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C* algebras, states, finite graphs

  1. Jan 7, 2013 #1


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    I recently got (re)interested in C* algebras. Poking around, I gathered that there is some way of constructing a C* algebra corresponding to a finite graph. I'll put some links here in case anyone knows anything about this. At the moment I'm ignorant but hope to find out more.
    No idea in advance whether this leads to some interesting satisfying math, or not. The links I'm turning up are in no particularly useful order.

    Graph C*-algebras [LECTURE NOTES with elementary definitions]
    Rainer Matthes, Wojciech Szymanski notes taken by: Pawel Witkowski
    August 17, 2005
    16 pages [but much of it very technical and the last 7 pages are K-theory]

    KMS states on finite-graph C*-algebras
    Tsuyoshi Kajiwara, Yasuo Watatani
    (Submitted on 24 Jul 2010)
    We study KMS states on finite-graph C*-algebras with sinks and sources. We compare finite-graph C*-algebras with C*-algebras associated with complex dynamical systems of rational functions. We show that if the inverse temperature β is large, then the set of extreme β-KMS states is parametrized by the set of sinks of the graph. This means that the sinks of a graph correspond to the branched points of a rational funcition from the point of KMS states. Since we consider graphs with sinks and sources, left actions of the associated bimodules are not injective. Then the associated graph C*-algebras are realized as (relative) Cuntz-Pimsner algebras studied by Katsura. We need to generalize Laca-Neshevyev's theorem of the construction of KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules are not injective.

    KMS states on the C*-algebras of finite graphs
    Astrid an Huef, Marcelo Laca, Iain Raeburn, Aidan Sims
    (Submitted on 10 May 2012)
    We consider a finite directed graph E, and the gauge action on its Toeplitz-Cuntz-Krieger algebra, viewed as an action of R. For inverse temperatures larger than a critical value βc, we give an explicit construction of all the KMSβ states. If the graph is strongly connected, then there is a unique KMSβc state, and this state factors through the quotient map onto the C*-algebra C*(E) of the graph. Our approach is direct and relatively elementary.

    KMS states on C*-algebras associated to higher-rank graphs
    Astrid an Huef, Marcelo Laca, Iain Raeburn, Aidan Sims
    (Submitted on 31 Dec 2012)
    Consider a higher-rank graph of rank k. Both the Cuntz-Krieger algebra and the Toeplitz-Cuntz-Krieger algebra of the graph carry natural gauge actions of the torus Tk, and restricting these gauge actions to one-parameter subgroups of Tk, gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures β, the simplex of KMSβ states on the Toeplitz-Cuntz-Krieger algebra has dimension d one less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a critical inverse temperature βc: for β larger than βc, there is a d-dimensional simplex of KMS states; when β = βc and the one-parameter subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz-Krieger algebra. As in previous studies for k=1, our main tool is the Perron-Frobenius theory for irreducible nonnegative matrices, though here we need a version of the theory for commuting families of matrices.
    Last edited: Jan 7, 2013
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