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Dirichlet Forms on Noncommutative Spaces

  • Chapter
Quantum Potential Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1954))

Abstract

We show how Dirichlet forms provide an approach to potential theory of noncommutative spaces based on the notion of energy. The correspondence with KMS-symmetric Markovian semigroups is explained in details and applied to the dynamical approach to equilibria of quantum spin systems. Second part focuses on the differential calculus underlying a Dirichlet form. Applications are given in Riemannian Geometry to a potential theoretic characterization of spaces with positive curvature and to the construction of Fredholm modules in Noncommutative Geometry.

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  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-20133, Milano, Italy

    Fabio Cipriani

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  1. Fabio Cipriani
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  1. Département de mathématiques de Besancon, Université de Franche-Comté, 16, route de Gray, 25030, Besancon cedex, France

    Uwe Franz

  2. Graduate School of Information Sciences, Tohoku University, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai, 980-8579, Japan

    Uwe Franz

  3. Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität Greifswald, Jahnstrasse 15a, 17487, Greifswald, Germany

    Michael Schürmann

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Cipriani, F. (2008). Dirichlet Forms on Noncommutative Spaces. In: Franz, U., Schürmann, M. (eds) Quantum Potential Theory. Lecture Notes in Mathematics, vol 1954. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69365-9_5

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