$C^{\ast }$-Algebraic Covariant Structures

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