}(g)$ and φ₊Y₁ + φ₋Y₂, φ₋Y₁ + φ₊Y₂, φ₊(Y₁Y₂) + P₋, and φ₊(Y₂Y₁ + P₋ are homotopic in the above sense for all $Y_1,Y_2\in \mathcal{S}(g)$.

• e) Let s ∈ S such that sa = as. Then

  $$sb=bs\hspace{0.17em},f(sc,c)f(s,c)=-{\mathit{\alpha }}_1{\mathit{\alpha }}_2\hspace{0.17em},$$

  $$f(sa,c)f{(c,sa)}^{*}=-1_E\hspace{0.17em},f(a,s)f{(s,a)}^{*}=f(b,s)f{(s,b)}^{*} .$$

• f) If sa = as for every s ∈ S then the map

  $$S\times ({\mathbb{Z}}_{\text{2}}\times {\mathbb{Z}}_{\text{2}})⟶T,\hspace{1em}(s,r)⟼s(\mathit{\omega }r)$$

  is an injective group homomorphism.

• h) If T is generated by $S\cup \mathit{\omega }\left({\mathbb{Z}}_{\text{2}}\times {\mathbb{Z}}_{\text{2}}\right)$ and sa = as for every s ∈ S then φ₊ and ψ₋ are E-C*-isomorphisms with inverse

  $$P_{\pm }\mathcal{S}(f)P_{\pm }⟼\mathcal{S}(g),\hspace{1em}Z⟼2{\displaystyle \sum _{s\in S}(Z_s\tilde{\otimes }{1}_L)V_s^g,}$$

  where

  $${\mathit{\psi }}_{-}:\mathcal{S}(g)⟶P_{-}\mathcal{S}(f)P_{-},\hspace{1em}Y⟼P_{-}Y{P}_{-} .$$

• h) If sa = as and f(a, s) = f(s, a) for every s ∈ S then $X\in \mathcal{S}{(g)}^c$, φ₋Y = P₋Y for every $Y\in \mathcal{S}(g)$, and there is a unique $\mathcal{S}(g)$-C**-homomorphism $\varphi :\mathcal{S}{(g)}_{2,2}⟶\mathcal{S}(f)$ such that

  $$\varphi \left[\begin{array}{cc}0& 0\\ ({\mathit{\alpha }}_1{\mathit{\beta }}_1^2)\otimes 1_L& 0\end{array}\right]=X .$$

  ϕ is injective and

  $$\varphi \left[\begin{array}{cc}{V}_1^g& 0\\ 0& 0\end{array}\right]=P_{+}\hspace{0.17em},\varphi \left[\begin{array}{cc}0& 0\\ 0& V_1^g\end{array}\right]=P_{-} .$$

• i) If sa = as and f(a, s) = f(s, a) for all s ∈ S and if T is generated by $S\cup \mathit{\omega }\left({\mathbb{Z}}_{\text{2}}\times {\mathbb{Z}}_{\text{2}}\right)$ then ϕ is an $\mathcal{S}(g)$-C*-isomorphism and

  $${\varphi }^{-1}{V}_1^f=\left[\begin{array}{cc}{1}_E\otimes 1_L& 0\\ 0& 1_E\oplus 1_L\end{array}\right],{\varphi }^{-1}{V}_c^f=\left[\begin{array}{cc}\mathit{\gamma }*\otimes 1_L& 0\\ 0& -\mathit{\gamma }*\oplus 1_L\end{array}\right],$$

  $${\varphi }^{-1}{V}_a^f=\left[\begin{array}{cc}0& -{\mathit{\beta }}_1^{*}\otimes 1_L\\ ({\mathit{\beta }}_2{\gamma }^{*})\otimes 1_L& 0\end{array}\right]\hspace{0.17em},$$

  $${\varphi }^{-1}{V}_b^f=\left[\begin{array}{cc}0& -{\mathit{\beta }}_2^{*}\otimes 1_L\\ ({\mathit{\beta }}_1{\gamma }^{*})\otimes 1_L& 0\end{array}\right]\hspace{0.17em},$$

  $${\varphi }^{-1}{P}_{+}=\left[\begin{array}{cc}{V}_1^g& 0\\ 0& 0\end{array}\right]\hspace{0.17em},{\varphi }^{-1}{P}_{-}=\left[\begin{array}{cc}0& 0\\ 0& V_1^g\end{array}\right]\hspace{0.17em},$$

  and for every s∈ S

  $${\varphi }^{-1}{V}_s^f=\left[\begin{array}{cc}{V}_s^g& 0\\ 0& V_s^g\end{array}\right] .$$

