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Amenable actions on ill-behaved simple C*-algebras

Yuhei Suzuki

TL;DR

This work constructs amenable actions of non-amenable groups on ill-behaved simple C*-algebras, yielding the first examples where the acting group is amenable in action but not in itself, and the ambient algebra is tensorially prime (neither stably finite nor purely infinite). The authors combine Rørdam’s projections-based framework with their amenable-action method to produce an amenable, pointwise outer action $\alpha$ of a countable group $\Gamma$ on a simple, separable, nuclear C*-algebra $A$ satisfying the UCT, with $A$ containing both a finite and an infinite projection and with $A\rtimes_{\mathrm{r},\alpha}\Gamma$ sharing these properties. For free groups, unital realizations are available. The paper further extends the construction to exact groups to produce unital, simple, non-exact algebras carrying amenable actions, still with both finite and infinite projections in the crossed products, offering new insight into the structure and classification of crossed products by ill-behaved simple algebras. These results impact the understanding of symmetry in non-classifiable C*-algebras and illuminate how amenable dynamical systems can exist on highly non-commutative, non-classifiable objects.

Abstract

By combining Rørdam's construction and the author's previous construction, we provide the first examples of amenable actions of non-amenable groups on simple separable nuclear C*-algebras that are neither stably finite nor purely infinite. For free groups, we also provide unital examples. We arrange the actions so that the crossed products are still simple with both a finite and an infinite projection.

Amenable actions on ill-behaved simple C*-algebras

TL;DR

This work constructs amenable actions of non-amenable groups on ill-behaved simple C*-algebras, yielding the first examples where the acting group is amenable in action but not in itself, and the ambient algebra is tensorially prime (neither stably finite nor purely infinite). The authors combine Rørdam’s projections-based framework with their amenable-action method to produce an amenable, pointwise outer action of a countable group on a simple, separable, nuclear C*-algebra satisfying the UCT, with containing both a finite and an infinite projection and with sharing these properties. For free groups, unital realizations are available. The paper further extends the construction to exact groups to produce unital, simple, non-exact algebras carrying amenable actions, still with both finite and infinite projections in the crossed products, offering new insight into the structure and classification of crossed products by ill-behaved simple algebras. These results impact the understanding of symmetry in non-classifiable C*-algebras and illuminate how amenable dynamical systems can exist on highly non-commutative, non-classifiable objects.

Abstract

By combining Rørdam's construction and the author's previous construction, we provide the first examples of amenable actions of non-amenable groups on simple separable nuclear C*-algebras that are neither stably finite nor purely infinite. For free groups, we also provide unital examples. We arrange the actions so that the crossed products are still simple with both a finite and an infinite projection.
Paper Structure (5 sections, 5 theorems, 81 equations)

This paper contains 5 sections, 5 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ingredients of the constructions
  4. Theorem \ref{['Thm:1']}
  5. Theorem \ref{['Thm:2']}

Key Result

Theorem 1

Let $\Gamma$ be a countable group. Then there is an amenable pointwise outerAn automorphism $\alpha$ on a C$^\ast$-algebra $A$ is said to be outer, if there is no unitary multiplier element $u\in \mathcal{M}(A)$ with $\alpha=\mathrm{Ad}(u)$. We say an action $\alpha \colon \Gamma \curvearrowright A$ Moreover, when $\Gamma$ is a free group, one can arrange $A$ to be unital.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Proposition 3: Ror, Proposition 4.5
  • Lemma 4
  • Remark 5
  • proof : Proof of Lemma \ref{['Lem:fp']}
  • Remark 6: Some properties of $\psi_{s, j}$
  • proof : Proof of Theorem \ref{['Thm:1']}
  • Claim 7
  • Claim 8
  • ...and 17 more