Applications of ternary rings to $C^{\ast }$-algebras

Abstract

We show that there is a functor from the category of positive admissible ternary rings to the category of $\ast$-algebras, which induces an isomorphism of partially ordered sets between the families of $C^{\ast }$-norms on the ternary ring and its corresponding $\ast$-algebra. We apply this functor to obtain Morita-Rieffel equivalence results between cross sectional $C^{\ast }$-algebras of Fell bundles, and to extend the theory of tensor products of $C^{\ast }$-algebras to the larger category of full Hilbert $C^{\ast }$-modules. We prove that, like in the case of $C^{\ast }$-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 $C^{\ast }$-algebras.