  We study the relations between (tight) logarithmic Sobolev inequalities,
entropy decay and spectral gap inequalities for Markov evolutions on von
Neumann algebras. We prove that log-Sobolev inequalities (in the
non-commutative form defined by Olkiewicz and Zegarlinski) imply spectral gap
inequalities, with optimal relation between the constants. Furthermore, we show
that a uniform exponential decay of a proper relative entropy is equivalent to
a modified version of log-Sobolev inequalities; this entropy decay turns out to
be implied by the usual log-Sobolev inequality adding some regularity
conditions on the quadratic forms.
