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 \(l_2 \bar \otimes A\).
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The first author is supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Grant No. 10XNJ033, “Study of Dirichlet forms and quantum Markov semigroups based on Hilbert C*-modules”)
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Zhang, L.C., Guo, M.Z. The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert W*-module. Acta. Math. Sin.-English Ser. 29, 857–866 (2013). https://doi.org/10.1007/s10114-013-2262-5
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DOI: https://doi.org/10.1007/s10114-013-2262-5