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Dirichlet Forms on Noncommutative Spaces

  • Fabio Cipriani
Chapter
Part of the Lecture Notes in Mathematics book series (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.

Keywords

Dirichlet Form Continuous Semigroup Dirichlet Space Markovian Semigroup Cyclic Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Fabio Cipriani
    • 1
  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

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