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Hilbert--Schmidt stability for graph products

Pieter Spaas

Abstract

In this short note we prove Hilbert--Schmidt stability for graph products of abelian groups and $C^*$-algebras on chordal graphs. In particular, this shows that right-angled Artin groups on chordal graphs are Hilbert--Schmidt stable.

Hilbert--Schmidt stability for graph products

Abstract

In this short note we prove Hilbert--Schmidt stability for graph products of abelian groups and -algebras on chordal graphs. In particular, this shows that right-angled Artin groups on chordal graphs are Hilbert--Schmidt stable.
Paper Structure (6 sections, 5 theorems, 14 equations)

This paper contains 6 sections, 5 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results

Key Result

Lemma 4

Suppose $A$ is generated as a $C^*$-algebra by elements $\{b_1, b_2, \ldots\}$. Let $(A_i,\tau_i)_{i\in I}$ be a family of tracial $C^*$-algebras, let $\mathcal{U}$ be an ultrafilter on $I$, and let $\theta:A\to\prod_\mathcal{U} (A_i,\tau_i)$ be a $^*$-homomorphism. Then the following are equivalent

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 4: HS16
  • Definition 5
  • Definition 6
  • Lemma 7: Inductive Construction of Chordal Graphs
  • Theorem 8
  • proof
  • Theorem 9
  • ...and 3 more