Rodrigo Frausino | Searcharxiv

arXiv2022

Reconstruction of topological graphs and their Hilbert bimodules

We show that the Hilbert bimodule associated to a compact topological graph can be recovered from the C*-algebraic triple consisting of the Toeplitz algebra of the graph, its gauge action and the commutative subalgebra of functions on the vertex space of the graph. We discuss connections with work of Davidson-Katsoulis and of Davidson-Roydor on local conjugacy of topological graphs and isomorphism of their tensor algebras. In particular, we give a direct proof that a compact topological graph can be recovered up to local conjugacy from its Hilbert bimodule, present an example of nonisomorphic locally conjugate compact topological graphs with isomorphic Hilbert bimodules. We also give an elementary proof that for compact topological graphs with totally disconnected vertex space the notions of local conjugacy, Hilbert bimodule isomorphism, isomorphism of C*-algebraic triples, and isomorphism all coincide.

math.OA↗

arXiv2018

Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States

Given a 0-1 infinite matrix $A$ and its countable Markov shift $\Sigma_A$, one of the authors and M. Laca have introduced a kind of {\it generalized countable Markov shift} $X_A=\Sigma_A \cup Y_A$, where $Y_A$ is a special set of finite admissible words. For some of the most studied countable Markov shifts $\Sigma_A$, $X_A$ is a compactification of $\Sigma_A$, and always it is at least locally compact. We developed the thermodynamic formalism on the space $X_A$, exploring the connections with standard results on $\Sigma_A$. New phenomena appear, such as new conformal measures and a {\it length-type phase transition}: the eigenmeasure lives on $\Sigma_A$ at high temperature and lives on $Y_A$ at low temperature. Using a pressure-point definition proposed by M. Denker and M. Yuri for iterated function systems, we proved that the Gurevich pressure is a natural definition for the pressure function in the generalized setting. For the gauge action, the Gurevich entropy is a critical temperature for the existence of new conformal measures (KMS states) living on $Y_A$. We exhibit examples with infinitely (even uncountable) many new extremal conformal measures, undetectable in the usual formalism. We prove that conformal measures always exist at low temperatures when the potential is coercive enough. We characterized a basis of the topology of $X_A$ to study the weak$^*$ convergence of measures on $X_A$, and we show some cases where the conformal measure living on $Y_A$ converges to a conformal one living on $\Sigma_A$. We prove the equivalence among several notions of conformality for locally compact Hausdorff second countable spaces, including quasi-invariant measures for generalized Renault-Deaconu groupoids.

math-ph↗