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In this work we undertake an extension of various aspects of the potential theory of Dirichlet forms from locally compact spaces to noncommu- tative C  -algebras with trace. In particular we introduce finite-energy states, potentials and multipliers of Dirichlet spaces. We prove several results among which the celebrated Deny's embedding theorem and the Deny's inequality, the fact that the carre du champ of bounded potentials are finite-energy func- tionals and the relative supply of multipliers. 1. Introduction and description of the results. In the present work we develop further the potential theory of Dirichlet forms on noncommutative C ∗ -algebras with trace. We introduce and investigate finite- energy states, potentials and multipliers, objects naturally associated to Dirichlet spaces and which are meant to encode or reveal the geometric nature of the latter. In a companion work the results here obtained will be crucial to construct on C ∗ - algebras endowed with a Dirichlet form, the building blocks of a metric differential geometry (Dirac operators and Spectral Triples) and topological invariants (sum- mable Fredholm modules in K-homology) in the framework of the Noncommutative Geometry developed by A. Connes (Co). Classical potential theory, studying harmonic functions on Euclidean spaces R n , finite-energy measures and their potentials, was based on the properties of kernel |x − y| −1 , the so called Green function, to understood as the integral kernel of the

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