Applications of ternary rings to $C^*$-algebras

TL;DR

This paper explores how ternary rings relate to $C^*$-algebras through a functor, leading to new insights on Morita-Rieffel equivalence, tensor products, and properties like nuclearity and exactness.

Contribution

It introduces a functor from positive admissible ternary rings to $*$-algebras, establishing isomorphisms of $C^*$-norms and extending tensor product theory to Hilbert $C^*$-modules.

Findings

Established a functor inducing isomorphism of $C^*$-norms

Proved existence of maximal and minimal tensor products for Hilbert modules

Provided simple proofs of invariance of nuclearity and exactness

Abstract

We show that there is a functor from the category of positive admissible ternary rings to the category of $*$-algebras, which induces an isomorphism of partially ordered sets between the families of $C^*$-norms on the ternary ring and its corresponding $*$-algebra. We apply this functor to obtain Morita-Rieffel equivalence results between cross sectional $C^*$-algebras of Fell bundles, and to extend the theory of tensor products of $C^*$-algebras to the larger category of full Hilbert $C^*$-modules. We prove that, like in the case of $C^*$-algebras, there exist maximal and minimal tensor products. As applications we give simple proofs of the invariance of nuclearity and exactness under Morita-Rieffel equivalence of $C^*$-algebras.