$C^*$-Algebraic Covariant Structures

Abstract

We introduce covariant structures {(A , κ), (a, α), (ã, α)} formed of a separable C * -algebra A , a measurable twisted action (a, α) of the second-countable locally compact group G , a measurable twisted action (ã, α) of another second-countable locally compact group G and a strictly continuous function κ : G × G → UM(A ) suitably connected with (a, α) and (ã, α) . 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.