Finite approximation properties of $C^{*}$-modules

TL;DR

This paper explores finite approximation properties and nuclearity for $C^*$-modules, extending classical results to modules over inverse semigroups with applications to operator algebras.

Contribution

It introduces module versions of nuclearity and exactness, extending finite approximation properties to $C^*$-modules over inverse semigroups.

Findings

Proves module versions of Kirchberg and Choi-Effros results.

Extends finite dimensional approximation properties to $C^*$-algebras on inverse semigroups.

Demonstrates these properties in the context of $C^*$-modules with compatible actions.

Abstract

We study the notions of nuclearity and exactness for module maps on $C^{*}$-algebras which are $C^*$-module over another $C^*$-algebra with compatible actions and examine finite approximation properties of such $C^*$-modules. We prove module versions of the results of Kirchberg and Choi-Effros. As a concrete example we extend the finite dimensional approximation properties of reduced $C^*$-algebras and von Neumann algebras on countable discrete groups to these operator algebras on countable inverse semigroups with the module structure coming from the action of the $C^*$-algebras on the subsemigroup of idempotents.