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A note on ideal C$^\ast$-completions and amenability

Tomasz Kochanek

Abstract

For a discrete group $G$, we consider certain ideals $\mathcal{I}\subset c_0(G)$ of sequences with prescribed rate of convergence to zero. We show that the equality between the full group C$^\ast$-algebra of $G$ and the C$^\ast$-completion $\mathrm{C}_{\mathcal{I}}^\ast(G)$ in the sense of Brown and Guentner implies that $G$ is amenable.

A note on ideal C$^\ast$-completions and amenability

Abstract

For a discrete group , we consider certain ideals of sequences with prescribed rate of convergence to zero. We show that the equality between the full group C-algebra of and the C-completion in the sense of Brown and Guentner implies that is amenable.
Paper Structure (2 sections, 5 theorems, 15 equations)

This paper contains 2 sections, 5 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Results

Key Result

Proposition 1

For any ideal $\mathcal{D}\subset \ell^\infty(G)$, $\mathrm{C}^\ast_{\mathcal{D}}(G)$ has a faithful $\mathcal{D}$-representation.

Theorems & Definitions (7)

  • Proposition 1: see brown_guentner
  • Theorem 2: brown_guentner
  • Theorem 3
  • Lemma 4
  • proof
  • proof : Proof of Theorem \ref{['thm1']}
  • Corollary 5