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
for all r, s, t ∈ T. The projective representation of T twisted by f is a unital C*-subalgebra of the C*-algebra 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 and is replaced by , the C*-algebra of adjointable operators of . 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), 1K ≔ idK ≔ identity map of K, E is a unital C*-algebra (resp. a W*-algebra), 1E is its unit, denotes the set E endowed with its canonical structure of a Hilbert right E-module ([1] Proposition 5.6.1.5),
([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
is an isomorphism of C*-algebras with inverse We identify E with 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) 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
ℤ denotes the group of integers and for every n ∈ ℕ we put ℤn ≔ ℤ / (nℤ).2) For every set denotes the set of subsets of 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 AB denotes the set of maps of B in A.
5) If A, B are topological spaces then denotes the set of continuous maps of A into B. If A is locally compact space and E is a C*-algebra then (resp. ) 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 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 ei ≔ (δi,j)j ∈ I ∈ l2(I).
8) If E, F are vector spaces in duality then EF 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:
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 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**- …”.
12) If F is a C*-algebra then we denote for every n ∈ ℕ by Fn,n the C*-algebra of n × n matrices with entries in F. If T is finite then FT,T has a corresponding signification.
14) Let F be a W*-algebra and H, K Hilbert right F-modules. We put for and (ξ, η) ∈ H × K,
and denote by the closed vector subspace of the dual H′ of H generated by and by the closed vector subspace of generated by If H is selfdual then is the predual of ([1] Theorem 5.6.3.5 b)) and 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 . If G is a W*-algebra a C*-homomorphism is called a W*-homomorphism if the map is continuous; in this case denotes the pretranspose of φ.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))
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
such that f(1, 1) = 1E and for all r,s,t ∈ T. We denote by the set of Schur E-functions for T and put for every .Schur functions are also called normalized factor set or multiplier or two-co-cycle (for T with values in Un Ec) 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 Ec can be replaced in this section by an arbitrary commutative multiplicative group (with * replaced by −1).
PROPOSITION 1.1.2 Let .
a) Putting s = 1 in the equation of f we obtain
so for all r, t ∈ T. HencePutting r = t and s = t−1 in the equation of f we get
By the above,
b) Putting r = s−1 in the equation of f, by a),
Putting now t = s−1 in the equation of f, by a) again,
DEFINITION 1.1.3 We put
and for every λ ∈ Λ (T, E).PROPOSITION 1.1.4
a) is a subgroup of the commutative multiplicative group (Un Ec)T×T such that f* is the inverse of f for every .
a) is obvious.
b) For r, s, t ∈ T,
so . For ,c) For r, s, t ∈ T,
so . For and s,t ∈ T, so δ is a group homomorphism. The other assertions are obvious.PROPOSITION 1.1.5 Let t ∈ T, m, n ∈ ℤ, and .
a) We may assume m ∈ ℕ because otherwise we can replace t by t−1. Put
From
it follows for all k ∈ ℤ.We prove the assertion by induction. P(m, 0) follows from Proposition 1.1.2 a). By the above
for all k ∈ ℤ. Thus Q(1) holds.Assume Q(p) holds for all p ∈ ℕm−1. Then P(m, p) holds for all p ∈ ℕm−1 ∪ {0}. Let. n ∈ ℤ. There is a k ∈ ℤ such that
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
we get by a),Thus the formula holds also for m + 1.
c) If m, n ∈ ℕ then by b),
If m, n ∈ ℕ, n ≤ m − 1 then by b),
If m, n ∈ ℕ, n ≥ m then by b), For all m, n ∈ ℕ put By the above and by Proposition 1.1.2 a), b), so R(1, 1) holds. Let now m, n ∈ ℕ and assume R(m, n) holds. Then 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
is a surjective group homomorphism with kernel 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
such that for all α, β ∈ E and x, y ∈ F, If F, G are E-modules then a C*-homomorphism φ : F ⟶ G is called E-linear if for all (α, x) ∈ E × F, For all (α, x) ∈ E × F, so 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 Ec defined with respect to E coincides with Ec 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
shows that an E-C*-homomorphism need not be unital.If we put , and
and if we denote by λ the Lebesgue measure on then L∞(λ) is an E-C*-algebra, x ∈ Un E, and x is homotopic to 1E in Un L∞(λ) but not in .DEFINITION 1.2.3 We denote by ℭE (resp. by ) 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.
b)The maps
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 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 the category of adapted E-modules for which the morphism are the E-linear C*-homomorphisms.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
is an injective involutive algebra homomorphism, is closed, F is adapted, and for all α ∈ E and x ∈ F, In particular every closed ideal of an E-C*-algebra is adapted and ℭE is a full subcategory of .e) A closed ideal G of an adapted E-module F, which is at the same time an E-submodule of F, is adapted.
a) and b) are easy to see.
c) Since λ and ι are injective and,
we get the first and the last two inequalities as well as the identity ‖(0, x)‖ = ‖x‖. It followsd) It is easy to see that φ is an injective involutive algebra homomorphism. Let . There are sequences (αn)n ∈ ℕ and (xn)n ∈ ℕ in E and F, respectively, such that
it followsThus is closed, which proves the assertion by pulling back the norm of E × G.
e) By c), F is a closed ideal of so G is a closed ideal of (use an approximate unit of F). Since G is an E-submodule of F its structure of E-module is inherited from . By d), G is adapted.
f) The map
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
for all (α, x) ∈ E × F, endowed with this norm is complete. For (α, x) ∈ E × F,For ,
soSince the map maps into itself and
we have by the above,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) and b) are easy to see.
c) If denotes the unitization of G then by b), 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
b) Let F1, F2, F3 be E-modules and let φ : F1 ⟶ F2 and ψ : F2 ⟶ F3 be E-linear C*-homomorphisms. Then .
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 denotes the quotient map (where F is identified to {(0, x) |x ∈ F}) then there is an E-C*-isomorphism such that .
a) is easy to see.
b) Let and let . 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 with respect to θ we see that G/F is adapted.
LEMMA 1.2.8 Let {(Fi)i∈ I, (φij)i,j ∈ I)} be an inductive system in the category of C*-algebras, {F,(φi)i ∈ I} its inductive limit, G a C*-algebra, for every i ∈ I, ψi : Fi ⟶ G a C*-homomorphism such that ψj ο φji = ψi for all i,j ∈ I, i ≤ j, and ψ : F ⟶ G the resulting C*-homomorphism. If Ker ψi ⊂ Ker φi for every i ∈ I then ψ is injective.
Let i ∈ I. Since Ker φi ⊂ Ker ψi is obvious, we have Ker φi = Ker ψi. Let ρ : Fi ⟶ Fi/Ker ψi be the quotient map and
the injective C*-homomorphisms withThen
For x ∈ Fi, since ψi′ and φi′ are norm-preserving,
Thus ψ preserves the norms on ∪i∈ Iφi(Fi). Since this set is dense in F, ψ is injective.
PROPOSITION 1.2.9 Let {(Fi)i ∈ I,(φij)i,j ∈ I} be an inductive system in the category and let (F,(φi)i ∈ I) be its inductive limit in the category of E-modules (Proposition 1.2.4 c)).
a) F is adapted.
b) Let (G,(ψi)i ∈ I) be the inductive limit in the category of the inductive system (Proposition 1.2.6 a),b)) and let be the unital C*-homomorphism such that for every i ∈ I. Then ψ is an E-C*-isomorphism.
a) Put
F0 is an involutive unital subalgebra of . p is a norm and by Proposition 1.2.4 c), exists and for every .Let (α, x) ∈ F0. Let further i ∈ I, xi, yi ∈ Fi with φixi = x, φiyi = α*x + x*α + x*x. Then
soFor ε > 0 there is a j ∈ I, i ≤ j, with
We get
By taking the infimum on the right side it follows, since ε is arbitrary,
and this shows that p is a C*-norm. It is easy to see that q is a C*-norms. By the above inequalities, endowed with the norm q is complete, i.e. is a C*-algebra and F is adapted.b) Let i ∈ I and let . Then
soBy Lemma 1.2.8, ψ is injective.
Let and let ε > 0. There are i ∈ I and x ∈ Fi with ‖φix − y‖ < ε. Then
Thus ψ(G) is dense in and ψ is surjective. Hence ψ is a C*-isomorphism.
COROLLARY 1.2.10 We put for every E-module F and similarly for every E-linear C*-homomorphism φ.
a) ΦE is a covariant functor from the category in the category .
b) The categories and possess inductive limits and the functor ΦE 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 ℭE does not possess inductive limits in general. This happens for instance if φij = 0 for all i, j ∈ I.
