Quantum Potential Theory: Dirichlet Forms on Noncommutative Spaces

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