Modular Completely Dirichlet forms as Squares of Derivations

TL;DR

This paper establishes a connection between closable derivations on GNS Hilbert spaces and GNS-symmetric semigroups of completely positive maps on von Neumann algebras, advancing the understanding of non-tracial weights.

Contribution

It introduces a novel framework linking derivations to symmetric semigroups in the context of non-tracial weights on von Neumann algebras.

Findings

Derivations induce GNS-symmetric semigroups

Semigroups are contractive and completely positive

Framework applies to non-tracial weights

Abstract

We prove that certain closable derivations on the GNS Hilbert space associated with a non-tracial weight on a von Neumann algebra give rise to GNS-symmetric semigroups of contractive completely positive maps on the von Neumann algebra.