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arXiv · 1612.08476

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

Fernando Abadie·Damián Ferraro
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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.

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Fernando Abadie, Damián Ferraro. 2016-12-27. Applications of ternary rings to $C^*$-algebras. https://doi.org/10.22034/aot.1612-1085

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