This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on two aspects: vector valued feature maps and universal kernels. In particular we characterize the structure of translation invariant kernels on abelian groups and we relate it to the universality problem.

For a quantum Markov semigroup T on the algebra B(h) with a faithful invariant state ρ, we can define an adjoint T with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterising adjoints T that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H,We study the adjoint semigroup with respect to both scalar products a, b = tr(ρa * b) and a, b = tr(ρ 1/2 a * ρ 1/2 b).

We find the structure of generators of norm continuous quantum Markov semigroups on B(h) that are symmetric with respect to the scalar product tr(ρ 1/2 x * ρ 1/2 y) induced by a faithful normal invariant state invariant state ρ and satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the socalled standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.

The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra is established providing a natural decomposition of a Markovian open quantum system into its noiseless (decoherencefree) and irreducible (ergodic) components. This leads to a new characterisation of the structure of invariant states and a new method for finding decoherence-free subsystems and subspaces. Examples are presented to illustrate these results.