Module tensorizing maps on $ C^*$-algebras

TL;DR

This paper explores the structure of module tensor products of $C^*$-algebras over a common algebra, introducing new concepts like module tensorizing maps and module nuclear pairs, with examples from inverse semigroup algebras.

Contribution

It introduces the notions of module tensorizing maps, module exactness, and module nuclear pairs for $C^*$-algebras, expanding the theory of tensor products in the module setting.

Findings

Established relations between amalgamated module tensor products and standard $C^*$-tensor products.

Defined and analyzed module tensorizing maps and module nuclear pairs.

Provided concrete examples involving $C^*$-algebras on inverse semigroups.

Abstract

For $ C^*$-algebras $ \mathfrak{A}, A$ and $ B $ where $ A $ and $ B $ are $ \mathfrak{A} $-bimodules with compatible actions, we consider amalgamated $ \mathfrak{A} $-module tensor product of $ A $ and $ B $ and study its relation with the C*-tensor product of $A$ and $B$ for the min and max norms. We introduce and study the notions of module tensorizing maps, module exactness, and module nuclear pairs of $ C^*$-algebras in this setting. We give concrete examples of $C^*$-algebras on inverse semigroups.