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

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 $l_2 \bar \otimes A$.