Tomforde, Mark2019-09-132019-09-13May 20172017-05May 2017https://hdl.handle.net/10657/4555Naimark’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.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).Naimark's problemGraph C*-algebraAF algebra2019-09-13Thesisborn digital