Entropy decay and log-Sobolev inequalities for non-commutative functions.

We investigate some contractivity properties for quantum Markov semigroups defined on the algebra B(h) of bounded operators on a separable Hilbert space h. In particular we concentrate on a possible generalization to the non-commutative context of some well-established results for commutative functions, about the relations linking hypercontractivity and log-Sobolev inequalities with a uniform exponentially fast decay of the entropy.