KMS Dirichlet forms, coercivity and super-bounded Markovian semigroups on von Neumann algebras

TL;DR

This paper constructs Dirichlet forms on von Neumann algebras linked to the Araki modular Hamiltonian, analyzes their properties, introduces superboundedness, and applies these concepts to quantum Ornstein-Uhlenbeck semigroups.

Contribution

It introduces a novel construction of Dirichlet forms associated with eigenvalues of the Araki modular Hamiltonian and explores superboundedness of related semigroups.

Findings

Established coercivity bounds for the Dirichlet forms.

Proved superboundedness for certain positivity preserving semigroups.

Applied the framework to quantum Ornstein-Uhlenbeck semigroups of CCR

Abstract

We introduce a construction of Dirichlet forms on von Neumann algebras M associated to any eigenvalue of the Araki modular Hamiltonian of a faithful normal non tracial state, providing also conditions by which the associated Markovian semigroups are GNS symmetric. The structure of these Dirichlet forms is described in terms of spatial derivations. Coercivity bounds are proved and the spectral growth is derived. We introduce a regularizing property of positivity preserving semigroups (superboundedness) stronger than hypercontractivity, in terms of the symmetric embedding of M into its standard space L2(M) and the associated noncommutative Lp(M) spaces. We prove superboundedness for a special class of positivity preserving semigroups and that some of them are dominated by the Markovian semigroups associated to the Dirichlet forms introduced above, for type I factors M. These tools are applied to a general construction of the quantum Ornstein-Uhlembeck semigroups of the Canonical Commutation Relations CCR and some of their non-perturbative deformations.