Dirichlet spaces as Banach algebras and applications

We study regular Dirichlet forms on locally compact Hausdorff spaces X in the framework of the theory of commutative Banach algebras. We prove that, suitably normed, the Dirichlet algebra of continuous functions vanishing at infinity in the extended domain Fe of a Dirichlet form (E,F) is a semisimple Banach algebra. This implies that two strongly local Dirichlet forms (E1,F1), (E2,F2) are quasi-equivalent if and only if they have the same domain. We describe the ideal structure of Be, showing that the algebraic K-theory. K(Be) of the Dirichlet algebra Be is isomorphic to the topological K-theory K(X). This allows the construction of Dirichlet structures on (sections of) finite-dimensional, locally trivial vector bundles over X.