arXiv subjects
Publications and source records attributed to Klaus Thomsen.
We study the KMS states and $KMS_{\infty}$ states of generalized gauge actions on the $C^*$-algebra of a pointed Cayley graph. Our results provide information for any finitely generated group, but they are only complete for nilpotent groups.
KMS weights for generalized gauge actions on graph C*-algebras are studied and a complete description of the structure is obtained for the gauge action when the graph is strongly connected and has at most countably many exits. The structure is surprisingly rich.
The paper contains a description of a connection between diagonal actions and certain KMS weights on groupoid $C^{*}$-algebras. It furthermore contains the realization of a graph $C^{*}$-algebra of a countable graph as the groupoid $C^{*}$-algebra of a local homeomorphism and applies the theory obtained on these algebras.
We exhibit a one-parameter group of automorphism on the Cuntz-algebra O_2 such that the simplex of KMS states changes abruptly at a certain critical temperature from infinitely many to one, and then none. The factor types of the extremal KMS states are also determined.
We determine the factor types of the extremal KMS weights for generalized gauge actions on a graph algebra, and the ground states for the restriction of the action to a corner defined from a vertex. The assumptions on the graph and the action varies a bit, but they all cover, at the very least, the case of the gauge action on a strongly connected graph with at most countably many exits. These are the graphs for which all the KMS weights of the gauge action were determined in the first part of this work.
The paper develops the general theory for the items in the title, assuming that the matrix is countable and cofinal.
The paper introduces a general method to construct conformal measures for a local homeomorphism on a locally compact non-compact Hausdorff space, subject to mild irreducibility-like conditions. Among others the method is used to give necessary and sufficient conditions for the existence of eigenmeasures for the dual Ruelle operator associated to a locally compact non-compact irreducible Markov shift equipped with a uniformly continuous potential function. As an application to operator algebras the results are used to determine for which $β$ there are gauge invariant $β$-KMS weights on a simple graph $C^*$-algebra when the one-parameter automorphism group is given by a uniformly continuous real-valued function on the path space of the graph.
The paper contains a description of the KMS weights for the one-parameter action on the reduced C*-algebra of a second countable locally compact Hausdorff etale groupoid, arising from a continuous real valued homomorphism satisfying two conditions. The results are subsequently applied to identify the KMS weights for the gauge action on a simple graph algebra. The von Neumann algebra generated by the GNS-representation of an extremal KMS weight is a factor, and tools are developed to determine its type. The paper concludes with three examples to illustrate the results.
We describe the KMS-states and the ground states for the gauge action on the C*-algebra of the oriented transformation groupoid of a continuous piecewise monotone and exact map of the circle.
We consider a construction of C*-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the C*-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the K-theory groups and turn them into an algorithmic method for Markov maps.
From a non-constant holomorphic map on a connected Riemann surface we construct an 'etale second countable locally compact Hausdorff groupoid whose associated groupoid C*-algebra admits a one-parameter group of automorphisms with the property that its KMS states corresponds to conformal measures in the sense of Sullivan. In this way certain quadratic polynomials give rise to quantum statistical models with a phase transition arising from spontaneous symmetry breaking.
We study the C*-algebra of an affine map on a compact abelian group and give necessary and sufficient conditions for strong transitivity when the group is a torus. The structure of the C*-algebra is completely determined for all strongly transitive affine maps on a torus of dimension one, two or three.
This paper contains a quite detailed description of the C*-algebra arising from the transformation groupoid of a rational map of degree at least two on the Riemann sphere. The algebra is decomposed stepwise via extensions of familiar C*-algebras whose nature depend on the structure of the Julia set and the stable regions in the Fatou set, as well as on the behaviour of the critical points.
The complex exponential function is a local homeomorphism and gives therefore rise to an 'etale groupoid and a C*-algebra. We determine this algebra, as well as the alge bra of the complex conjugate of the exponential function.
It is shown that the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism is purely infinite when it is simple, provided the endomorphism does not leave a trace state invariant and maps the unit to a full projection.
We obtain a description of the C*-algebras which can occur as a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension.
Let $A$, $A'$ be separable $C^*$-algebras, $B$ a stable $σ$-unital $C^*$-algebra. Our main result is the construction of the pairing $[[A',A]]\times\operatorname{Ext}^{-1/2}(A,B)\to\operatorname{Ext}^{-1/2}(A',B)$, where $[[A',A]]$ denotes the set of homotopy classes of asymptotic homomorphisms from $A'$ to $A$ and $\operatorname{Ext}^{-1/2}(A,B)$ is the group of semi-invertible extensions of $A$ by $B$. Assume that all extensions of $A$ by $B$ are semi-invertible. Then this pairing allows us to give a condition on $A'$ that provides semi-invertibility of all extensions of $A'$ by $B$. This holds, in particular, if $A$ and $A'$ are shape equivalent. A similar condition implies that if $\operatorname{Ext}^{-1/2}$ coincides with $E$-theory (via the Connes-Higson map) for $A$ then the same holds for $A'$.
We prolonge the list of C*-algebras for which all extensions by any stable separable C*-algebra are semi-invertible. In particular, we handle certain amalgamations, both of C*-algebras and of groups. Concerning groups we consider both reduced and full group C*-algebras.