Relative Entropy and Fisher Information

We show that in finite dimension the set of generates satisfying a stable version of the log-sobolev inequality for the Fisher information is dense. The results is based on  a new algebraic property , valid for subordinates semigroups  for sublabplacians  on compact Riemann manifolds which is then transferred to matrix algebras. Even in the commutative setting the inequalities for subordinated sublaplacians are entirely new. We also found counterexample for why a naive approach via hypercontractivity is not expected to work in a matrix-valued setting, similar  to results by Bardet and collaborators.