• j) The above results still hold for an arbitrary subgroup S of T if we replace $\mathcal{S}$ with ${\mathcal{S}}_{\Vert \cdot \Vert }$.

a) By the equation of the Schur functions,

$$f(a,a)=f(a,c)f(a,b)\hspace{0.17em}, f(a,b)f(c,a)=f(a,c)f(b,a)\hspace{0.17em}, f(a,b)f(c,b)=f(b,b)\hspace{0.17em},$$

$$f(b,a)f(c,b)=f(b,c)f(a,b)\hspace{0.17em},f(a,b)f(c,c)=f(a,a)f(b,c)$$

and so

$${\mathit{\alpha }}_1=f(a,c)\hspace{0.17em},f(c,a)=-f(a,c)=-{\mathit{\alpha }}_1\hspace{0.17em},f(c,b)={\mathit{\alpha }}_2\hspace{0.17em},$$

$$-{\mathit{\alpha }}_2=-f(c,b)=f(b,c)\hspace{0.17em},f(c,c)={\mathit{\alpha }}_{1}f(b,c)=-{\mathit{\alpha }}_1{\mathit{\alpha }}_2 .$$

For s ∈ S, by Proposition 2.1.2 b),

$$V_c^f{V}_s^f=(f(c,s)\tilde{\otimes }{1}_K)V_{cs}^f=(f(s,c)\tilde{\otimes }{1}_K)V_{sc}^f=V_s^f{V}_c^f$$

and so $V_c^f\in \mathcal{S}{(g)}^c$ (by Proposition 2.1.2 d)).

b) By Proposition 2.1.2 b),d),e) (and Corollary 2.1.22 c)),

$$X^{*}=\frac{1}{2}((({\mathit{\alpha }}_1^{*}{\mathit{\beta }}_1^{*})\tilde{\otimes }{1}_K)V_a^f+(({\mathit{\alpha }}_2^{*}{\mathit{\beta }}_2^{*})\tilde{\otimes }{1}_K)V_b^f)\hspace{0.17em},$$

$$P_{+}=\frac{1}{4}(2V_1^f+(({\mathit{\alpha }}_1^{*}{\mathit{\beta }}_1^{*}{\mathit{\beta }}_2)\tilde{\otimes }{1}_K)V_c^f-(({\mathit{\alpha }}_2^{*}{\mathit{\beta }}_2^{*}{\mathit{\beta }}_1)\tilde{\otimes }{1}_K)V_c^f)=\frac{1}{2}(V_1^f+(\gamma \tilde{\otimes }{1}_K)V_c^f)\hspace{0.17em},$$

$$P_{-}=\frac{1}{4}(2V_1^f+(({\mathit{\beta }}_1{\mathit{\alpha }}_2^{*}{\mathit{\beta }}_2^{*})\tilde{\otimes }{1}_K)V_c^f-(({\mathit{\beta }}_2{\mathit{\alpha }}_1^{*}{\mathit{\beta }}_1^{*})\tilde{\otimes }{1}_K)V_c^f)=\frac{1}{2}(V_1^f-(\gamma \tilde{\otimes }{1}_K)V_c^f) .$$

By a),

$$P_{\pm }^{*}=\frac{1}{2}(V_1^f\pm ({\gamma }^{*}\tilde{\otimes }{1}_K)((-{\mathit{\alpha }}_1^{*}{\mathit{\alpha }}_2^{*})\tilde{\otimes }{1}_K)V_c^f)=P_{\pm }\hspace{0.17em},$$

$$P_{\pm }^2=\frac{1}{4}(V_1^f\pm 2(\gamma \tilde{\otimes }{1}_K)V_c^f+({\gamma }^2\tilde{\otimes }{1}_K)((-{\mathit{\alpha }}_1{\mathit{\alpha }}_2)\tilde{\otimes }{1}_K)V_1^f)=$$

$$=\frac{1}{2}(V_1^f\pm (\gamma \tilde{\otimes }{1}_K)V_c^f)=P_{\pm }\hspace{0.17em},$$

so, by a) again, $P_{\pm }\in \mathcal{S}{(g)}^c\cap Pr\hspace{0.17em}\mathcal{S}(f)$. By Proposition 2.1.2 b),d),

$$X^2=\frac{1}{4}((({\mathit{\beta }}_1^2{\mathit{\alpha }}_1+{\mathit{\beta }}_2^2{\mathit{\alpha }}_2)\tilde{\otimes }{1}_K)V_1^f+(({\mathit{\beta }}_1{\mathit{\beta }}_2)\tilde{\otimes }{1}_K)(V_a^f{V}_b^f+V_b^f{V}_a^f))=0\hspace{0.17em},$$

$${(X+X^{*})}^2=X^2+X{X}^{*}+X^{*}X+X^{*2}=P_{+}+P_{-}=V_1^f .$$

For the last relation we remark that by the above,

$$XYX=X(P_{+}+P_{-})YX=X{P}_{+}YX=XY{P}_{+}X=0 .$$

c) follows from b) and Lemma 1.3.2.

