$C^*$-Algebraic Covariant Structures

TL;DR

This paper introduces covariant structures involving $C^*$-algebras and twisted group actions, developing a framework for twisted crossed products that generalizes Abelian Takai duality.

Contribution

It defines covariant structures with dual group actions, constructs twisted crossed products, and shows isomorphisms among resulting $C^*$-algebras, extending Takai duality to non-commutative settings.

Findings

Developed a new framework for covariant structures with dual group actions.

Constructed twisted crossed products and demonstrated their properties.

Established isomorphisms among various $C^*$-algebras derived from the structures.

Abstract

We introduce {\it covariant structures} $\left\{(\A,\k),(\a,\aa),\(\ha,\haa\)\right\}$ formed of a separable $C^*$-algebra $\A$, a measurable twisted action $(\a,\aa)$ of the second-countable locally compact group $\G$\,, a measurable twisted action $(\ha,\haa)$ of another second-countable locally compact group $\hG$ and a strictly continuous function $\k:\G\times\hG\to\U\M(\A)$ suitably connected with $(\a,\aa)$ and $\(\ha,\haa\)$\,. Natural notions of covariant morphisms and representations are considered, leading to a sort of twisted crossed product construction. Various $C^*$-algebras emerge by a procedure that can be iterated indefinitely and that also yields new pairs of twisted actions. Some of these $C^*$-algebras are shown to be isomorphic. The constructions are non-commutative, but are motivated by Abelian Takai duality that they eventually generalize.