1.3. Some topologies
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
If is a Hausdorff topology on then for every , denotes the set endowed with the relative topology and denotes the closure of in . Moreover denotes the sum with respect to .LEMMA 1.3.2 For x ∈ E, by the above identification of E with ,
is well-defined and belongs to .b) Assume E is a W*-algebra. Then for every , the family is summable in and for every x ∈ E,
Thus φ is a W*-homomorphism ([1] Theorem 5.6.3.5 d)) with where denotes the pretranspose of φ.c) If we consider E as a canonical unital C**-subalgebra of by using the embedding of a) then is an E-C**-algebra.
a) follows from [6] page 37 (resp. [3] Proposition 1.4).
b) We have
Thus the family is summable in and
If denotes the transpose of φ then
By continuity and φ is a unital W*-homomorphism.
c) Let x ∈ Ec and . By [1] Proposition 5.6.3.17 d),
Thus for ,
and so .DEFINITION 1.3.3 We put for all ξ, η ∈ H (resp. and )
and denote, respectively, by the topologies on generated by the set of seminormsMoreover ‖⋅‖ denotes the norm topology on .
Of course . In the C*-case, 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 and ξ, η ∈ H (resp. and ).
a) From
it followsb) .
c) We have
By Schwarz’ inequality ([1] Proposition 2.3.3.9),
sod) The first equation follows from
and the second fromLEMMA 1.3.5 Let n ∈ ℕ and a family in E. Then
We prove the relation by induction with respect to n. By [1] Corollary 4.2.2.4 and by the hypothesis of the induction,
LEMMA 1.3.6 Let n ∈ ℕ, x ∈ En,n, and for every j ∈ ℕn put
ThenFor , by Lemma 1.3.5,
For i, j ∈ ℕn,
soCOROLLARY 1.3.7
b) norm topology.
e) is complete in the C*-case.
f) If T is finite then is the norm topology in the C*-case.
g) is dense in .
a) follows from Proposition 1.3.4 a).
b) follows from Proposition 1.3.4 b),c). 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 be a Cauchy filter on . Put
where the limits are considered in the norm topology of H. For ξ, η ∈ H, so and Z = Y *. Thus converges to Y in and is complete.f) follows from b) and Lemma 1.3.6.
g) Let and ξ ∈ H. For every put
and let be the upper section filter or . Then for every and in H (resp. in ) ([1] Proposition 5.6.4.1 e) (resp. [1] Proposition 5.6.4.6 c))). Thus with respect to the topology . Since the same holds for X*, it follows that X belongs to the closure of in .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 ⇒ 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
a) By Lemma 1.3.8 a ⇒ b, F# is dense in so is dense in and is the extension of 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 .
b) By a) and [3] Proposition 1.4 e), the map
is an injective C*-homomorphism. By [3] Proposition 1.9 b), its image is dense in .c) Denote by N the vector subspace of generated by
By a) and [3] Proposition 1.9 a), N is dense in so by Corollary 1.3.7 c),
For and , by Proposition 1.3.4 c),
where a = x|a| is the polar representation of a, so the map is continuous.LEMMA 1.3.10 Let , and
Then ‖x‖ = ‖ξ‖.For and i ∈ ℕn,
On the other hand if we put then for i ∈ ℕn, LEMMA 1.3.11 Let F,G be unital C**-algebras, φ : F ⟶ G a surjective C**-homomorphism, I a set, and for every ξ ∈ L put .a) For ,
It follows . Moreover for i ∈ I,
b)
Case 1 {i ∈ I | ηi ≠ 0} is finite
For simplicity we assume for some n ∈ ℕ. We put
θ is obviously a surjective C*-homomorphism. So if we put then there is an x ∈ Fn,n with θ x = y, ‖x‖ = ‖y‖ ([5] Theorem 10.1.7). If we put and then and by [1] Theorem 5.6.6.1 a), ‖z‖ ≤ ‖x‖. We get for i ∈ ℕn,By a) and Lemma 1.3.10,
Case 2 η arbitrary in the W*-case
We may assume ‖η‖ = 1. We put for every ,
By Case 1, for every there is a ξJ ∈ L with and ‖ξJ‖ = ‖ηJ‖ ≤ 1. Let be an ultrafilter on finer than the upper section filter of . By [1] Proposition 5.6.3.3 a ⇒ b,
exists in . For i ∈ I, so . By a), , so ‖ξ‖ = ‖η‖.Case 3 η arbitrary in the C*-case
We put for every and every ζ ∈ M,
Moreover we denote by the upper section filter of , set
and denote by the vector subspace of generated by the setLet be the vector subspace of generated by the set
is an involutive subalgebra of . Let (αq)q ∈ Q, (βq)q ∈ Q be finite families in L such thatLet further α ′, β ′ ∈ M0. By Case 1, there are α, β ∈ L with and we get by a),
It follows ([1] Proposition 5.6.4.1 e))
Thus the linear map
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
Let
and let . By Case 1, there is an α ∈ L# with . By a),Since M0 is dense in M ([1] Proposition 5.6.4.1 e)), it follows
Step 2 is dense in
Let α, β ∈ M. By [1] Proposition 5.6.4.1 e),
so by [1] Proposition 5.6.5.2 a), which proves the assertion.Step 3 ψ is a surjective C*-homomorphism
By Step 1, ψ is a C*-homomorphism. Since its image contains (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 with
From we see that we may assume ThenIf we put ξ ≔ u(1F ⊗ ej) ∈ L then u = ξ 〈 ⋅ |1F ⊗ ej,〉 ‖η‖ = ‖u‖ = ‖ξ‖,
c) Let . By b), there is a ξ0 ∈ L with . By a), for ξ ∈ L,
We put
and denote by θ ′ : M′ ⟶ L′ its transpose. By the above, . Since θ ′ is continuous, and this proves the assertion.PROPOSITION 1.3.12 We use the notation of Lemma 1.3.11.
a) For i ∈ I, so by Lemma 1.3.11 a),
By [1] Proposition 5.6.4.1 e) (resp. [1] Proposition 5.6.4.6 c) and [1] Proposition 5.6.3.4 c)),
so by Lemma 1.3.11 a) (resp. c)),
b) For and ξ, η ∈ L, by Lemma 1.3.11 a),
By Lemma 1.3.11 b), , , and , i.e. the map is a C*-homomorphism.
For and ξ, η ∈ L (resp. and ), by Lemma 1.3.11 a),
so by Lemma 1.3.11 b), the map is continuous with respect to the topology . The proof for the other topologies is similar.c) For ζ ∈ L, by Lemma 1.3.11 a),
so by Lemma 1.3.11 b), The last assertion follows now from b).SCHUR PRODUCTS
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,
If we want to emphasize the role of f then we put instead of Vt. For x ∈ E,
PROPOSITION 2.1.2 Let s, t ∈ T, x ∈ E, , and ξ ∈ H.
a) Vtξ ∈ H.
b) .
c) Vt(ζ ⊗ es) = (f(t, s)ζ) ⊗ ets.
d)
e)
f)
g) If T is infinite and denotes the filter on T of cofinite subsets, i.e.
a) For ,
so Vtξ ∈ H.b) For r ∈ T,
soc) For r ∈ T,
sod) We have
soe) For η ∈ H, by Proposition 1.1.2 a),b),
so with . By b) and d),f) follows from c).
g) Let us consider first the C*-case. Let ξ, η ∈ H, t ∈ T, and ε > 0. There is an such that ║ηeT\S║ < ε. By e),
soFrom
it followsThe W*-case can be proved similarly.
Remark. By e), cannot be replaced by in g).
PROPOSITION 2.1.3 Let s, t ∈ T.
a) For and ξ ∈ H,
sob) For , by a),
soc) For and r ∈ T, by a),
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))) .d) For ξ ∈ H (resp. and ) and , by c),
and the assertion follows.PROPOSITION 2.1.4 Let s, t ∈ T and x ∈ E.
a) For , by Proposition 2.1.2 c),
so Vs ut = ust f(s, t).b) For and r ∈ T, by Proposition 2.1.2 c) and Proposition 2.1.3 a),
soc) For ,
so .d) follows from c).
DEFINITION 2.1.5 We put for all s, t ∈ T (Proposition 2.1.3 a))
and set Xt ≔ φt, 1 X for every .PROPOSITION 2.1.6 Let s, t ∈ T.
a) follows from Proposition 2.1.3 a),b).
b) We have
c)
The C*-case
By b), for ,
The W*-caseLet and let a = x|a| be its polar representation. By b), for ,
d) For ,
so φt, t is involutive. For n ∈ ℕ, , and , ([1] Theorem 5.6.6.1 f) and [1] Theorem 5.6.1.11 c1 ⇒ c2) so φt, 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),
f) By e) (and Proposition 1.1.2 a)),
g) By Proposition 2.1.4 c),d),
DEFINITION 2.1.7 We put
and call the Schur product associated to f. Moreover we put in the C*-case and in the W*-case. If F is a subset of E then we put and use similar notation for the other .By Proposition 2.1.2 b),d),e), is an involutive unital E-subalgebra of (with V1 as unit). In particular is an E-C*-subalgebra of . If T is finite then . By Corollary 1.3.7 e), is complete.
