The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert \(W^\ast\)-module
type: Item
instance of: scholarly article
title: The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert \(W^\ast\)-module
zbMATH Open document ID: 1287.47055
author: Lunchuan Zhang
author: Maozheng Guo
publication date: 2013-05-14T00:00:00Z
review text: Based on the Hilbert \(W^*\)-module \(l_2\otimes A\), the authors introduce the concept of operator-valued quadratic form, characterize the positive cone and order relation on \(l_2\otimes A\), and establish a one to one correspondence between the set of positive self-adjoint regular module operators on \(l_2\otimes A\) and the set of regular quadratic forms, where \(A\) is a finite and \(\sigma\)-finite von Neumann algebra. They also show that a strict continuous symmetric regular module operator semigroup \(\{T_t\}_{t \in \mathbb{R}^+} \subset \mathcal{L}(l_2\otimes A)\) is Markovian if and only if the associated \(A\)-valued quadratic form is a Dirichlet form, where \(\mathcal{L}(l_2\otimes A)\) is the von Neumann algebra of all adjointable module maps on \(l_2\otimes A\).
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47L05
Mathematics Subject Classification ID: 46C50
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 6162871
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: Hilbert \(W^\ast\)-module
zbMATH Keywords: quantum Markov semigroup
zbMATH Keywords: operator-valued Dirichlet forms
MaRDI profile type: Publication
full work available at URL: s10114-013-2262-5
OpenAlex ID: W2080085029
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DOI: 10.1007/S10114-013-2262-5
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6162871