Naimark's Problem for Graph C*-Algebras



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Naimark’s problem asks if a C*-algebra with exactly one irreducible representation up to unitary equivalence is isomorphic to K(H), the algebra of compact operators on some Hilbert space H. A C*-algebra that satisfies the premise of this question but not its conclusion is a counterexample to Naimark’s problem. It is known that neither separable C*-algebras nor Type I C*-algebras can be counterexamples to Naimark’s problem. In 2004, Akemann and Weaver constructed an aleph_1-generated counterexample using Jensen’s ♦ axiom (pronounced “diamond axiom”), which is known to be independent of ZFC. In fact, they showed that the existence of an aleph_1-generated counterexample is independent of ZFC. The general problem remains open. In this thesis we focus on Naimark’s problem for a subclass of C*-algebras called graph C*-algebras. We show that approximately finite-dimensional (denoted AF) graph C*-algebras cannot be counterexamples to Naimark’s problem. We also show that, as a consequence, C*-algebras of row-countable graphs cannot be counterexamples to Naimark’s problem. Since C*-algebras with unique irreducible representations up to unitary equivalence must be simple, and since simple graph C*-algebras are either AF or purely infinite, a complete answer to Naimark’s problem for all graph C*-algebras now hinges on an examination of the class of purely infinite graph C*-algebras.



Naimark's problem, Graph C*-algebra, AF algebra