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

#### 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.

#### Article information

Source
Adv. Oper. Theory, Volume 2, Number 3 (2017), 293-317.

Dates
Accepted: 4 May 2017
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431678

Digital Object Identifier
doi:10.22034/aot.1612-1085

Mathematical Reviews number (MathSciNet)
MR3730056

Zentralblatt MATH identifier
1383.46040

Subjects
Primary: 46L08: $C^*$-modules
Secondary: 46L06: Tensor products of $C^*$-algebras

#### Citation

Abadie, Fernando; Ferraro, Damián. Applications of ternary rings to $C^*$-algebras. Adv. Oper. Theory 2 (2017), no. 3, 293--317. doi:10.22034/aot.1612-1085. https://projecteuclid.org/euclid.aot/1512431678

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