Sup-norm-closable bilinear forms and Lagrangians

TL;DR

This paper studies the conditions under which symmetric bilinear forms on function algebras are closable in the supremum norm, linking them to Dirichlet forms and energy measures, with implications for functional analysis.

Contribution

It establishes new conditions for supremum-norm closability of bilinear forms and connects them to Dirichlet forms and energy measures, extending previous work to more general settings.

Findings

Any Dirichlet form induces a sup-norm closable bilinear form.

Under mild conditions, such bilinear forms admit finitely additive energy measures.

With an energy dominant measure, they can be turned into Dirichlet forms with a carré du champ.

Abstract

We consider symmetric non-negative definite bilinear forms on algebras of bounded real valued functions and investigate closability with respect to the supremum norm. In particular, any Dirichlet form gives rise to a sup-norm closable bilinear form. Under mild conditions a sup-norm closable bilinear form admits finitely additive energy measures. If, in addition, there exists a (countably additive) energy dominant measure, then a sup-norm closable bilinear form can be turned into a Dirichlet form admitting a carr\'e du champ. Moreover, we can always transfer the bilinear form to an isometrically isomorphic algebra of bounded functions on the Gelfand spectrum, where these measures exist. Our results complement a former closability study of Mokobodzki for the locally compact and separable case.