PROPOSITION 2.1.8 For and s, t ∈ T,
Let be a filter on converging to X in the -topology. By Proposition 2.1.6 c),f) (and Corollary 1.3.7 d)),
THEOREM 2.1.9 Let .
d)
i) If E is a W*-algebra then may be identified canonically with a unital C*-subalgebra of by using the map of Proposition 1.3.9. b). By this identification generates as W*-algebra.
j) If F is a closed ideal of E (resp. of ) then is a closed ideal of (resp. of ).
l) .
a) By Proposition 2.1.6 c),e),
b&c&dBy Proposition 2.1.3 d), Corollary 1.3.7 d), Proposition 2.1.8, and Proposition 2.1.4 b),d),
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)),
By a),
By Step 1 and Proposition 2.1.2 e) (and Proposition 1.1.2 a)),
Together with Step 1 this proves
In particular
e) By b) and Corollary 1.3.7 b), in the C*-case,
The proof is similar in the W*-case.
f) For ξ ∈ H and s ∈ T, by e),
g) By b), Corollary 1.3.7 b),d), and Proposition 2.1.2 b),d),
Since by d),
for every s ∈ T we get , again by d). By Corollary 1.3.7 b) and Proposition 2.1.6 c),e),By the above, c), and Proposition 1.1.2 b),
It follows by Proposition 1.1.2 a),
h) By c) and g), is an involutive unital subalgebra of . Being closed (resp. closed in (d) and Corollary 1.3.7 c))) it is a C**-subalgebra of (resp. generated by [1] Theorem 5.6.3.5 b) and [1] Corollary 4.4.4.12 a) and by [1] Corollary 6.3.8.7 is dense in , 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), is a closed set of .
i) The assertion follows from h), Proposition 1.3.9 b), and Lemma 1.3.8 c) ⇒ a).
j) For , , and t ∈ T, by g), so is an ideal of . The closure properties follow from Proposition 2.1.6 c).
k) By c) and g), is a unital involutive subalgebra of and by Proposition 2.1.6 c), is a C**-subalgebra of . The last assertion follows from the fact that the image of the map contains .
l) There are with
For t ∈ T,
By g),
soRemark. It may happen that by the identification of i), (Remark of Proposition 2.1.23).
COROLLARY 2.1.10
a) For , by Proposition 2.1.2 e),
and the assertion follows.b) By Theorem 2.1.9 c), and
for all t ∈ T soBy Theorem 2.1.9 g), and
for every t ∈ T soc) is easy to see.
Remark. There may exist for which is not norm summable, as it is known from the theory of trigonometric series (see Proposition 819). In particular the inclusion 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 φ1,1 is a positive continuous (resp. W*-continuous) unital trace.
b) If τ is faithful then τ ο φ1,1 is faithful and V1 is finite.
c) In the W*-case, SW(f) is finite iff E is finite.
a) Let . By Theorem 2.1.9 g) (and Proposition 1.1.2 a)),
Thus τ ο φ1,1 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), φ1,1 is faithful, so τ ο φ is also faithful. Let with X*X = V1. By a),
so and V1 is finite.c) By b), if E is finite then is also finite. The reverse implication follows from the fact that E ⊗̅ 1K is a unital W*-subalgebra of (Theorem 2.1.9 h)).
COROLLARY 2.1.12 Assume T finite and for every x′ ∈ (E′)T put
a) and
for every x′ ∈ (E′)T and the map is an isomorphism of involutive vector spaces such that ([1] Proposition 2.2.7.2) for every x ∈ E and x′ ∈ (E′)T.b) If E is a W*-algebra then the map
COROLLARY 2.1.13 Assume T finite and let M be a Hilbert right –module. M endowed with the right multiplication
and with the inner-product is a Hilbert right E-module denoted by , is a unital C*-subalgebra of , and M is selfdual if is so.By Proposition 2.1.6 d),g) and Theorem 2.1.9 g),l), for and x ∈ E,
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 be an E-C*-homomorphism. Then (φ Vt)i, j ∈ Ec for all t ∈ T and all i, j ∈ ℕn.
For x ∈ E, by Proposition 2.1.2 d) and Theorem 2.1.9 h),
so (φ Vt)i, j ∈ Ec.COROLLARY 2.1.15 Let S be a group and . If we put
then .The assertion follows from Theorem 2.1.9 h).
COROLLARY 2.1.16 Let .
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),
Then for every α ∈ l∞(T, E) and the map is continuous and W*-continuous.
a) In the C*-case the family ((Xs ⊗ 1K)Vs)s ∈ S is summable since 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
is well-defined, linear, and continuous. The assertion follows by continuity.c) Let x ∈ E, S ⊂ T, and α ≔ xeS. For ξ, η ∈ H and , by a) and Lemma 1.3.2 b) (and Theorem 2.1.9 b)),
Let G be the involutive subalgebra {α ∈ l∞(T, E)|α(T) is finite} of l∞(T, E) and let 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 , where F ≔ l∞(T, E).
Let α ∈ l∞(T, E)# and let be a filter on G# converging to α in ([1] Corollary 6.3.8.7). By the above (and by Theorem 2.1.9 h)),
in and so . The assertion follows.COROLLARY 2.1.17 Let S be a subgroup of T. Put
a) is obvious.
b) By Theorem 2.1.9 c),g), is an involutive unital subalgebra of 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 .
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 𝔖T the set of finite subgroups of T and call T locally finite if 𝔖T is upward directed and
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 fS ≔ f | (S × S) for every and identify with (Corollary 2.1.17 e)).
b) is the norm closure of and so it is canonically isomorphic to the inductive limit of the inductive system and for every the inclusion map is the associated canonical morphism.
a) There is a with . Let with . By Corollary 2.1.17 b), for with S ⊂ R,
sob) follows from a).
Remark. The C*-algebras of the form 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.
By Proposition 2.1.2 b),d),e),
soThus a) is equivalent to and , which is equivalent to b).
COROLLARY 2.1.21 Let t ∈ T such that t2 = 1 and . Then
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,
anda) 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),
d1 ⇔ d2 is known.d1 ⇔ d3. By a),
By Proposition 2.1.20 d1) is equivalent to so, by the above, since α*βf(s, st)* − β*αf(t, st)* is normal, it is equivalent to
d3 ⇔ d4 follows from b).PROPOSITION 2.1.23 Let .
a) follows from Theorem 2.1.9 g).
b) By a),
and by Proposition 2.1.6 a),c) Let n ≔ Card T and for every t ∈ T put Xt ≔ 1E, ξt ≔ 1E. Then
For t ∈ T, by Theorem 2.1.9 e),
sod) 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 (xt)t ∈ T in E such that the family is summable in in the W*-case but not in the C*-case as the following example shows. Take , f constant, , and xt ≔ (δt, s)s ∈ T∈E for every t ∈ T. By Proposition 2.1.23 b), ((xt ⊗ 1K)Vt)t ∈ T is not summable in in the C*-case. In the W*-case for ξ ∈ H and s, t ∈ T,
Thus
Using the identification of Theorem 2.1.9 i), we get .
COROLLARY 2.1.24 Let .
a) iff Xt ∈ Ec for all t ∈ T.
d) .
e) If the conjugacy class of t ∈ T (i.e. the set {s−1ts|s ∈ T}) is infinite and X ∈{Vt|t ∈ T}c then Xt = 0.
For s, t ∈ T, x ∈ E, and , by Theorem 2.1.9 g),
a) follows from the above by putting s ≔ 1 (Proposition 1.1.2 a)).
b) follows from the above by putting x ≔ 1E 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),
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 ⇒ b. By Corollary 2.1.24 g1 ⇒ g2, T is commutative, E = Ec, and f(s, t) = f(t, s) for all s, t ∈ T. Since E is isomorphic with a C*-subalgebra of (Theorem 2.1.9 h)), E = Re E. By Proposition 2.1.2 e),
so by Theorem 2.1.9 a), , so t2 = 1.b ⇒ a. By Corollary 2.1.24 g2 ⇒ g1, . For and t ∈ T, by Theorem 2.1.9 c),
so X* = X (Theorem 2.1.9 a)).PROPOSITION 2.1.26 Let (Ei)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 Ei with the corresponding closed ideal of E (resp. of ) and put
c) In the C*-case, if I is finite then (resp. ) is isomorphic to (resp. ).
d) In the W*-case, is isomorphic to the C*-direct product of the family .
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, Ei ≔ 𝕂 for every i ∈ I, and E ≔ l∞ (i.e. E is the C*-direct product of the family (Ei)i ∈ I). For every n ∈ ℕ put . Assume there is an (resp. ) with (resp. ), where ψ and φ are the maps of b) and c), respectively. Then for all i, n ∈ ℕ and this implies , 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 such that g(s1, s2)∈Un Ec (where Un Ec is identified with ) for all s1, s2∈S and put
a) is obvious.
b) For and (s, t)∈S × T, by Theorem 2.1.9 c),g) and Proposition 2.1.6 g),
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 . For every s ∈ S putThen φ X = Z and φ is surjective.
