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

Abstract

Let A and B be simple separable nuclear monotracial C * -algebras, and let α and β be strongly outer actions of a countable discrete amenable group Γ on A and B, respectively. In this paper, we show that α ⊗ id W on A ⊗ W and β ⊗ id W on B ⊗ W are cocycle conjugate where W is the Razak-Jacelon algebra. Also, we characterize such actions by using the fixed point subalgebras of Kirchberg's central sequence C * -algebras.

Approximate representability of finite abelian group actions on the Razak-Jacelon algebra

“…By [21, Proposition 3.11], we have the following proposition. Note that [21,Proposition 3.11] is based on Matui and Sato's techniques [14], [15] and [16] (with pioneering works [25] and [26]). See also [27] and [30].…”

“…Therefore we can recognize difficulties due to stably finiteness by studying W-type actions. In [21], the author showed that if α is a strongly outer action of a countable discrete amenable group, then W-type actions are unique up to cocycle conjugacy. (Note that an action α of a discrete countable group on a simple separable monotracial C * -algebra A is strongly outer if the induce action α by α on π τA (A)…”