Dirichlet forms as Banach algebras and applications

Fabio Cipriani

Vol. 223 (2006), No. 2, 229–249

DOI: 10.2140/pjm.2006.223.229

Abstract

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 = C₀(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 (ℰ₁,ℱ₁), (ℰ₂,ℱ₂) are quasi-equivalent (that is, c⁻¹ ℰ₁ ≤ℰ₂ ≤ cℰ₁ 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.

Keywords

Dirichlet form, Banach algebra, K-theory

Mathematical Subject Classification 2000

Primary: 31C25

Secondary: 46J10, 46J20, 60J45

Milestones

Received: 1 October 2003

Revised: 10 August 2005

Accepted: 12 August 2005

Published: 1 February 2006