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

TL;DR

This paper demonstrates that certain strongly outer actions on simple nuclear monotracial C*-algebras become cocycle conjugate when tensoring with the Razak-Jacelon algebra, and characterizes these actions via fixed point subalgebras.

Contribution

It establishes equivariant Kirchberg-Phillips type absorption results for the Razak-Jacelon algebra and provides a characterization of actions using fixed point subalgebras.

Findings

Strongly outer actions become cocycle conjugate after tensoring with the Razak-Jacelon algebra.

Characterization of actions via fixed point subalgebras of Kirchberg's central sequence C*-algebras.

Extension of absorption results to the setting of monotracial C*-algebras.

Abstract

Let $A$ and $B$ be simple separable nuclear monotracial C$^*$-algebras, and let $\alpha$ and $\beta$ be strongly outer actions of a countable discrete amenable group $\Gamma$ on $A$ and $B$, respectively. In this paper, we show that $\alpha\otimes\mathrm{id}_{\mathcal{W}}$ on $A\otimes\mathcal{W}$ and $\beta\otimes\mathrm{id}_{\mathcal{W}}$ on $B\otimes\mathcal{W}$ are cocycle conjugate where $\mathcal{W}$ is the Razak-Jacelon algebra. Also, we characterize such actions by using the fixed point subalgebras of Kirchberg's central sequence C$^*$-algebras.