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

Authors: Fabio Cipriani and Jean-Luc Sauvageot
Journal: Trans. Amer. Math. Soc. 367 (2015), 4837-4871
MSC (2010): Primary 46L57, 46L87
Published electronically: February 12, 2015
MathSciNet review: 3335402
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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$ 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.

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Additional Information

Fabio Cipriani
Affiliation: Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy

Jean-Luc Sauvageot
Affiliation: Institut de Mathématiques, CNRS-Université Pierre et Marie Curie, Université Paris VII, boite 191, 4 place Jussieu, F-75252 Paris Cedex 05
Address at time of publication: Institut de Mathématiques, CNRS-Université Denis Diderot, F-75205 Paris Cedex 13, France

Keywords: C$^*$ and von Neumann algebras, Dirichlet space multiplier, finite-energy functional, potential, carr\'e du champ.
Received by editor(s): March 22, 2013
Published electronically: February 12, 2015
Additional Notes: This work was supported by Italy I.N.D.A.M. – France C.N.R.S. G.D.R.E.-G.R.E.F.I. Geometrie Noncommutative and by Italy M.I.U.R.-P.R.I.N. project N. 2012TC7588-003
Dedicated: Dedicated to Gabriel Mokobodzki
Article copyright: © Copyright 2015 American Mathematical Society

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