# C* algebras, states, finite graphs

• marcus
In summary, the conversation discusses the topic of C* algebras and their relation to finite graphs. Various links are shared that explore this topic further, including lecture notes and research papers. The conversation also mentions the study of KMS states on these C* algebras and their connection to complex dynamical systems. The use of C* algebras in physics is also briefly mentioned.
marcus
Gold Member
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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.

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 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.

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 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.

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Thanks for the interesting reads. Wikipedia has some interesting examples of ##C^*-##algebras, which makes me wonder why they don't occur more often on our screens in PF. They are made for physics!

## 1. What are C* algebras?

C* algebras are mathematical structures that are used to study operator algebras and their representations. They are complex Banach algebras with additional properties that make them useful in the study of quantum mechanics, differential geometry, and other areas of mathematics.

## 2. What is a state in a C* algebra?

A state in a C* algebra is a linear functional that assigns a complex number to each element of the algebra. It must satisfy certain properties, such as being positive and norm-preserving. States are important because they can be used to define inner products and norms on the algebra, which are essential for studying its properties.

## 3. How are states related to finite graphs?

In the context of C* algebras, finite graphs can be represented by certain types of operator algebras known as graph C* algebras. States on these algebras can be used to define measures on the graphs, which can provide insights into their properties and behavior.

## 4. What is the significance of finite graphs in C* algebras?

Finite graphs play an important role in the study of C* algebras because they provide a concrete way to visualize and understand the algebraic structures involved. They can also provide a useful framework for studying more complex and abstract C* algebras.

## 5. Can C* algebras be applied to other fields of science?

Yes, C* algebras have applications in many areas of mathematics and science, including quantum mechanics, statistical mechanics, differential geometry, and more. They provide a powerful tool for studying and understanding complex mathematical structures and their applications in various fields.

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