Unique expectations for discrete crossed products

Abstract

Let G be a discrete group acting on a unital C * -algebra A by * -automorphisms. In this note, we show that the inclusion A ⊆ A ⋊rG has the pure extension property (so that every pure state on A extends uniquely to a pure state on A ⋊rG) if and only if G acts freely on A, the spectrum of A. The same characterization holds for the inclusion A ⊆ A ⋊G. This generalizes what was already known for A abelian.2010 Mathematics Subject Classification. Primary: 47L65, 46L55, 46L30 Secondary: 46L07.

Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness

“…Besides looking at weak conditional expectations, we also looked at pseudo-expectations, which take values in the injective hull of A. Here we were motivated by the theorem of Zarikian that a crossed product for a group action has a unique pseudoexpectation if and only if the action is aperiodic (see [61,Theorem 3.5]). The injective hull of a commutative C ˚-algebra is equal to its local multiplier algebra (see [26]).…”

“…Hence every M loc -expectation is a pseudoexpectation as well. These have been studied, for instance, in [52,53,61]. It is, however, often necessary to work with the reduced crossed product.…”

Free coactions of a finite dimensional $C^*$-Hopf algebra and strong Morita equivalence

“…We show that the above property is an extension of the freeness of discrete groups in [14] to coactions of finite dimensional C * -Hopf algebras.…”

“…Also, we discuss the relations of the Rokhlin property, the freeness, outerness and saturatedness of coactions of a finite dimensional C * -Hopf algebra on a C * -algebra. Furthermore, we show that strong Morita equivalence for coactions preserves the freeness of coactions of a finite dimensional C * -Hopf algebra on a C * -algebra using the result similar to [14,Theorem 3.1.2].…”

“…We shall introduce a notion of free coactions of a finite dimensional C * -Hopf algebra on a C * -algebra modifying a notion of free actions of a discrete group on a C * -algebra, which is defined in Zarikian [14] and we shall give a result similar to [14,Theorem 3.1.2]. Also, we discuss the relations of the Rokhlin property, the freeness, outerness and saturatedness of coactions of a finite dimensional C * -Hopf algebra on a C * -algebra.…”