PROPOSITION 2.1.28 If T is infinite and then X(H#) is not precompact.
Let t ∈ T with Xt ≠ 0. There is an (resp. ) with . We put t1 ≔ 1 and construct a sequence (tn)n ∈ ℕ recursively in T such that for all m, n ∈ ℕ, m < n,
Let n ∈ ℕ \ {1} and assume the sequence was constructed up to n − 1. Since (Proposition 2.1.23 a))
for all m ∈ ℕn − 1 there is a tn ∈ T with for all m ∈ ℕn − 1. By Schwarz’ inequality ([1] Proposition 2.3.4.6 c)) for m ∈ ℕn − 1,This finishes the recursive construction.
For r, s ∈ T, by Theorem 2.1.9 e),
For m, n ∈ ℕ, m < n, it follows
Thus the sequence 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,
Then and we define for every X ∈ A and Y ∈ B,
Then A (resp. B) is a unital C*-subalgebra of (resp. of ) are C*-isomorphisms, and φ = ψ−1.It is easy to see that A (resp. B) is a unital C*-subalgebra of (resp. of ) 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),
so φ is a C*-homomorphism and we haveMoreover for Y ∈ B,
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)
It is easy to see that Uλ is well-defined, , and the map is an injective group homomorphism with (Proposition 1.1.4 c)). Moreover for all λ, μ ∈ Λ (T, E).PROPOSITION 2.2.2 Let and λ ∈ Λ (T, E).
By Proposition 1.1.4 c), for every λ ∈ Λ (T, E).
a1 ⇒ a2& b. For s, t ∈ T and , by Proposition 2.1.2 c),
so (by Proposition 2.1.2 e)) Thus the map 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
so (φX)t = λ(t)*Xt.a2 ⇒ a1. Put h ≔ fδλ. By the above, for t ∈ T,
so and this implies g = h.c) follows from a).
Remark. Not every E-C*-isomorphism is an isomorphism (see Remark of Proposition 3.2.3).
COROLLARY 2.2.3 Let
and for every λ ∈ Λ0(T, E) putThen the map λ ⟼ φλ is an injective group homomorphism.
By Proposition 1.1.4 c), Λ0(T, E) is the kernel of the map
so by Proposition 2.2.2, φλ is well-defined. Thus only the injectivity of the map has to be proved. For t ∈ T and , by Proposition 2.1.2 c),So if φλ is the identity map then λ(t) = 1E for every t ∈ T.
PROPOSITION 2.2.4 Let F be a unital C**-algebra, φ : E ⟶ F a surjective C**-homomorphism, , and . We put for all ξ ∈ H, η ∈ L, and ,
where ζ ∈ H with (Lemma 1.3.11 a),b) and Proposition 1.3.12 a)). Then for every and the map is a surjective C**-homomorphism, continuous with respect to the topologies , k∈{1, 2, 3} such thatFor s, t ∈ T and ξ ∈ H,
so by Lemma 1.3.11 b),By Theorem 2.1.9 b),
so by the above and by Proposition 1.3.12 b),By Proposition 1.3.12 b), is a surjective C**-homomorphism, continuous with respect to the topologies . 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 Ec) ⊂ Fc, , and . Then the map
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
is a C*-homomorphism and by Theorem 2.1.9 k), the corresponding map is also a C*-homomorphism. The assertion follows from .PROPOSITION 2.2.6 Let T′ be a group, K′ ≔ l2(T′), , ψ : T ⟶ T′ a surjective group homomorphism such that
and such that f′ ο (ψ × ψ) = f. If we put for every and t′ ∈ T′ then the family is summable in for every and the map is a surjective E-C**-homomorphism.We may drop the hypothesis that ψ is surjective if we replace by .
Let . By Corollary 2.1.16 a), since ψ is surjective and
it follows that the family is summable in and therefore .Let . By Theorem 2.1.9 c),g), for t′ ∈ T′,
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 and in an obvious way then .
For and t′∈T′,
soPROPOSITION 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
Moreover we denote for all s, t ∈ T by , and the corresponding operators associated with F . Let such that for every ξ ∈ HF and put where ξ′ ∈ HF with , and (by the canonical identification of F with ) for every t ∈ T.a&b&c are easy to see.
d) By a) and Proposition 2.1.6 b),
In particular
and by Proposition 2.1.8,e) By c) and Proposition 2.1.3 d), for ξ ∈ HF,
By d) and Proposition 2.1.4 b),d),
by Proposition 2.1.3 d), again. Thusf) For s, t ∈ T, by d),
so by b) and e),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 for every ξ ∈ H, where
a) follows from the density of φ(E) in (Lemma 1.3.8 a ⇒ c).
b) For x ∈ E, t ∈ T, and ξ ∈ H,
soLet now . By Theorem 2.1.9 b),
so by the above and by Proposition 1.3.9 c) (and Theorem 2.1.9 d)), so . By Theorem 2.1.9 a), for every t ∈ T.Since φ (E) is dense in (Lemma 1.3.8 a) ⇒ c)) it follows that
so is dense in and therefore generates 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, , and
a) By [1] Corollary 6.3.8.7, there is a filter on converging to z in . By Lemma 1.3.2 b), for ,
which proves the assertion.b) Let (ai, bi)i ∈ I be a finite family in . For x ∈ E,
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 . We denote by ⊗σ the spatial tensor product and put
c) In the C*-case, and .
d) In the W*-case, if z ∈ G# and (t, s)∈T × S then belongs to the closure of in
e) In the W*-case, .
a) is obvious.
Let us treat the C*-case first. For ξ, ξ′ ∈ H and η, η′ ∈ L,
so the linear map 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 (xi, yi)i ∈ I in E × F such that
Then so . It follows that φ is surjective and so H ⊗ L ≈ M.The proof for the inclusion can be found in [6] page 37.
Let us now discus the W*-case. follows from [2] Proposition 1.3 e), follows from [3] Corollary 2.2, and follows from [2] Theorem 2.4 d) or [3] Theorem 2.4.
b) For t1, t2∈T, s1, s2∈S, , and , by Proposition 2.1.2 f) and [3] Corollary 2.11,
We put
By the above, u(ζ ⊗ er) = 0 for all and r ∈ T × S.
Let us consider the C*-case first. Since is dense in , we get u(z ⊗ er) = 0 for all and r ∈ T × S. For ζ ∈ M, by [1] Proposition 5.6.4.1 e),
which proves the assertion in this case.Let us consider now the W*-case. Let z ∈ G# and r ∈ T × S and let be a filter on converging to z in ([1] Corollary 6.3.8.7). For η ∈ M, , and r ∈ T × S,
in . Since is continuous ([1] Proposition 5.6.3.4 c)), we get by the above u(z ⊗ er) = 0. For ζ ∈ M it follows by [1] Proposition 5.6.4.6 c), which proves the assertion in the W*-case.c) By b), so by a),
Let z ∈ G#, (t, s)∈T × S, and ε > 0. There is a finite family (xi, yi)i ∈ I in E × F such that
By b),
and so by a),d) By a) and Lemma 2.2.10 a), there is a filter on
converging to in . For ξ, η ∈ M and , which proves the assertion.e) By Theorem 2.1.9 h),
By b), , so by Lemma 2.2.10 b),
By [3] Proposition 2.5,
For x ∈ E, y ∈ F, and (t, s)∈T × S, by b),
Let z ∈ G#. By d), there is a filter on
converging to in , so by the aboveWe get
COROLLARY 2.2.12 Let n ∈ ℕ anda) Take F ≔ 𝕂n, n and S ≔ {1} in Proposition 2.2.11. Then G ≈ En, n and
By Proposition 2.2.11 c),e),
b) By Theorem 2.1.9 b),
so by Theorem 2.1.9 a).COROLLARY 2.2.13 Let n ∈ ℕ. If 𝕂 = ℂ (resp. if n = 4m for some m ∈ ℕ) then there is an (resp. ) such that
By [1] Proposition 7.1.4.9 b),d) (resp. [1] Theorem 7.2.2.7 i),k)) there is a (resp. ) such that If we put then by Proposition 2.2.11 a),e), (resp. ) and COROLLARY 2.2.14 Let F be a unital C**-algebra, , and Then and COROLLARY 2.2.15 If E is a W*-algebra then the following are equivalent:a ⇒ b. Assume first that there are a finite W*-algebra F and a Hilbert space L such that . Put
By Corollary 2.2.14, By Corollary 2.1.11 c), is finite and so 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 (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 and put L ≔ l2(S), M ≔ l2(S × T), and
Then and the map is an -C*-isomorphism.For , , and (s, t)∈S × T, by Theorem 2.1.9 c), g),
so and φ is an -C*-homomorphism.If with φ X = 0 then for (s, t)∈S × T,
so φ is injective.Let x ∈ E and (s, t)∈S × T. Put
Then for (r, u)∈S × T, so and φ is surjective.PROPOSITION 2.2.17 Let S be a finite subgroup of T and g ≔ f|(S × S). We identify with the E-C**-subalgebra of (Corollary 2.1.17 e)). Let , P+ ≔ X*X, and P− ≔ XX* and assume .
a) follows from the hypothesis on X.
b) follows from a).
c) Let with Z = P+Y P+. By the hypotheses of the Proposition,
and ψ is a C*-homomorphism. The other assertions follow fromd) By b) and c),
e) follows from b), c), and Lemma 1.3.2.
f) follows from Corollary 2.1.17 d).
