Phase transition on Exel crossed products assocaited to dilation matrices

Abstract

An integer matrix $A\in M_d(\mathbb{Z})$ induces a covering ${\sigma }_A$ of $\T^d$ and an endomorphism ${\alpha }_A:f\mapsto f\circ {\sigma }_A$ of $C(\T^d)$ for which there is a natural transfer operator $L$. In this paper, we compute the KMS states on the Exel crossed product $C(\T^d)\rtimes_{α_A,L}\N$ and its Toeplitz extension. We find that $C(\T^d)\rtimes_{α_A,L}\N$ has a unique KMS state, which has inverse temperature $\beta =\log |\det A|$. Its Toeplitz extension, on the other hand, exhibits a phase transition at $\beta =\log |\det A|$, and for larger $\beta$ the simplex of KMS${}_{\beta }$ states is isomorphic to the simplex of probability measures on $\T^d$.

Cited by in corpus (2)

• Exel and Stacey crossed products, and Cuntz-Pimsner algebras
• Phase transitions on the Toeplitz algebras of Baumslag-Solitar semigroups