  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.