Remark. Even if φ± is injective is not an E-C*-subalgebra of .
THEOREM 2.2.18 Let S be a finite subgroup of T, L ≔ l2(S), g ≔ f|(S × S), an injective group homomorphism such that ,
β1, β2∈Un Ec such that , We assume f(s, c) = f(c, s) and cs = sc for every s ∈ S, and f(a, b) = −f(b, a) = 1E. Moreover we consider as an E-C**-subalgebra of (Corollary 2.1.17 e)).d) The maps
are orthogonal injective E-C**-homomorphisms and φ+ + φ− is an injective E-C*-homomorphism. If (resp. ) then (resp. ). Moreover the map is an E-C**-isomorphism such that If 𝕂 = ℂ then X + X* is homotopic to in and ψ is homotopic to the identity map of . Using this homotopy we find that φ+Y is homotopic in the above sense to φ−Y for every and φ+Y1 + φ−Y2, φ−Y1 + φ+Y2, φ+(Y1Y2) + P−, and φ+(Y2Y1 + P− are homotopic in the above sense for all .j) The above results still hold for an arbitrary subgroup S of T if we replace with .
a) By the equation of the Schur functions,
and soFor s ∈ S, by Proposition 2.1.2 b),
and so (by Proposition 2.1.2 d)).b) By Proposition 2.1.2 b),d),e) (and Corollary 2.1.22 c)),
By a), so, by a) again, . By Proposition 2.1.2 b),d),For the last relation we remark that by the above,
c) follows from b) and Lemma 1.3.2.
d) By b) and c), the map φ± is an E-C**-homomorphism. Let with φ±Y = 0. By b), so by Proposition 2.1.2 b),d) and Theorem 2.1.9 b),
which implies Ys = 0 for every s ∈ S (Theorem 2.1.9 a)). Thus φ± is injective. It follows that φ+ + φ− is also injective.Assume first . By b),
Similarly . The case is easy to see.
By b), ψ is an E-C**-isomorphism with
Moreover for ,
Assume now 𝕂 = ℂ. By b), . Being selfadjoint its spectrum is contained in {−1, +1} and so it is homotopic to in .
e) We have sb = sac = asc = acs = bs. By a),
soFrom
we getf) Since S and ω(ℤ2 × ℤ2) commute, the map is a group homomorphism. If s(ωr) = 1 for then , which implies s = 1 and r = (0, 0). Thus this group homomorphism is injective.
g) By e) and the hypothesis of f), for every t ∈ T there are uniquely s ∈ S and d∈{1, a, b, c} with t = sd. Let . By b) and Theorem 2.1.9 b) (and Corollary 1.3.7 d)),
By Proposition 2.1.2 b), and so by Theorem 2.1.9 a), By e), Zsa = Zsb = 0 for every s ∈ S. We get (by a), d), and Proposition 2.1.2 b)) Thus φ± is an E-C*-isomorphism with the mentioned inverse.h) is a long calculation using e).
i) follows from h).
j) follows from Corollary 2.1.17 d).
Remark. An example in which the above hypotheses are fulfilled is given in Theorem 4.1.7.
2.3. The functor
In this section we present the construction in the frame of category theory. Some of the results still hold for T locally finite.
DEFINITION 2.3.1 The above construction of can be done for an arbitrary E-module F, in which case we shall denote the result by . Moreover we shall write instead of in this case.
If F is an E-module then is canonically an E-module. If in addition F is adapted then is adapted and isomorphic to . If F is an E-C*-algebra then is also an E-C*-algebra.
PROPOSITION 2.3.2 If F, G are E-modules and φ : F ⟶ G is an E-linear C*-homomorphism then the map
is an E-linear C*-homomorphism, injective or surjective if φ is so.The assertion follows from Theorem 2.1.9 a),c),g).
COROLLARY 2.3.3 Let F1, F2, F3 be E-modules and let φ : F1 ⟶ F2, ψ : F2 ⟶ F3 be E-linear C*-homomorphisms.
a) is obvious.
b) Let . For every t ∈ T, Yt ∈ Ker ψ = Im φ . If we identify F1 with Im φ then Yt ∈ F1. It follows , .
c) follows from b) and Proposition 2.3.2.
COROLLARY 2.3.4 Let F be an adapted E-module and put
Then the sequence
is split exact.PROPOSITION 2.3.5 The covariant functor (resp. ) (Proposition 2.3.2, Corollary 2.3.3 a)) is continuous with respect to the inductive limits (Proposition 36 a),b)).
Let {(Fi)i ∈ I, (φ ij)i, j ∈ I} be an inductive system in the category (resp. ) and let {F, (φi)i ∈ I} be its limit in the category (resp. ). Then is an inductive system in the category (resp. ). Let {G, (ψi)i ∈ I} be its limit in this category and let be the E-linear C*-homomorphism such that for every i ∈ I. In the case, for α ∈ E and i ∈ I,
so that ψ is an E-C*-homomorphism.Let i ∈ I and let . Then φiXt = 0 for every t ∈ T. Since T is finite, for every ε > 0 there is a j ∈ I, j ≥ i, with
for every t ∈ T. ThenIt follows
By Lemma 1.2.8, ψ is injective. Since
Imψ is dense in . Thus ψ is surjective and so an E-C*-isomorphism.PROPOSITION 2.3.6 Let θ : F ⟶ G be a surjective morphism in the category . We use the notation of Theorem 2.2.18 and mark with an exponent if this notation is used with respect to F or to G. For every , there is a such that
By Proposition 2.3.2 c), is surjective and so there is a with ‖Z0‖ = 1 and . Put
By Theorem 2.2.18 b),
Since
We get
PROPOSITION 2.3.7 Let F be an adapted E-module and Ω a locally compact space. We define for (see Corollary 1.2.5 d)) and ,
Then are E-linear C*-isomorphisms and φ = ψ−1.Let ω0 ∈ Ω and assume F is an E-C*-algebra. Then the above maps φ and ψ induce the following E-C*-isomorphisms
Let and . By Proposition 2.1.23 b) and Corollary 2.1.10 a),
and it is easy to see that φ and ψ are E-linear. By Theorem 2.1.9 c),g), for t ∈ T and ω ∈ Ω, so and φ is a C*-homomorphism. Similarly so and ψ is a C*-homomorphism. Moreover so ψφ X = X and φ ψY = Y which proves the assertion.The last assertion is easy to see.
PROPOSITION 2.3.8 Let F be an adapted E-module,
the associated exact sequences (Proposition 1.2.4 h)), andThen φ is an injective E-C*-homomorphism and .
PROPOSITION 2.3.9 If E is commutative and F is an E-module then the map
is a surjective C*-homomorphism. If in addition E = 𝕂 then φ is a C*-isomorphism with inverseIt is obvious that φ is surjective. For and x, y ∈ F, by Theorem 2.1.9 c),g) and Proposition 2.1.2 b),d),e),
so φ is a C*-homomorphism.Assume now E = 𝕂 and let and x ∈ F. Then
which proves the last assertion (by using the first assertion).EXAMPLES
We draw the reader's attention to the fact that in additive groups the neutral element is denoted by 0 and not by 1.
3.1. T ≔ ℤ2
PROPOSITION 3.1.1
b)
d) If 𝕂 = ℂ and if A is a connected and simply connected compact space or a totally disconnected compact space then for every there is a with x = ey.
a) follows from Proposition 1.1.2 a) (and Proposition 1.1.4 a)).
b) follows from Definition 1.1.3.
c) For , by Theorem 2.1.9 c),g) (and Proposition 1.1.2 a)),
so i.e. φ is an E-C*-homomorphism. φ is obviously injective.Let (y, z) ∈ E × E. If we take with
then φX = (y, z), i.e. φ is surjective.d) is known.
e1) follows by using the spectrum of Ec.
e2) Put
For , by Theorem 2.1.9 c),g), so ψ is an E-C*-homomorphism. Thus by c), φ is an E-C*-homomorphism. The bijectivity of φ is easy to see.f) By Proposition 2.1.2 e), V1 is unitary so its spectrum is contained in { eiθ | θ ∈ ℝ }. For θ ∈ ℝ and ,
Thus X is the inverse of eiθV0 − V1 iff X0 = eiθX1 and eiθX0 − f11X1 = 1E, i.e. (e2iθ − f11)X1 = 1E. Therefore eiθV0 − V1 is invertible iff does not vanish on σ(Ec).COROLLARY 3.1.2 Assume 𝕂 ≔ ℝ and let S be a group, F a unital C*-algebra, , and
a) .
b) .
