KMS States on the Operator Algebras of Reducible Higher-Rank Graphs

Abstract

We study the equilibrium or KMS states of the Toeplitz C * -algebra of a finite higher-rank graph which is reducible. The Toeplitz algebra carries a gauge action of a higher-dimensional torus, and a dynamics arises by choosing an embedding of the real numbers in the torus. Here we use an embedding which leads to a dynamics which has previously been identified as "preferred", and we scale the dynamics so that 1 is a critical inverse temperature. As with 1-graphs, we study the strongly connected components of the vertices of the graph. The behaviour of the KMS states depends on both the graphical relationships between the components and the relative size of the spectral radii of the vertex matrices of the components.We test our theorems on graphs with two connected components. We find that our techniques give a complete analysis of the KMS states with inverse temperatures down to a second critical temperature β c < 1.

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

“…Composing these gauge actions with an embedding t → e itr of R in T k gives actions α r of R on C * (Λ) and T C * (Λ) (the dynamics). Then one naturally wonders about the KMS states of these dynamics, and, as usual, the results turn out to be interesting [9,10,6,7].…”

“…In [10], we studied the KMS states of C * (Λ) for periodic irreducible graphs, and we found that the behaviour of the KMS states at the critical inverse temperature 1 is dictated in a very concrete way by the periodicity [10,Theorem 7.1]. We recently investigated graphs with several irreducible components which are themselves aperiodic [11,7]. We were pleasantly surprised that we were able to get rather complete descriptions of the KMS states for 1-graphs [11,Theorem 5.3], and that we were then able to extend many of the key arguments of [11] to higher-rank graphs.…”

“…Unfortunately, even for the preferred dynamics, we ran into new difficulties when the graph had three components. The program of [7] involves reducing problems to smaller graphs by describing what happens when we remove hereditary components. The Toeplitz algebras of these smaller graphs are quotients of the Toeplitz algebra T C * (Λ).…”

KMS states on the Toeplitz algebras of higher-rank graphs

“…For a finite k-graph Λ, a Toeplitz-Cuntz-Krieger Λ-family consists of partial isometries {S λ : λ ∈ Λ} subject to the conditions: [3,6,15]. The Toeplitz algebra T C * (Λ) of Λ is then the C * -algebra generated by a universal Toeplitz-Cuntz-Krieger Λ-family.…”

Symmetries of the KMS simplex

“…This is also the case for the articles about KMS states on C * -algebras of higher rank graphs that have appeared the last several years, e.g. [4], [5], [6] and [7]. In [7] the authors come to the conclusion that the simplex of KMS states for the C * -dynamical systems they consider is "highly symmetric" in the sense that there is an abelian group that acts transitively and freely on the extremal points of the simplex.…”