Conditions \(C_p\), \(C'_p\), and \(C"_p\) for \(p\)-operator spaces

Abstract

Conditions \(C\), \(C'\), and \(C"\) were introduced for operator spaces in an attempt to study local reflexivity and exactness of operator spaces (Effros and Ruan, 2000). For example, it is known that an operator space \(W\) is locally reflexive if and only if \(W\) satisfies condition \(C"\) (Effros and Ruan, 2000) and an operator space \(V\) is exact if and only if \(V\) satisfies condition \(C'\) (Effros and Ruan, 2000). It is also known that an operator space \(V\) satisfies condition \(C\) if and only if it satisfies conditions \(C'\) and \(C"\) (Effros and Ruan, 2000, and Han, 2007). In this paper, we define \(p\)-operator space analogues of these definitions, which will be called conditions \(C_p\), \(C'_p\), and \(C"_p\), and show that a \(p\)-operator space on \(L_p\) space satisfies condition \(C_p\) if and only if it satisfies both conditions \(C'_p\) and \(C"_p\). The \(p\)-operator space injective tensor product of \(p\)-operator spaces will play a key role.