Put E ≔ ℝ in the above Proposition and define by f(1,1) = −1 (Proposition 3.1.1 a)). By this Proposition e2), . Thus by Proposition 2.2.11 c),e),
DEFINITION 3.1.3 We put
EXAMPLE 3.1.4 Let and with
If we put for every then the map is an isomorphism of C*-algebras (but not an E-C*-isomorphism).For , by Theorem 2.1.9 c),g),
so for z ∈ 𝕋, i.e. φ is a C*-homomorphism. If φX = 0 then for z ∈ 𝕋, so, successively, and φ is injective.Put
Let and take with Then so . Since is dense in E, and φ is surjective.DEFINITION 3.1.5 For every which does not take the value 0 we put
If A is a connected compact space and γ is a cycle in A (i.e. a continuous map of 𝕋 in A), which is homologous to 0 (or more generally, if a multiple of γ is homologous to 0), then for every we have w(x ∘ γ) = 0. If A is a compact space and such that w(x ∘ γ) = 0 for every cycle γ in A then there is a with x = ey.
EXAMPLE 3.1.6 Let , , and n ≔ w(f(1, 1)).
b) If n is odd then is isomorphic to E.
d) There is a complex unital C*-algebra E and a family in such that for distinct , .
Put
Since w(f(1, 1)α−n) = 0, there is a y ∈ Un E with w(y) = 0 and f(1, 1)α−n = y2.a) If we put then and f(1,1) = x2 and the assertion follows from Proposition 3.1.1 c).
b) We put . Then f(1, 1) = αx2. Take with g(1,1) = α and λ ∈ Λ (ℤ2, E) with (δλ)(1,1) = x2 (Proposition 3.1.1 a),b)). Then f = gδλ. By Example 3.1.4, is isomorphic to E and by Proposition 2.2.2 a1 ⇒ a2, is also isomorphic to E.
c) follows from Proposition 3.1.1 b) and Proposition 2.2.2 a),c).
d) Denote by E the C*-direct product of the sequence and for every β ∈ {0, 1}ℕ define by
By a) and b), for distinct β, γ ∈{0, 1}ℕ, (Proposition 2.1.26 a)).EXAMPLE 3.1.7 Let I, J be finite disjoint sets and for all i ∈ I ∪ J and j ∈ J put Ai ≔ Bj ≔ 𝕋. We define the compact spaces A and B in the following way. For A we take first the disjoint union of the spaces Ai for all i ∈ I ∪ J and identify then the points 1 ∈ Ai for all i ∈ I ∪ J. For B we take first the disjoint union of all the spaces Ai for all i ∈ I ∪ J and of the spaces Bj for all j ∈ J and identify first the points 1 ∈ Ai for all i ∈ I ∪ J and identify then also the points −1 ∈ Ai for all i ∈ I and 1 ∈ Bj for all j ∈ J.
Let and with
For every define by Then the map is an isomorphism of C*-algebras.Let . By Theorem 2.1.9 c),g),
For z ∈ Ai with i ∈ I, For z ∈ Aj or z ∈ Bj with j ∈ J, Thus φ is a C*-homomorphism. Assume . For z ∈ Ai with i ∈ I, so, successively, For z ∈ Aj with j ∈ J, so Thus φ is injective.Let such that for every i ∈ I there is a family with
for all z ∈ Ai. Define X0, X1 ∈ E in the following way. If z ∈ Ai with i ∈ I we put If z ∈ Aj with j ∈ J then we put z′ ≔ z ∈ Bj, It is easy to see that X0 and X1 are well defined. Then for all z ∈ Ai with i ∈ I and for all z ∈ Aj ∈ Bj with j ∈ J. Since the elements x of the above form are dense in , φ is surjective.EXAMPLE 3.1.8 Let and with
Then the maps are isomorphisms of C*-algebras.Remark. and are isomorphic but not E-C*-isomorphic.
EXAMPLE 3.1.9 Let and with
If we put for every then the map is an injective unital C*-homomorphism with In particular is isomorphic to E.Let . By Theorem 2.1.9 c),g),
so for (z1, z2) ∈ 𝕋2, i.e. φ is a unital C*-homomorphism. If then for (z1, z2) ∈ 𝕋2, so, successively, and φ is injective.The inclusion is obvious. Let , and
Define Then . Since the elements of the above form are dense in , .If we consider the equivalence relation ~ on 𝕋2 defined by
then the quotient space 𝕋2 / ∼ is homeomorphic to 𝕋2. Thus is isomorphic to E.EXAMPLE 3.1.10 Let .
a) For x ∈ Un E and z ∈ 𝕋, w(x(·, z)) and w(x(z, ·)) do not depend on z, where w denotes the winding number (Definition 3.1.5).
a) follows by continuity.
b) follows from a).
c) Let with
Then By b), there is an x ∈ Un E with f(1,1) = x2g(1,1). By Proposition 3.1.1 b) and Proposition 2.2.2 a1 ⇒ a2, .c1) Assume m even and put
If with then g(1,1) = y2h(1,1). By Proposition 3.1.1 b) and Proposition 2.2.2 a1 ⇒ a2, and by Example 3.1.8 a1 ⇒ a2, . Thus .c2) If we put
then g(1,1) = y2 and the assertion follows from Proposition 3.1.1 c).c3) We put
and take with then g(1,1) = y2h(1,1) so by Proposition 3.1.1 b) and Proposition 2.2.2 a1 ⇒ a2, . By Example 3.1.9 , so .Remark. In a similar way it is possible to show that for every n ∈ ℕ, is isomorphic to (ℤ2)n and
EXAMPLE 3.1.11 Let I, J, K be finite pairwise disjoint sets and for every i ∈ I ∪ J ∪ K and k ∈ K put Ai ≔ Bk ≔ 𝕋2. We define the compact spaces A and B in the following way. For A we take first the disjoint union of the spaces Ai with i ∈ I ∪ J ∪ K and then identify the points (1, 1) ∈ Ai for all i ∈ I ∪ J ∪ K. For B we take first the disjoint union of the spaces Ai with i ∈ I ∪ J ∪ K and of the spaces Bk with k ∈ K. Then we identify the points (1,1) ∈ Ai for all i ∈ I ∪ J ∪ K and then we identify for every j ∈ J the points (z1, z2) ∈ Aj with the points (−z1, −z2) ∈ Aj and finally we identify the points (−1, 1) ∈ Ai for all i ∈ I ∪ J with the points (1, 1) ∈ Bk for all k ∈ K.
Let and such that
We define for every a map by Then the map is an isomorphism of C*-algebras.The proof is similar to the proof of Example 3.1.7.
EXAMPLE 3.1.12 If n ∈ ℕ, , and then is isomorphic either to or to .
EXAMPLE 3.1.13 Assume , where A denotes Moebius's band (resp. Klein's bottle), i.e. the topological space obtained from [0,2π] × [−π, π] by identifying the points (0, α) and (2π, −α) for all α ∈ [−π, π] (resp. and the points (θ, −π) and (θ, π) for all θ ∈ [0, 2π]). We put B ≔ 𝕋 × [−𝜫, 𝜫] (resp. B ≔ 𝕋2) and
for every x ∈ E.a) is well-defined and belongs to for every x ∈ E.
c) Let x ∈ Un E. If w(x(·, 0)) = 0 (where w denotes the winding number) then there is a y ∈ E with ey = x.
d) Let x ∈ Un E and put n ≔ w(x(θ, 0)). Then there is a y ∈ E with ey = e−inθ x.
e) The group is isomorphic to ℤ2.
f) If w(f1,1(·,0)) is even (resp. odd) then is isomorphic to E × E (resp. to ).
a) For α ∈ [−π, π],
so is well-defined. Moreover and in the case of Klein's bottle i.e. .b) For and (θ, α) ∈ [0,2π] × [−π, π], by Theorem 2.1.9 c),g),
i.e. φ is a C*-homomorphism. If φX = 0 then for α ∈ [−π, π], so for θ ∈ [0, π], replacing θ by θ + π and α by −α in the second relation, It follows successively Thus φ is injective.Let . Put
For α ∈ [−π, π], so X0, X1 ∈ E. Moreover for (θ, α) ∈ [0,2π] × [−π, π], i.e. φ is surjective.c) If A is Moebius's band then the assertion is obvious so assume A is Klein's bottle. The winding numbers of
are equal by homotopy, but their sum is equal to 0. Thus these winding numbers are equal to 0. The paths θ and α on A generate the homotopy group of A. Thus the winding number of x on any path of A is 0 and the assertion follows.d) The winding number of
is 0 and the assertion follows from c).e) The assertion follows from d) and Proposition 3.1.1 b).
f) The assertion follows from b), d), Proposition 2.2.2 a1 ⇒ a2, and Proposition 3.1.1 c).
