Acta Mathematica Sinica, English Series

, Volume 29, Issue 5, pp 857–866 | Cite as

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert W*-module

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Abstract

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module \(l_2 \bar \otimes A\), and give a one to one correspondence between the set of positive self-adjoint regular module operators on \(l_2 \bar \otimes A\) and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup \(\{ T_t \} _{t \in \mathbb{R}^ + } \subset L(l_2 \bar \otimes A)\) is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L(\(l_2 \bar \otimes A\)) is the von Neumann algebra of all adjointable module maps on \(l_2 \bar \otimes A\).

Keywords

Hilbert W*-module quantum Markov semigroup operator-valued Dirichlet forms 

MR(2010) Subject Classification

47L05 46C50 
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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Information ScienceRenmin University of ChinaBeijingP. R. China
  2. 2.School of Mathematical SciencePeking UniversityBeijingP. R. China

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