Finite group actions on certain stably projectionless C*-algebras with the Rohlin property

Abstract

Abstract. We introduce the Rohlin property and the approximate representability for finite group actions on stably projectionless C * -algebras and study their basic properties. We give some examples of finite group actions on the Razak-Jacelon algebra W 2 and show some classification results of these actions. This study is based on the work of Izumi, Robert's classification theorem and Kirchberg's central sequence C * -algebras. Introduction A C* -algebra A is said to be stably projectionless if A ⊗ K has no non-zero projections where K is the C * -algebra of compact operators on an infinite-dimensional separable Hilbert space. Note that every stably projectionless C * -algebra does not have a unit. In this paper we study finite group actions on stably projectionless C * -algebras. In the theory of operator algebras, the study of group actions on operator algebras is one of the most fundamental subjects and has a long history. We refer the reader to [14]

Equivariant ⁎-homomorphisms, Rokhlin constraints and equivariant UHF-absorption

“…We use this results to classify actions of finite groups on separable C*-algebras with the Rokhlin property. Our results complement and extend those obtained by Izumi in [20] and [21] in the unital setting, and by Nawata in [24] for C*-algebras A that satisfy A ⊆ GL( A).…”

“…Other results have been obtained by Elliott and Su in [13] for direct limit actions of Z 2 on AF-algebras, and by Izumi in [20] and [21], where he proved a number of classification results for actions of finite groups on arbitrary unital separable C*-algebras with the Rokhlin property, as well as for approximately representable actions. The classification result of Izumi regarding actions with the Rokhlin property has been extended recently by Nawata in [24] to cover actions on certain not-necessarily unital separable C*-algebras (specifically for algebras A such that A ⊆ GL( A)). It should be emphasized that the classification of group actions on C*-algebras is a far less developed subject than the classification of C*-algebras and even farther less developed than the classification of group actions on von Neumann algebras.…”

Model theory and Rokhlin dimension for compact quantum group actions

“…A rigidity result for Rokhlin actions of coexact second countable compact quantum groups, generalizing results for finite and compact from [20,22,29,42] and for finite quantum groups from [33], has been obtained in [2,Theorem 5.10]. In the rest of this section, we observe here that such a result holds for arbitrary (not necessarily coexact) second countable compact quantum groups.…”

The spatial Rokhlin property for actions of compact quantum groups

“…The third feature is duality theory; a result of Izumi [15] shows that an action of a finite abelian group on a separable, unital C * -algebra has the Rokhlin property if and only if the dual action is approximately representable, and vice versa. This has been generalized to the non-unital case by Nawata [25], and to actions of compact abelian groups by the first two authors [4]; see also [8].…”