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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Authors: | Lun Chuan Zhang Mao Zheng Guo |
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Institution: | 12262. School of Information Science, Renmin University of China, Beijing, 100090, P. R. China 22262. School of Mathematical Science, Peking University, Beijing, 100871, P. R. China
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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 $l_2 \bar \otimes A$ . |
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Keywords: | Hilbert W*-module quantum Markov semigroup operator-valued Dirichlet forms |
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