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The RFD property for graph C*-algebras

Guillaume Bellier, Tatiana Shulman

TL;DR

The paper determines when a finite graph C*-algebra is residually finite-dimensional, proving $C^*(G)$ is RFD if and only if no cycle has an entry. The key strategy is to decompose graph algebras as amalgamated free products over finite-dimensional subalgebras, enabling an application of the Li–Shen criterion for RFD. This approach not only yields the main characterization but also provides a general, potentially independent, decomposition tool for graph C*-algebras. The results strengthen known connections between quasidiagonality and RFD in the finite-graph setting and offer practical methods to construct finite-dimensional representations separating elements of $C^*(G)$.

Abstract

It is proved that the graph C*-algebra of a finite graph is residually finite-dimensional (RFD) if and only if no cycle has an entry. To obtain this result we prove that C*-algebras of graphs with finitely many vertices often admit a convenient decomposition into amalgamated free products.

The RFD property for graph C*-algebras

TL;DR

The paper determines when a finite graph C*-algebra is residually finite-dimensional, proving is RFD if and only if no cycle has an entry. The key strategy is to decompose graph algebras as amalgamated free products over finite-dimensional subalgebras, enabling an application of the Li–Shen criterion for RFD. This approach not only yields the main characterization but also provides a general, potentially independent, decomposition tool for graph C*-algebras. The results strengthen known connections between quasidiagonality and RFD in the finite-graph setting and offer practical methods to construct finite-dimensional representations separating elements of .

Abstract

It is proved that the graph C*-algebra of a finite graph is residually finite-dimensional (RFD) if and only if no cycle has an entry. To obtain this result we prove that C*-algebras of graphs with finitely many vertices often admit a convenient decomposition into amalgamated free products.
Paper Structure (8 sections, 10 theorems, 68 equations, 2 figures)

This paper contains 8 sections, 10 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph C*-algebras
  4. Amalgamated free products.
  5. Decomposition theorems
  6. Case $G^0_2 = G^0$.
  7. Case $G^0_2 \neq G^0$.
  8. Graphs with RFD C*-algebras

Key Result

Theorem 2.1

(Raeburn, TomfordeThe terminology in Tomforde is different from Raeburn, in particular what is a source in Raeburn is a sink in Tomforde.) If $G$ is a finite graph with no cycles, and $v_1, \ldots, v_m$ are the sources of $G$, then This isomorphism sends each source to a rank one projection.This follows from the construction of the isomorphism.

Figures (2)

  • Figure 1: Examples
  • Figure 2: Embeddings

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Definition 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 8 more