Markov semigroups KMS-symmetric for a weight
type: Item
instance of: scholarly article
title: Markov semigroups KMS-symmetric for a weight
zbMATH Open document ID: 0932.46055
author: Stanisław Goldstein
author: J. Martin Lindsay
published in: Mathematische Annalen
publication date: 2000-03-08T00:00:00Z
review text: Let \(A\) be a von Neumann algebra with faithful normal semifinite weight \(\varphi\), \(M\) be the crossed product of \(A\) with the modular automorphism group \(\sigma_t(x)= h^{it}xh^{-it}\) of \(\varphi\), and let \(S\) be an operator on \(L^p(A)\) with domain \(\text{Dom}(S)\), where \(p\in [1,\infty]\). \(S\) is called Markov on \(L^p(A)\), \(h^{(1/p)}\), or Markov with respect to \(h^{1/p}\), if (i) \([0,h^{1/p}]_{L^p}\subset \text{Dom}(S)\), (ii) \(S([0, h^{1/p}]_{L^p})\subset [0, h^{1/p}]_{L^p}\); \(S\) is Markov on \((A,\varphi)\) if \((\text{i}')\) \([0,1]_m\subset \text{Dom}(S)\), where \(m=\{x\in A:\varphi(|x|)<\infty\}\). \((\text{ii}')\) \(S([0, 1]\cap \text{Dom}(S))\subset [0,1]_A\), An operator \(T\) on \(A\) (with domain \(\text{Dom}(T)\)) is called KMS-symmetric (with respect to \(\varphi\)) if \(m\subset\text{Dom}(T)\) and \(\text{tr}(Ta: i(b))= \text{tr }i(a)\), for all \(a,b\in m\). In the paper, the authors consider Markov semigroups which are KMS-symmetric with respect to a faithful normal semifinite weight, and show that these correspond to selfadjoint contraction semigroups on its standard Hilbert space whose form generator is Dirichlet. It is reminded that a nonnegative quadratic form \(Q\) in \(L^2(A)\) is called Dirichlet (with respect to \(\varphi\)) if \(Q\cdot \pi_I\leq Q\), where \(\pi_I\) is the nearest point projection onto the closed convex set \([0,h^{1/2}]_{L^2}\).
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 1336971
zbMATH Keywords: Dirichlet form
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: faithful normal semifinite weight
zbMATH Keywords: modular automorphism group
zbMATH Keywords: KMS-symmetric
zbMATH Keywords: Markov semigroups
zbMATH Keywords: selfadjoint contraction semigroups
zbMATH Keywords: quadratic form
zbMATH Keywords: nearest point projection
reviewed by: A. A. Rakhimov
MaRDI profile type: Publication
full work available at URL: s002080050249
OpenAlex ID: W1978356724
DOI: 10.1007/S002080050249
description: scientific article; zbMATH DE number 1336971