Sup-norm-closable bilinear forms and Lagrangians

We consider symmetric nonnegative 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 carré 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.