Change of base for operator space modules

Abstract

We prove a change of base theorem for operator space modules over C*-algebras, analogous to the change of rings for algebraic modules. We demonstrate how this can be used to show that the category of (right) matrix normed modules and completely bounded module maps has enough injectives.

Exact Structures for Operator Modules

“…When is a C*-algebra, ( ) supplies (OMod ∞ , Ex ) with enough injectives, however it loses its role when is a general operator algebra ([30, Example 3.5]) and in fact, the question seems to be open. For C*-algebras, [28,Proposition 4.13] answers the question affirmatively for (mnMod ∞ , Ex…”

“…When is a unital operator algebra and M is the class of admissible monomorphisms in the exact category (OMod ∞ , Ex ), it is unclear whether there are enough Minjectives. The canonical object ( , ), where is injective in Op ∞ , which is the analogue of the canonical injective object in algebraic module categories, lies in the larger category of matrix normed modules [10], [28]. We will discuss this issue at the end of the present section.…”

“…Suppose Γ is a non-amenable discrete group; then the full C*-algebra * (Γ) is not nuclear and hence its bidual * (Γ) * * not injective as a C*-algebra, hence not injective in Op ∞ by [14,Theorem 3.2]. By duality, * (Γ) * * = ( , C) where = (Γ) is the Fourier-Stieltjes algebra and ( , ) is always Ex -injective for an injective operator space (by the operator algebra version of [28,Corollary 4.11]) and thus automatically Ex -injective.…”

Exact structures for operator modules

“…When 𝐴 is a unital operator algebra and M is the class of admissible monomorphisms in the exact category (OMod ∞ 𝐴 , Ex 𝑚𝑎𝑥 ), it is unclear whether there are enough Minjectives. The canonical object 𝐶𝐵( 𝐴, 𝐼), where 𝐼 is injective in Op ∞ , which is the analogue of the canonical injective object in algebraic module categories, lies in the larger category of matrix normed modules [10], [29]. We will discuss this issue at the end of the present section.…”

“…Comparing with immediately tells us that is a full subcategory of (see [6, Example 3.1.5]). The category is, e.g., used in [3, 10, 29].…”