Finite approximation properties of C∗-modules

Abstract

We study the notions of nuclearity and exactness for module maps on $C^{\ast }$-algebras which are $C^{\ast }$-module over another $C^{\ast }$-algebra with compatible actions and examine finite approximation properties of such $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-algebras on the subsemigroup of idempotents.