Rohlin Actions of Finite Groups on the Razak–Jacelon Algebra

Abstract

Let A be a simple separable nuclear C * -algebra with a unique tracial state and no unbounded traces, and let α be a strongly outer action of a finite group G on A. In this paper, we show that α ⊗ id on A ⊗ W has the Rohlin property, where W is the Razak-Jacelon algebra. Combing this result with the recent classification results and our previous result, we see that such actions are unique up to conjugacy.2010 Mathematics Subject Classification. Primary 46L55, Secondary 46L35; 46L40.

Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra

“…Then ̺ maps B ω ∩ Φ(A) ′ into M(Φ(A), B). Essentially the same proof as [34, Theorem 3.1] (see also [25,Theorem 3.3] and [38,Proposition 3.4]) shows the following proposition.…”

“…Since Rohlin actions of a finite group on O 2 are unique up to conjugacy, we can regard this result as an equivariant version of the Kirchberg-Phillips type absorption for outer actions of finite groups. As an analog of this result, the author showed that if α is a strongly outer action of a finite group on a simple separable nuclear monotracial C * -algebra, then α ⊗ id W on A ⊗ W has the Rohlin property in [38]. In [56], Szabó generalized Izumi's result to actions of countable discrete amenable groups.…”

“…Hence [37, Proposition 4.2] and the proposition above imply the following proposition. By [38,Section 4], there exists a homomorphism ρ from…”

“…In Section 4, we shall consider properties of the fixed point subalgebra F (A⊗ W) α⊗idW where α is a strongly outer action of a countable discrete amenable group on a simple separable nuclear monotracial C * -algebra A. This section is based on the author's previous works in [37] and [38]. In Section 5, we shall consider a slight variant of Szabó's approximate cocycle intertwining argument, which is our main technical tool in this paper.…”

On a categorical framework for classifying C*-dynamics up to cocycle conjugacy

“…The Rokhlin property for actions of finite groups was introduced by Izumi [41,42], who also showcased its rigid behavior by giving a satisfactory classification theory up to conjugcy. Results of this type were subsequently found for not necessarily unital C *algebras [81,26,82], actions of compact groups [36,23,24], and compact quantum groups [1]. The Rokhlin property for flows was introduced by Kishimoto in [59], who provided evidence why one should expect that these can be classified up to cocycle conjugacy [65,6].…”