Measures and Dirichlet forms under the Gelfand transform

Using the standard tools of Daniell–Stone integrals, Stone–Čech compactification and Gelfand transform, we show explicitly that any closed 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–Čech 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.