Spatial realisations of KMS states on theC⁎-algebras of higher-rank graphs

Abstract

Several authors have recently been studying the equilibrium or KMS states on the Toeplitz algebras of finite higher-rank graphs. For graphs of rank one (that is, for ordinary directed graphs), there is a natural dynamics obtained by lifting the gauge action of the circle to an action of the real line. The algebras of higher-rank graphs carry a gauge action of a higher-dimensional torus, and there are many potential dynamics arising from different embeddings of the real line in the torus. Previous results show that there is nonetheless a "preferred dynamics" for which the system exhibits a particularly satisfactory phase transition, and that the unique KMS state at the critical inverse temperature can then be implemented by integrating vector states against a measure on the infinite path space of the graph. Here we obtain a similar description of the KMS state at the critical inverse temperature for other dynamics. Our spatial implementation is given by integrating against a measure on a space of paths which are infinite in some directions but finite in others. Our results are sharpest for the algebras of rank-two graphs.

KMS states on the Toeplitz algebras of higher-rank graphs

“…Furthermore the continuous one-parameter group α r corresponds to the one arising from the groupoid homomorphism c r (x, n, y) = r·n in the groupoid picture of T C * (Λ), and the topology on G Λ makes the map Φ : G Λ → Z k given by Φ(x, n, y) := n continuous. Since the definition of the topology on G Λ is not crucial for our exposition, we refer the reader to Appendix B in [5] or [17] for the details.…”

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

“…If α r is the preferred dynamics, then r ∈ (0, ∞) k implies that ρ(A i ) > 1 for 1 ≤ i ≤ k; this rules out the case in which some coordinate graph is a disjoint union of cycles (see [11,Lemma A.1]). The main results of [12,13] concern the preferred dynamics, and in [10] we studied more general dynamics satisfying (4.1).…”

“…A more general result is stated in [10, Proposition 4.1], but unfortunately that result is not true as stated: if Λ is reducible, it is quite possible that there are KMS β states with β < 1 (see §7.3 below). We discuss the flawed proof in [10] in the following remark.…”

“…Then Λ Λ\D = Λ C . At this point we have a complete analysis of the KMS states of (T C * (Λ), α r ) for β > β c because we know all the KMS states of the coordinatewise irreducible graph Λ C from [10].…”

Wavelets and Graph C ∗-Algebras

“…since the vectors {c m,v } form an orthonormal set with respect to the inner product (23). Thus, the functions {f m,v } are an orthonormal set in L 2 (Λ ∞ , M).…”