d) By b) and c), the map φ_± is an E-C**-homomorphism. Let $Y\in \mathcal{S}(g)$ with φ_±Y = 0. By b), $Y=\mp Y(\gamma \tilde{\otimes }{1}_K)V_c^f$ so by Proposition 2.1.2 b),d) and Theorem 2.1.9 b),

$${\displaystyle \sum _{s\in S}(Y_s\tilde{\otimes }{1}_K)V_s^f=\mp Y(\gamma \tilde{\otimes }{1}_K)V_c^f}=\mp {\displaystyle \sum _{s\in S}((Y_s\gamma f(s,c))\tilde{\otimes }{1}_K)V_{sc}^f,}$$

which implies Y_s = 0 for every s ∈ S (Theorem 2.1.9 a)). Thus φ_± is injective. It follows that φ₊ + φ₋ is also injective.

Assume first $Y_1,Y_2\in Un \mathcal{S}(g)$. By b),

$${({\mathit{\phi }}_{+}{Y}_1+{\mathit{\phi }}_{-}{Y}_2)}^{*}({\mathit{\phi }}_{+}{Y}_1+{\mathit{\phi }}_{-}{Y}_2)=({\mathit{\phi }}_{+}{Y}_1^{*}+{\mathit{\phi }}_{-}{Y}_2^{*})({\mathit{\phi }}_{+}{Y}_1+{\mathit{\phi }}_{-}{Y}_2)=$$

$$={\mathit{\phi }}_{+}(Y_1^{*}{Y}_1)+{\mathit{\phi }}_{-}(Y_2^{*}{Y}_2)=P_{+}+P_{-}=V_1^f .$$

Similarly $({\mathit{\phi }}_{+}{Y}_1+{\mathit{\phi }}_{-}{Y}_2){({\mathit{\phi }}_{+}{Y}_1+{\mathit{\phi }}_{-}{Y}_2)}^{*}=V_1^f$. The case $Y_1,Y_2\in Pr \mathcal{S}(g)$ is easy to see.

By b), ψ is an E-C**-isomorphism with

$${\mathit{\psi }}^{-1}=\mathit{\psi }\hspace{0.17em},\mathit{\psi }{P}_{+}=(X+X^{*})X^{*}X(X+X^{*})=X{X}^{*}X{X}^{*}=P_{-} .$$

Moreover for $Y\in \mathcal{S}(g)$,

$$\mathit{\psi }{\mathit{\phi }}_{+}Y=(X+X^{*})P_{+}Y{P}_{+}(X+X^{*})=XY{X}^{*}={\mathit{\phi }}_{-}Y .$$

Assume now 𝕂 = ℂ. By b), $X+X^{*}\in Un \mathcal{S}(f)$. Being selfadjoint its spectrum is contained in {−1, +1} and so it is homotopic to $V_1^f$ in $Un \mathcal{S}(f)$.

e) We have sb = sac = asc = acs = bs. By a),

$$f(s,c)f(sc,c)=f(s,1)f(c,c)=-{\mathit{\alpha }}_1{\mathit{\alpha }}_2\hspace{0.17em},$$

$$f(s,a)f(sa,c)=f(s,b)f(a,c)={\mathit{\alpha }}_{1}f(s,b)\hspace{0.17em},$$

$$f(c,as)f(a,s)=f(c,a)f(b,s)=-{\mathit{\alpha }}_{1}f(b,s)\hspace{0.17em},$$

$$f(c,bs)f(b,s)=f(c,b)f(a,s)={\mathit{\alpha }}_{2}f(a,s)\hspace{0.17em},$$

$$f(s,c)f(sc,b)=f(s,a)f(c,b)={\mathit{\alpha }}_{2}f(s,a)\hspace{0.17em},$$

$$f(c,s)f(cs,b)=f(c,sb)f(s,b)$$

so

$$f(sa,c)f{(c,as)}^{*}=-f(s,b)f{(s,a)}^{*}f{(b,s)}^{*}f(a,s)=$$

$$=-f(c,s)f(cs,b)f{(c,sb)}^{*}{\mathit{\alpha }}_2$$