A higher category approach to twisted actions on c^* -algebras

Abstract

Abstract. C * -algebras form a 2-category with * -homomorphisms or correspondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions, and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby-Smith twisted actions and equivalence of such actions, covariant representations, and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green and Busby-Smith.The Packer-Raeburn Stabilisation Trick implies that all Busby-Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalise this to actions of strict 2-groupoids.

A bicategorical interpretation for relative Cuntz-Pimsner algebras

“…Example A.4. The correspondence bicategory C is defined in [6] as the bicategory with C * -algebras as objects, correspondences as arrows, and correspondence isomorphisms as 2-arrows. The unit arrow 1 A on a C * -algebra A is A viewed as a Hilbert that is, 2-arrows α x,y (f ) : α y F x,y (f ) ⇒ G x,y (f )α x for all x, y ∈ ob B.…”

Morita Enveloping Fell Bundles

“…The unitaries introduced in Definition A.2 can be used as 2-arrows for this weak 2-category. This works similarly to the 2-category of C * -algebras with correspondences as arrows introduced in [9].…”

“…Following [9], we view a saturated Fell bundle B as an action by equivalences of G on the C * -algebra B e (the unit fiber). A non-saturated Fell bundle is viewed as a partial action by equivalences of G on B e .…”

“…In particular, a Fell bundle is a continuous Banach bundle, by definition. However, the main axiom concerning the continuity of the bundle, namely, the continuity of the norm function B → [0, ∞), b → b , is somehow automatic, see[9, Lemma 3.16]. A similar observation holds for every Hilbert B-bundle: the continuity of the norm function x → x on X follows from the continuity of the norm function on B because x = x, x B…”

Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles

“…This is why the conventions for ϕ B are used, for instance, in [8]. So the conventions in [8] would associate the equivalence bimodule A ϕ to ϕ −1 and not to ϕ.…”