3.2. T ≔ ℤ2 × ℤ2
PROPOSITION 3.2.1 Let E be a unital C*-algebra and let a, b, c be the three elements of (ℤ2 × ℤ2) / {(0,0)}. Put
and for every ϱ ∈ A and σ ∈ (Un Ec)3 denote by fϱ and gσ the functions defined by the following tables:a) is a long calculation.
b) is easy to verify.
c1 ⇒ c2 By Proposition 2.2.2 a2 ⇒ a1 there is a λ ∈ Λ (ℤ2 × ℤ2, E) with . By b), there is a σ ≔ (x, y, z) ∈ (Un Ec)3 with . We get ε = ε′ and
It follows xyz = αα′* ββ′* γγ′* soc2 ⇒ c3 is trivial.
c3 ⇒ c2 If we put z ≔ xyγ*γ′ then
c2 ⇒ c1 follows from b) and Proposition 2.2.2 a1 ⇒ a2.
d) follows from Corollary 2.1.24 b).
e) follows from Corollary 2.1.24 c).
f) follows from Theorem 2.1.9 c),g).
g) follows from f).
COROLLARY 3.2.2 We use the notation of Proposition 3.2.1 and take .
The proof is a long calculation using Proposition 3.2.1 f).
Remarks. d) is contained in Proposition 3.2.3 c). An example with ε = 1E but different from a) is presented in Proposition 3.3.2.
PROPOSITION 3.2.3 We use the notation of Proposition 3.2.1 and take .
Remark. Take , . By c), and by Proposition 3.2.1 c1 ⇒ c2, implies the existence of an x ∈ Un Ec with x2 = γ′.
COROLLARY 3.2.4 We use the notation of Proposition 3.2.3 and take , α = 1, and β = γ = ε = −1. Let S be a group, F a unital C*-algebra, , and
By Proposition 3.2.3 c), , so by Proposition 2.2.11 c),e),
EXAMPLE 3.2.5 Let and .
Put
where w denotes the winding number. By Proposition 2.2.2 a1 ⇒ a2, we may assume α = zm, β = zn, γ = zp.a) If n + p is even then the assertion follows from Proposition 3.2.3 c). If n + p is odd then either m + p or m + n is even and the assertion follows again from Proposition 3.2.3 c).
b) follows from Proposition 2.2.2 a),c).
Remark. Assume 𝕂 ≔ ℝ and let E be the real C*-algebra ([1] Theorem 4.1.1.8 a)), ε = −1E,
and . Then by Corollary 3.2.2 c), .EXAMPLE 3.2.6 We put , γ ≔ 1E,
and (with the notation of Proposition 3.2.1) .a) By Proposition 3.2.1 d), is not commutative. Assume and let us use the notation of Proposition 3.2.3 a).
Assume {Aa ≠ 0} ∩ {Ab ≠ 0} ≠ ∅. By Proposition 3.2.3 a),
so Ba ≠ 0 and Bb ≠ 0 on this set. We put with . By Proposition 3.2.3 a), , , Since 2AaAb = −(BaCb + BbCa) there is a with which is a contradiction.The assertion follows from Step 1 by symmetry.
The assertion follows from Steps 1 and 2 and from |Aa| + |Ab| + |Ac| ≠ 0.
By Step 3 and by the symmetry, the sets {Aa ≠ 0}, {Ab ≠ 0}, and {Ac ≠ 0} are clopen and by |Aa| + |Ab| + |Ac| ≠ 0 their union is equal to 𝕋2. So there is exactly one of these sets equal to 𝕋2 which implies
and no one of these identities can hold.b) is a direct verification.
c) follows from a) and b).
3.3. with n ∈ ℕ
EXAMPLE 3.3.1 Assume f constant and put
for all s,t ∈ T (where is identified with {0,1}) and for all t ∈ T. Then the map is an E-C*-isomorphism.For r,s,t ∈ T,
For t ∈ T and , by Theorem 2.1.9 c),g), so φt and φ are E-C*-homomorphisms.We have
We want to prove for all s ∈ T, s ≠ 0, by induction with respect to . Let with s(i) ≠ 0 and put r ≔ s + ei, Then But if r = 0. By the hypothesis of the induction if r ≠ 0 (with replaced by , since r(i) = 0). This finishes the proof by induction.For r ∈ T and , by the above,
Hence φ is bijective.EXAMPLE 3.3.2 Let , denote by z ≔ (z1, z2, ⋯, zn) the points of , and put for every . We identify with by using the bijection
and denote by the addition on corresponding to this identification. We put for every and Then and, if we put for every , the map is an isomorphism of C*-algebras.Let . By Theorem 2.1.9 c),g),
so φ is a C*-homomorphism.We put for , , and ,
and Let and for every put Then φX = x and so . Since is dense in E, it follows that φ is surjective.We prove that φ is injective by induction with respect to n ∈ ℕ. The case n = 1 was proved in Example 3.1.4. Assume the assertion holds for n − 1. Let X ∈ Ker φ. Then
By replacing zn by −zn in the above relation, we get and so By the induction hypothesis, we get XI = 0 for all and so X = 0. Thus φ is injective and a C*-isomorphism.EXAMPLE 3.3.3 Let , put
and denote by g the element of defined by g(1, 1) ≔ f(s, s) Proposition 3.1.1 a).a) There is a family in (Un Ec)4 such that f is given by the attached table and such that for every and
c) If ε1 = − 1E, ε2 =ε 4 =γ 1 = 1E, and there is an x ∈ Ec with then .
a) is a long calculation.
b) and c) follow from a) and Theorem 2.2.18 e).
d) is a long calculation using a).
3.4. with n ∈ ℕ
PROPOSITION 3.4.1 Put A ≔ Un Ec and for every α ∈ An−1 put
where and are canonically identified and αn ≔ 1E.EXAMPLE 3.4.2 Let , , z: the canonical inclusion, and
where and are canonically identified. Then . Let further S be the subgroup of generated by , where is the quotient map, and for every . Then the map is an E-C*-isomorphism.The next example shows that the set is not reduced by restricting the Schur functions to have the form indicated in Example 3.4.2.
EXAMPLE 3.4.3 Let and . Put
where we take a fixed (but arbitrary) branch of log. If we define then there is a such that g = fδλ, where f is the Schur function defined in Example 3.4.2. In particular .3.5.
EXAMPLE 3.5.1 Let .
By Corollary 1.1.6 c) and Proposition 2.2.2 a1 ⇒ a2, we may assume f constant. By Proposition 2.2.10 c),e), we may assume . Let be the inclusion map. Then
is an isomorphism of Hilbert spaces. If we identify these Hilbert spaces using this isomorphism then V1 becomes the multiplicator operator so is an injective, involutive algebra homomorphism. The assertion follows.CLIFFORD ALGEBRAS
4.1. The general case
T is a subgroup of . We canonically associate to every element t ∈ T in a bijective way the ”word” , where
and use sometimes this representation instead of t (to 1 ∈ T we associate the ”empty word”).PROPOSITION 4.1.1
a) Let be a finite sequence of letters with for every j and use transpositions of successive letters with distinct indices in order to bring these indices in an increasing order. If τ denotes the number of used transpositions then (−1)τ does not depend on the manner in which this operation was done.
b) Let s, t ∈ T and let
be the canonically associated words of s and t, respectively. We put for every k ∈ I, if there is a with k = ij and if the above condition is not fulfilled and define in a similar way. Moreover we put (Proposition 1.1.2 a)) where τ denotes the number of transpositions of successive letters with distinct indices in the finite sequence of letters in order to bring the indices in an increasing order. Then .
a) We define a new total order relation on the indices of the given word by putting for all
Let P be a sequence of transpositions of successive letters in order to bring the indices in an increasing form with respect to the new order and let τ′ be the number of used transpositions. Then τ − τ′ is even and so (−1)τ = (−1)τ′. By the theory of permutations (−1)τ′ does not depend on P, which proves the assertion.b) By a), f is well-defined. Let r, s, t ∈ T and let
be the words canonically associated to r, s, and t, respectively. There are α, β ∈ {−1, +1} such that Write the finite sequence of letters and use transpositions of successive letters with distinct indices in order to bring the indices in an increasing order. We can do this acting first on the letters of r and s only and then in a second step also on the letters of t. Then α = (−1)μ, where μ denotes the number of all performed transpositions. For β we may start first with the letters of s and t and then in a second step also with the letters of r. Then β = (−1)ν, where ν is the number of all effectuated transpositions. By a), α = (−1)μ = (−1)ν = β. The rest of the proof is obvious.c) follows from Corollary 2.1.17 d).
