  We study relations between non-commutative Ruelle transfer operators over the
C$^*$-algebra $B(\mathcal{H})$ of linear bounded operators over separable
Hilbert spaces $\mathcal{H}$ (infinite-dimensional) and other completely
positive maps. Transfer operators possess a simple description in terms of the
so called non-commutative Radon-Nikodym derivatives. We describe the problem of
existence of a largest positive eigenvalue associated to a positive
eigenfunction and uniform convergence of sequences of iterates of transfer
operators over $B(\mathcal{H})$. Part of the proof related to the
Ruelle-Perron-Frobenius theorem is obtained by adapting results from quantum
spin chain analysis.
