We study the notions of nuclearity and exactness for module maps on -algebras which are -module over another -algebra with compatible actions and examine finite approximation properties of such -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 -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 -algebras on the subsemigroup of idempotents.
"Finite approximation properties of -modules." Illinois J. Math. 66 (3) 315 - 348, September 2022. https://doi.org/10.1215/00192082-10059123