Ferraro, D., Abadie, F. (2017). Applications of ternary rings to $C^*$-algebras. Advances in Operator Theory, 2(3), 293-317. doi: 10.22034/aot.1612-1085
Damian Ferraro; Fernando Abadie. "Applications of ternary rings to $C^*$-algebras". Advances in Operator Theory, 2, 3, 2017, 293-317. doi: 10.22034/aot.1612-1085
Ferraro, D., Abadie, F. (2017). 'Applications of ternary rings to $C^*$-algebras', Advances in Operator Theory, 2(3), pp. 293-317. doi: 10.22034/aot.1612-1085
Ferraro, D., Abadie, F. Applications of ternary rings to $C^*$-algebras. Advances in Operator Theory, 2017; 2(3): 293-317. doi: 10.22034/aot.1612-1085
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.