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Tracially amenable actions and purely infinite crossed products

Eusebio Gardella, Shirly Geffen, Julian Kranz, Petr Naryshkin, Andrea Vaccaro

Abstract

We introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C$^*$-algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group $F_2$, we show that outer, tracially amenable actions on simple, unital, $\mathcal{Z}$-stable C$^*$-algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.

Tracially amenable actions and purely infinite crossed products

Abstract

We introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C-algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group , we show that outer, tracially amenable actions on simple, unital, -stable C-algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.
Paper Structure (7 sections, 16 theorems, 58 equations)

This paper contains 7 sections, 16 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Tracially amenable actions
  3. Examples
  4. Purely infinite crossed products
  5. From dynamical comparison to pure infiniteness of crossed products
  6. Groups with weak paradoxical towers
  7. Establishing dynamical comparison from tracial amenability

Key Result

Theorem A

Let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a countable, discrete, exact group $G$ on a separable, unital, tracial C$^*$-algebra $A$. The following are equivalent:

Theorems & Definitions (43)

  • Definition 1: \ref{['df:TracialAmenAct']}
  • Theorem A: \ref{['thm:TracAm']}
  • Theorem B: \ref{['cor:main']}
  • Example 1: \ref{['eg:TracAmenEll']}; \ref{['prop:ExOnAFAlg']}
  • Lemma 2.1
  • Definition 1
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Lemma 2.2
  • ...and 33 more