Variations in noncommutative potential theory: Finite-energy states, potentials and multipliers

Abstract

In this work we undertake an extension of various aspects of the potential theory of Dirichlet forms to noncommutative C ∗ ^* -algebras with trace. In particular we introduce finite-energy states, potentials and multipliers of Dirichlet spaces. We prove several results among which are the celebrated Deny’s embedding theorem, Deny’s inequality, the fact that the carré du champ of bounded potentials are finite-energy functionals and the fact that bounded eigenvectors are multipliers. Deny’s embedding theorem and Deny’s inequality are also crucial to prove that the algebra of finite-energy multipliers is a form core and that it is dense in A A provided the resolvent has the Feller property. Examples include Dirichlet spaces on group C ∗ ^* -algebras associated to negative definite functions, Dirichlet forms arising in free probability, Dirichlet forms on algebras associated to aperiodic tilings, Dirichlet forms of Markovian semigroups on locally compact spaces, in particular on post critically finite self-similar fractals, and Bochner and Hodge-de Rham Laplacians on Riemannian manifolds.