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Naimark’s problem for graph \(C^\ast\)-algebras. (English) Zbl 1414.46043
Naĭmark proved that any two irreducible representations of \(K(\mathcal{H})\) are unitarily equivalent [M. A. Naĭmark, Usp. Mat. Nauk 3, No. 5(27), 52–145 (1948; Zbl 0036.07701)], and later asked whether this property characterizes \(K(\mathcal{H})\) up to an isomorphism [M. A. Naĭmark, Usp. Mat. Nauk 6, No. 6(46), 160–164 (1951; Zbl 0045.21402)]. To be precise, Naĭmark’s problem is stated as follows: “If \(A\) is a \(C^*\)-algebra with only one irreducible representation up to unitary equivalence, is \(A\) isomorphic to \(K(\mathcal{H})\) for some (non necessarily separable) Hilbert space \(\mathcal{H}\)?”
From the point of view of a classical approach, the problem asks to what extent the isomorphism class of a \(C^*\)-algebra is determined by its representation theory. From a modern point of view, it can be seen as an early inquiry about classification of \(C^*\)-algebras: must a (non necessarily separable) \(C^*\)-algebra that is Morita equivalent to \(K(\mathcal{H})\) for some (non necessarily separable) Hilbert space \(\mathcal{H}\) be in fact isomorphic to \(K(\mathcal{H})\)?
From 1951 on, various partial solutions to Naĭmark’s problem were obtained, due to I. Kaplansky [Trans. Am. Math. Soc. 70, 219–255 (1951; Zbl 0042.34901)], A. Rosenberg [Am. J. Math. 75, 523–530 (1953; Zbl 0053.25903)], J. M. G. Fell [Ill. J. Math. 4, 221–230 (1960; Zbl 0094.09702)], J. Dixmier [Bull. Soc. Math. Fr. 88, 95–112 (1960; Zbl 0124.32403)] and J. Glimm [Ann. Math. (2) 73, 572–612 (1961; Zbl 0152.33002)]. On the other hand, Akemann and Weaver, using Jensen’s diamond axiom, constructed a counterexample to Naĭmark’s problem generated by \(\aleph_1\) elements [C. Akemann and N. Weaver, Proc. Natl. Acad. Sci. USA 101, No. 20, 7522–7525 (2004; Zbl 1064.46034)]; moreover, they showed that the existence of such a counterexample is independent of ZFC (the independence of Naĭmark’s problem from ZFC remains unknown). In view of this result, it is reasonable to consider restrictions of the problem to particular classes of \(C^*\)-algebras.
In the paper under review, the authors consider the problem when restricted to the class of graph \(C^*\)-algebras over (non necessarily countable) directed graphs. In this case, they show that the existence of a unique irreducible representation (up to unitary equivalence) is tightly related to the algebra being AF (Propositions 3.3 and 3.5), and they conclude that the answer to Naĭmark’s question is affirmative in the context of graph \(C^*\)-algebras (Theorems 4.1 and 4.2).

MSC:
46L35 Classifications of \(C^*\)-algebras
46K10 Representations of topological algebras with involution
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References:
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