Recent zbMATH articles in MSC 58Bhttps://zbmath.org/atom/cc/58B2021-05-28T16:06:00+00:00WerkzeugThe covariant Gromov-Hausdorff propinquity.https://zbmath.org/1459.460682021-05-28T16:06:00+00:00"Latrémolière, Frédéric"https://zbmath.org/authors/?q=ai:latremoliere.fredericSummary: We extend the Gromov-Hausdorff propinquity to a metric on Lipschitz dynamical systems which are given by strongly continuous actions of proper monoids on quantum compact metric spaces via Lipschitz morphisms. We prove that the resulting distance between two Lipschitz dynamical systems is zero if and only if there exists an equivariant full quantum isometry between them. We apply our results to convergence of the dual actions on fuzzy tori to the dual actions on quantum tori. Our framework is general enough to also allow for the study of the convergence of continuous semigroups of positive linear maps and other actions of proper monoids.Cyclic-homology Chern-Weil theory for families of principal coactions.https://zbmath.org/1459.580032021-05-28T16:06:00+00:00"Hajac, Piotr M."https://zbmath.org/authors/?q=ai:hajac.piotr-m"Maszczyk, Tomasz"https://zbmath.org/authors/?q=ai:maszczyk.tomasz.1Summary: Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern-Weil homomorphism. To realize the thus constructed Chern-Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of \textit{J.-L. Loday} [Cyclic homology. Berlin: Springer-Verlag (1992; Zbl 0780.18009)] and \textit{M. Wodzicki} [C. R. Acad. Sci., Paris, Sér. I 306, No. 9, 399--403 (1988; Zbl 0637.16014)]. We work with families of principal coactions, and instantiate our noncommutative Chern-Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of \textit{P. Feng} and \textit{B. Tsygan} [Commun. Math. Phys. 140, No. 3, 481--521 (1991; Zbl 0743.17020)] for the quantum-deformation family of the standard quantum Hopf fibrations.Entropy theory for the parametrization of the equilibrium states of Pimsner algebras.https://zbmath.org/1459.460552021-05-28T16:06:00+00:00"Kakariadis, Evgenios T. A."https://zbmath.org/authors/?q=ai:kakariadis.evgenios-t-aSummary: We consider Pimsner algebras that arise from C*-correspondences of finite rank, as dynamical systems with their rotational action. We revisit the Laca-Neshveyev classification of their equilibrium states at positive inverse temperature along with the parametrizations of the finite and the infinite parts simplices by tracial states on the diagonal. The finite rank entails an entropy theory that shapes the KMS-structure. We prove that the infimum of the tracial entropies dictates the critical inverse temperature, below which there are no equilibrium states for all Pimsner algebras. We view the latter as the entropy of the ambient C*-correspondence. This may differ from what we call strong entropy, above which there are no equilibrium states of infinite type. In particular, when the diagonal is abelian then the strong entropy is a maximum critical temperature for those. In this sense we complete the parametrization method of Laca-Raeburn and unify a number of examples in the literature.The modular Gromov-Hausdorff propinquity.https://zbmath.org/1459.460672021-05-28T16:06:00+00:00"Latrémolière, Frédéric"https://zbmath.org/authors/?q=ai:latremoliere.fredericSummary: Motivated by the quest for an analytic framework to study classes of \(\mathrm{C}^*\)-algebras and associated structures as geometric objects, we introduce a metric on Hilbert modules equipped with a generalized form of a differential structure, thus extending Gromov-Hausdorff convergence theory to vector bundles and quantum vector bundles -- not convergence as total space but indeed as quantum vector bundle. Our metric is new even in the classical picture, and creates a framework for the study of the moduli spaces of modules over \(\mathrm{C}^*\)-algebras from a metric perspective. We apply our construction, in particular, to the continuity of Heisenberg modules over quantum \(2\)-tori.