COROLLARY 4.1.2. then for all s, t ∈ T,
PROPOSITION 4.1.3. Let s, t ∈ T.
a) .
b) .
c) Assume . If is even or if is odd then f(s, t) = f(t, s). If in addition st = ts then VsVt = VtVs.
d) If Card I is an odd natural number and T is commutative then for every t ∈ T with .
a) For ,
sob) By Proposition 2.1.2 b),
Thus if st = ts then by a), Conversely, if this relation holds then by a), and we get st = ts by Theorem 2.1.9 a).c) follows from a) and b).
d) follows from c) (and Proposition 2.1.2 d)).
e) We have
By Proposition 2.1.2 b),e),PROPOSITION 4.1.4. Let S be a finite subset of T′ \ {1} such that st = ts and is odd for all distinct s, t ∈ S and for every t ∈ S let αt, εt ∈ Un Ec and Xt ∈ E be such that
a) By Proposition 4.1.3 b),e),
b) Remark first that β ∈ Un Ec and put
For u ∈ [0, 1], so by a), . MoreoverCOROLLARY 4.1.5. Let s,t ∈ T′ \ {1}, s≠t, st = ts, αs, αt, εs, εt ∈ Un Ec such that
and puta) ; we denote by Ps ∧ Pt the infimum of Ps and Pt in (by b) and c) it exists).
b) If VsVt ≠ VtVs then Ps ∧ Pt = 0.
c) If VsVt = VtVs then .
a) follows from Proposition 2.1.20 b ⇒ a.
b) By Proposition 4.1.3 b), VsVt = −VtVs. Let with X ≤ Ps and X ≤ Pt. By [1] Proposition 4.2.7.1 d ⇒ c,
so X = 0 and Ps ∧ Pt = 0.c) We have PsPt = PtPs so and PsPt = Ps ∧ Pt by [1] Corollary 4.2.7.4 a ⇒ b&d.
COROLLARY 4.1.6. Let m, , , , and for every i let ti ∈ T′ with and titj = tjti for all . If for every i ∈ ,
thenFor distinct ,
is odd. For every put . Then and the assertion follows from Proposition 4.1.4 a).THEOREM 4.1.7. Let such that is an ordered subset of I, , g ≔ f|(S × S), a, b ∈ T such that a2 = b2 = 1,
the (injective) group homomorphism defined by ω(1, 0) ≔ a, ω(0,1) ≔ b, α1 ≔ f(a, a), α2 ≔ f(b, b), β1, β2 ∈ Un Ec such that , We consider as an E -C**-subalgebra of (Corollary 2.1.17 e)).a) is easy to see.
b) follows from Proposition 4.1.3 b).
c) follows from a) and Theorem 2.2.18 b),h).
d) follows from Theorem 2.2.18 c).
e) By b) and c), the map is well-defined. The assertion follows now from Theorem 2.2.18 d),h).
f) follows from b),c), and Theorem 2.2.18 h).
g) follows from b) and Proposition 2.2.17 d).
h) follows from c), d), Proposition 3.2.1 a), Corollary 3.2.2 d), and Proposition 3.2.3 c).
i) follows from Theorem 2.2.18 f).
PROPOSITION 4.1.8. We use the notation and the hypotheses of Theorem 4.1.7 and assume , , and with .
If we take α ≔ β ≔ γ ≔ −δ ≔ β1 ≔ β2 ≔ 1E in b) then the map
is an E-C**-isomorphism.a) Use Corollary 4.1.2 and Proposition 2.1.2 b).
b) Use Proposition 6437 a) and Proposition 4.1.1.
c) follows from b) and Proposition 3.4.1 f1.
4.2. A special case
Throughout this section we denote by S a totally ordered set, put , and fix a map ρ: S ⟶ Un Ec. We define for every s ∈ S, by putting fs(1,1) = ρ(s) (Proposition 3.1.1 a)). Moreover we denote by fρ the Schur function f defined in Proposition 4.1.1 b) (with I replaced by S) and put .
Remark. If then so is a special case of the example treated in section 3.2. With the notation used in the left table of Proposition 3.2.1 this case appears for a ≔ (1,0) and b ≔ (0.1) exactly when ε = −1E, α = −ρ(b), β = ρ(a), and γ = 1E.
LEMMA 4.2.1. endowed with the composition law
is a locally finite commutative group (Definition 2.1.18) with as neutral element and the map is a group isomorphism with inverse We identify T with by using this isomorphism and write s instead of {s} for every s ∈ S. For A, B ∈ T, where τ is defined in Proposition 4.1.1 b).PROPOSITION 4.2.2. Assume S finite and let F be an E-C*-algebra. Let further (xs)s∈S be a family in F such that for all distinct s,t ∈ S and for every y ∈ E,
Then there is a unique E-C*-homomorphism such that φVs = xs for all s ∈ S. If the family is E-linearly independent (resp. generates F as an E-C*-algebra) then φ is injective (resp. surjective).Put
for every A ≔ {s1, s2, ⋯, sm}, where s1 < s2 < ⋯ < sm, and It is easy to see that (φ Vs)(φ Vt) = φ (VsVt) and y φ Vs = (φ Vs)y for all s, t ∈ S and y ∈ E (Proposition 2.1.2 b)). Let where the letters are written in strictly increasing order. Then by Proposition 4.1.3 e).For (by Theorem 2.1.9 c),g)),
(Proposition 4.1.3 e)) i.e. φ is an E-C*-homomorphism. The uniqueness and the last assertions are obvious (by Theorem 2.1.9 a)).PROPOSITION 4.2.3. Let m, , , , , ,
and Ai ≔ A ∪ {2n + i} for every A ⊂ S and .a) By Proposition 4.1.3 e),
b) follows from a) and Corollary 4.1.6.
c) By Proposition 4.1.3 c), for s ∈ S, for every so . By b), for distinct s, t ∈ S (Proposition 4.1.3 b)),
By Proposition 4.2.2 there is a unique E-C*-homomorphism with the given properties.Let with φX = 0. Then
and this implies XA = 0 for all A ⊂ S (Theorem 2.1.9 a)). Thus φ is injective.d)
Let . Then (by Proposition 2.1.2 b)) so for every A ⊂ S. If we put then Thus φ is surjective.Let . Then
For every B ⊂ S put
Then It follows for B ⊂ S, so by Proposition 4.1.3 a),b), . If we put then and so for B ⊂ S, Thus φX = Y and φ is surjective.Remark. If m = 3 then φ may be not surjective.
PROPOSITION 4.2.4 Let 𝕂 ≔ ℝ, n ∈ ℕ ∪ {0}, S ≔ ℕ2n, and
Let be the complexification of , considered as a real E-C*-algebra ([1] Theorem 4.1.1.8 a)) by using the embedding Then there is a unique E-C*-isomorphism such that for every s ∈ S andWe put
For s ∈ S, by Proposition 4.1.3 b), By Proposition 674 b),e), and the assertion follows from Proposition 4.2.2.PROPOSITION 4.2.5 Let , , , , α1, α2 ∈ UnEc, and
a) Put
for every s ∈ S and For distinct s, t ∈ S and i ∈ ℕ2, By Proposition 4.2.2 there is a unique E-C*-homomorphism satisfying the given conditions.We put for every A ⊂ S and i ∈ ℕ2
For A ⊂ S, Then for , It follows from the above identities that φ is bijective.b) By the above,
and the assertion follows.COROLLARY 4.2.6 Let m, , , , and
Then .PROPOSITION 4.2.7 Let 𝕂 ≔ ℝ, n ∈ ℕ ∪ {0}, S ≔ ℕ2n, S′ ≔ ℕ2n + 2, α1, α2 ∈ Un Ec, and
Then there is a unique E-C*-isomorphism such that where i, j, k are the canonical unitaries of .Put
For distinct s, t ∈ S and , by Proposition 4.1.3 b), By Proposition 4.2.2 there is a unique E-C*-homomorphism satisfying the given conditions.For ,
and so φ is bijective.PROPOSITION 4.2.8 Let , , A′ ≔ A ∪ {2n + 1} for every A ⊂ S,
, and defined by for every and A ⊂ S.a) By Proposition 4.1.3 d),e), ,
sob) By a), Proposition 4.1.3 c),d), Proposition 4.1.1 b), and Proposition 1.1.2 b),
c) For ,
so and By a) and b),d) For and A ⊂ S, by c),
e) For and A ⊂ S, by b) and c),
so .f) For and B ⊂ S, by a),b),c),e) and Proposition 4.1.1 b) (and Corollary 2.1.17 e)),
so by a),g) The assertion concerning φ as well as the identity in the definition of ψ follow from a),d),e), and f). Thus φ is a surjective E-C*-homomorphism. For Y ∈ Kerψ,
so by a) and e), and we get i.e. ψ is injective.