Naimark's problem for graph C*-algebras

TL;DR

This paper investigates Naimark's problem within graph C*-algebras, demonstrating that for certain classes like AF graph C*-algebras and those with countably many edges per vertex, the problem has an affirmative solution.

Contribution

The paper proves that Naimark's problem has an affirmative answer for AF graph C*-algebras and for graph C*-algebras with vertices emitting countable edges.

Findings

Naimark's problem is affirmatively resolved for AF graph C*-algebras.

The problem also has an affirmative answer for graph C*-algebras with countable edges per vertex.

Abstract

Naimark's problem asks whether a C*-algebra that has only one irreducible *-representation up to unitary equivalence is isomorphic to the C*-algebra of compact operators on some (not necessarily separable) Hilbert space. This problem has been solved in special cases, including separable C*-algebras and Type I C*-algebras. However, in 2004 Akemann and Weaver used the diamond principle to construct a C*-algebra with $\aleph_1$ generators that is a counterexample to Naimark's Problem. More precisely, they showed that the statement "There exists a counterexample to Naimark's Problem that is generated by $\aleph_1$ elements." is independent of the axioms of ZFC. Whether Naimark's problem itself is independent of ZFC remains unknown. In this paper we examine Naimark's problem in the setting of graph C*-algebras, and show that it has an affirmative answer for (not necessarily separable) AF graph C*-algebras as well as for C*-algebras of graphs in which each vertex emits a countable number of edges.