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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

http://toknotes.mimuw.edu.pl/sem1/files/Rainer_kgca.pdf

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]

http://arxiv.org/abs/1007.4248

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.

http://arxiv.org/abs/1205.2194

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 β_{c}, we give an explicit construction of all the KMS_{β}states. If the graph is strongly connected, then there is a unique KMS_{βc}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.

http://arxiv.org/abs/1212.6811

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 T^{k}, and restricting these gauge actions to one-parameter subgroups of T^{k}, 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 KMS_{β}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 β_{c}: for β larger than β_{c}, there is a d-dimensional simplex of KMS states; when β = β_{c}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.

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# C* algebras, states, finite graphs

Can you offer guidance or do you also need help?

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