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$\mathcal{W}$-absorbing actions of finite groups on the Razak-Jacelon algebra

Norio Nawata

Abstract

We say that a countable discrete group action $α$ on a C$^*$-algebra $A$ is \textit{$\mathcal{W}$-absorbing} if there exist a C$^*$-algebra $B$ and an action $β$ on $B$ such that $α$ is cocycle conjugate to $β\otimes \mathrm{id}_{\mathcal{W}}$ on $B\otimes \mathcal{W}$ where $\mathcal{W}$ is the Razak-Jacelon algebra. In this paper, we completely classify outer $\mathcal{W}$-absorbing actions of finite groups on $\mathcal{W}$ up to conjugacy and cocycle conjugacy.

$\mathcal{W}$-absorbing actions of finite groups on the Razak-Jacelon algebra

Abstract

We say that a countable discrete group action on a C-algebra is \textit{-absorbing} if there exist a C-algebra and an action on such that is cocycle conjugate to on where is the Razak-Jacelon algebra. In this paper, we completely classify outer -absorbing actions of finite groups on up to conjugacy and cocycle conjugacy.

Paper Structure

This paper contains 11 sections, 31 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite group actions on monotracial C$^*$-algebras
  4. Characteristic invariant and inner invariant
  5. Tracial state spaces and coboundaries
  6. Kirchberg's relative central sequence C$^*$-algebras
  7. Approximate cocycle morphisms and property W
  8. Model actions
  9. Approximate representability
  10. Absorption of actions with trivial characteristic invariant
  11. Classification

Key Result

Proposition 2.2

Let $A$ be a simple separable monotracial C$^*$-algebra, and let $\alpha$ be an outer action of a finite group $\Gamma$ on $A$. Then every tracial state on $A\rtimes_{\alpha}\Gamma$ is a restriction of a tracial state on $\pi_{\tau_{A}\circ E_{\alpha}}(A\rtimes_{\alpha}\Gamma)^{"}\cong \pi_{\tau_{A}

Theorems & Definitions (52)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Theorem 2.7
  • ...and 42 more