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

TL;DR

This paper extends potential theory of Dirichlet forms to noncommutative C*-algebras, introducing finite-energy states, potentials, and multipliers, and proves key theorems like Deny's embedding and inequality.

Contribution

It introduces noncommutative analogues of potential theory concepts and proves fundamental results in this new setting.

Findings

Proved Deny's embedding theorem in noncommutative setting

Established that the carré du champ of bounded potentials are finite-energy functionals

Demonstrated the relative abundance of multipliers in the noncommutative framework

Abstract

In this work we undertake an extension of various aspects of the potential theory of Dirichlet forms from locally compact spaces 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 the celebrated Deny's embedding theorem and the Deny's inequality, the fact that the carre' du champ of bounded potentials are finite-energy functionals and the relative supply of multipliers.