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Fall and Winter 2017 Naimark’s problem for graph $C^{*}$-algebras
Nishant Suri, Mark Tomforde
Illinois J. Math. 61(3-4): 479-495 (Fall and Winter 2017). DOI: 10.1215/ijm/1534924836

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.

Citation

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Nishant Suri. Mark Tomforde. "Naimark’s problem for graph $C^{*}$-algebras." Illinois J. Math. 61 (3-4) 479 - 495, Fall and Winter 2017. https://doi.org/10.1215/ijm/1534924836

Information

Received: 21 September 2017; Revised: 14 May 2018; Published: Fall and Winter 2017
First available in Project Euclid: 22 August 2018

zbMATH: 06932513
MathSciNet: MR3845730
Digital Object Identifier: 10.1215/ijm/1534924836

Subjects:
Primary: 46L55

Rights: Copyright © 2017 University of Illinois at Urbana-Champaign

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Vol.61 • No. 3-4 • Fall and Winter 2017
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