Traces on crossed products

We give a description of traces on C(X)\rtimes G in terms of measurable fields of traces on the C*-algebras of the stabilizers of the action of G on X.Comment: 5 pages, AMS-LaTe

Type III_1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers

We complete the analysis of KMS-states of the Toeplitz algebra of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature beta in the critical interval [1,2], the unique KMS_beta-state is of type III_1. We prove this by reducing the type classification from the Toeplitz algebra to that of the symmetric part of the Bost-Connes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of the Toeplitz algebra in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of the affine semigroup on the Nica spectrum, we can also recover all the KMS-states originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMS_beta-states for beta>2 that does not ostensibly come from an action on the C*-algebra.Comment: 15 pages, AMS-LaTe

KMS states of quasi-free dynamics on Pimsner algebras

A continuous one-parameter group of unitary isometries of a right Hilbert C*-bimodule induces a quasi-free dynamics on the Cuntz-Pimsner C*-algebra of the bimodule and on its Toeplitz extension. The restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial, and the corresponding KMS states of the Toeplitz-Cuntz-Pimsner and Cuntz-Pimsner C*-algebras are characterized in terms of traces on the algebra of coefficients. This generalizes and sheds light onto various earlier results about KMS states of the gauge actions on Cuntz algebras, Cuntz-Krieger algebras, and crossed products by endomorphisms. We also obtain a more general characterization, in terms of KMS weights, for the case in which the inducing isometries are not unitary, and accordingly, the restriction of the quasi-free dynamics to the algebra of coefficients is nontrivial

Classification of non-Kac compact quantum groups of SU(n) type

We classify up to isomorphism all non-Kac compact quantum groups with the same fusion rules and dimension function as $SU(n)$. For this we first prove, using categorical Poisson boundary, the following general result. Let $G$ be a coamenable compact quantum group and $K$ be its maximal quantum subgroup of Kac type. Then any dimension-preserving unitary fiber functor $Rep\ G \to Hilb_f$ factors, uniquely up to isomorphism, through $Rep\ K$. Equivalently, we have a canonical bijection $H^2(\hat G; T) \cong H^2(\hat K; T)$. Next, we classify autoequivalences of the representation categories of twisted $q$-deformations of compact simple Lie groups.Comment: 22 pages; v1: subsumes and strengthens the classification result from arXiv:1310.4407; v2: minor improvements, appendix corrected; v3: minor corrections, final versio

Entropy of automorphisms of II_1-factors arising from the dynamical systems theory

Let a countable amenable group G acts freely and ergodically on a Lebesgue space (X,mu), preserving the measure mu. If T is an automorphism of the equivalence relation defined by G then T can be extended to an automorphism alpha_T of the II_1-factor M=L^\infty(X,\mu)\rtimes G. We prove that if T commutes with the action of G then H(alpha_T)=h(T), where H(alpha_T) is the Connes- Stormer entropy of alpha_T, and h(T) is the Kolmogorov-Sinai entropy of T. We prove also that for given s and t, 0\le s\le t\le\infty, there exists a T such that h(T)=s and H(alpha_T)=t.Comment: LaTeX2e, 12 page