The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert C-module l2(A)

We characterize A -linear symmetric and contraction module operator semigroup { T t } t ∈ℝ+ ⊂ L ( l 2 ( A )), where A is a finite-dimensional C *-algebra, and L ( l 2 ( A )) is the C *-algebra of all adjointable module maps on l 2 ( A ). Next, we introduce the concept of operator-valued quadratic forms, and give a one to one correspondence between the set of non-positive definite self-adjoint regular module operators on l 2 ( A ) and the set of non-negative densely defined A -valued quadratic forms. In the end, we obtain that a real and strongly continuous symmetric semigroup { T t } t ∈ℝ+ ⊂ L ( l 2 ( A )) being Markovian if and only if the associated closed densely defined A -valued quadratic form is a Dirichlet form.