THE CURVATURE DEFECT FUNCTIONAL

We construct a canonical bounded self-adjoint operator Rφ —the curvature operator—on the L2-gradient module H(2)∇ associated with a completely Dirichlet form on a von Neumann algebra. The construction proceeds through a rigorous Reilly-type identity and yields the operator Weitzenböck decomposition L =∇∗H∇H+Rφ at the level of self-adjoint operators via Kato’s Representation Theorem.