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Dirichlet forms 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 B e = C 0 (X) ∩ F e of continuous functions vanishing at infinity in the extended domain F e of a Dirichlet form (E, F) is a semisimple Banach algebra. This implies that two strongly local Dirichlet forms (E 1 , F 1 ), (E 2 , F 2 ) are quasi-equivalent (that is, c-1 E 1 ≤E 2 ≤ c E 1 for some c > 0) if and only if they have the same domain. We describe the ideal structure of B e , showing that the algebraic K-theory K * (B e ) of the Dirichlet algebra B e 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.