# Journal of Operator Theory

Volume 83, Issue 1, Winter 2020 pp. 139-177.

A program for finding all KMS states on the Toeplitz algebra of a higher-rank graph

**Authors**:
James Fletcher (1), Astrid an Huef (2), Iain Raeburn (3)

**Author institution:**(1) School of Mathematics and Statistics, Victoria
University of Wellington, Wellington, 6140, New Zealand

(2) School of Mathematics and Statistics, Victoria University of Wellington,
Wellington, 6140, New Zealand

(3) School of Mathematics and Statistics, Victoria University of Wellington, Wellington, 6140, New Zealand

**Summary: **The Toeplitz algebra of a finite graph of rank $k$ carries a natural action of the torus $\mathbb{T}^k$, and composing with an embedding of $\mathbb{R}$ in $\mathbb{T}^k$ gives a dynamics on the Toeplitz algebra. In this paper we describe a program for determining the simplex of KMS states associated to this dynamics at all inverse temperatures. For inverse temperatures larger than a critical value, the KMS states are well-understood, and this analysis is the first step in our program. At the critical inverse temperature, much less is known, and the second step in our program is an analysis of the KMS states at the critical value. This is the main technical contribution of the present paper. The third step in our program shows that the problem of finding the KMS states at inverse temperatures less than the critical value is equivalent to our original problem for a smaller graph. We test our program on a wide range of examples, including a very general family of graphs with three strongly connected components.

**DOI: **http://dx.doi.org/10.7900/jot.2018aug23.2202

**Keywords: **$C^*$-algebras, higher-rank graphs, KMS states

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