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The Space of Traces of the Free Group and Free Products of Matrix Algebras

Joav Orovitz, Raz Slutsky, Itamar Vigdorovich

Abstract

We show that the space of traces of the free group $F_d$ on $2\leq d \leq \infty $ generators is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. This fails for many virtually free groups. The same result holds for free products of the form $C(X_1)*C(X_2)$ where $X_1$ and $X_2$ are compact metrizable spaces without isolated points. Using a similar strategy, we show that the space of traces of the free product of matrix algebras $M_n(\mathbb{C}) * M_n(\mathbb{C})$ is a Poulsen simplex as well, answering a question of Musat and R\ordam for $n \geq 4$. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.

The Space of Traces of the Free Group and Free Products of Matrix Algebras

Abstract

We show that the space of traces of the free group on generators is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. This fails for many virtually free groups. The same result holds for free products of the form where and are compact metrizable spaces without isolated points. Using a similar strategy, we show that the space of traces of the free product of matrix algebras is a Poulsen simplex as well, answering a question of Musat and R\ordam for . Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.
Paper Structure (20 sections, 34 theorems, 50 equations)

This paper contains 20 sections, 34 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tracial representations
  4. Approximation of traces
  5. Free products of commutative $C^*$-algebras
  6. Perturbations of representations
  7. Generators for $\mathbf{M}_n$
  8. Proof of Theorem \ref{['Thm:free products of commutative algebras']}
  9. Free products of matrix algebra
  10. Generating sets in von Neumann algebras
  11. An amplification lemma
  12. Representations of $\mathbf{M}_n$
  13. Representations of $\mathbf{M}_n*\mathbf{M}_n$
  14. Proof of Theorem \ref{['Second Theorem']}
  15. Factorizable quantum channels
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $G_1$ and $G_2$ be infinite countable abelian groups, and let $G_0$ be any countable group. Then the space of traces on $G=G_0*G_1*G_2$ is a Poulsen simplex.

Theorems & Definitions (70)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4: See Theorem \ref{['thm:free product of ET is not Poulsen']} for a generalization
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 2.1: See dixmier1977cbekka_harp
  • Lemma 2.2
  • ...and 60 more