The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert \(W^\ast\)-module
Based on the Hilbert \(W^*\)-module \(l_2\otimes A\), the authors introduce the concept of operator-valued quadratic form, characterize the positive cone and order relation on \(l_2\otimes A\), and establish a one to one correspondence between the set of positive self-adjoint regular module operators on \(l_2\otimes A\) and the set of regular quadratic forms, where \(A\) is a finite and \(\sigma\)-finite von Neumann algebra. They also show that a strict continuous symmetric regular module operator semigroup \(\{T_t\}_{t \in \mathbb{R}^+} \subset \mathcal{L}(l_2\otimes A)\) is Markovian if and only if the associated \(A\)-valued quadratic form is a Dirichlet form, where \(\mathcal{L}(l_2\otimes A)\) is the von Neumann algebra of all adjointable module maps on \(l_2\otimes A\).
scientific article; zbMATH DE number 6162871
scientific article