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

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module l2..A, and give a one to one correspondence between the set of positive self-adjoint regular module operators on l2..A and the set of regular quadratic forms, where A is a finite and .-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup {Tt}t.R+.L(l2..A) is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L(l2..A) is the von Neumann algebra of all adjointable module maps on l2..A.