Measures and Dirichlet forms under the Gelfand transform

Abstract

Using the standard tools of Daniell-Stone integrals, Stone-\v{C}ech compactification and Gelfand transform, we discuss how any Dirichlet form defined on a measurable space can be transformed into a regular Dirichlet form on a locally compact space. This implies existence, on the Stone-\v{C}ech compactification, of the associated Hunt process. As an application, we show that for any separable resistance form in the sense of Kigami there exists an associated Markov process.