KMS Dirichlet forms, coercivity and superbounded Markovian semigroups on von Neumann algebras

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)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(M)$$\\end{document} and the associated noncommutative Lp(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p(M)$$\\end{document} 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.