Applications of ternary rings to $C^*$-algebras
Fernando Abadie, Dami\'an Ferraro

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 -norms on the ternary ring and its corresponding -algebra. We apply this functor to obtain Morita-Rieffel equivalence results between cross sectional -algebras of Fell bundles, and to extend the theory of tensor products of -algebras to the larger category of full Hilbert -modules. We prove that, like in the case of -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 -algebras.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology
