Non-Hausdorff symmetries of C*-algebras

Abstract

Symmetry groups or groupoids of C * -algebras associated to nonHausdorff spaces are often non-Hausdorff as well. We describe such symmetries using crossed modules of groupoids. We define actions of crossed modules on C * -algebras and crossed products for such actions, and justify these definitions with some basic general results and examples.

Gerbes over posets and twisted C*-dynamical systems

“…denote the canonical morphisms. We consider covariant (non-degenerate) representations of β in the sense of [11], given by pairs π : γ(l, m))), the above representation is spatial. Thus taking the seminorm on A ⋊ α,ν,γ (G, Π) defined by these representations yields a surjective *-morphism…”

“…where the crossed product ⋊ β is in the sense of [11]. A more general version of this crossed product in the setting of C * -correspondences has been discussed in [13].…”

Crossed products for actions of crossed modules on C*-algebras

“…In the discrete case, this implies that C is equivalent to the trivial crossed module. An interesting example is the crossed module associated to a dense embedding Z → T, which acts on the corresponding noncommutative torus (see [1]). Being thin is an invariant of equivalence of crossed modules.…”

“…This proposition allows us to compute crossed products by 2-Abelian crossed modules in two more elementary steps: first take the crossed product C * (A) by the locally compact group G; then take the fibre at 1 ∈Ĥ for the canonical C 0 (Ĥ)-algebra structure on C * (A). , and Z acts by n → V −n (see [1]). The crossed product for this action is already computed in [1]: it is the C * -algebra of compact operators on the Hilbert space L 2 (T).…”

“…Our proof, in fact, generalises this idea to cover general extensions of crossed modules. Another special case is [1,Theorem 1], which says that (A⋊H)⋊C ∼ = A⋊G if C = (G, H, ∂, c) is a crossed module and A is a G-C * -algebra.…”