Hypercontractivity in finite-dimensional matrix algebras

Abstract

We obtain hypercontractivity estimates for a large class of semigroups defined on finite-dimensional matrix algebras Mn. These semigroups arise from Poisson-like length functions Ψ on Zn × Zn and provide new hypercontractive families of quantum channels when Ψ is conditionally negative. We also study the optimality of our estimates.