Logarithmic Sobolev inequality were introduced by Gross as a way to measure the smoothing properties of Markov semigroups. Log-Sobolev inequality is stable under tensorization, which makes it a basic ingredient in the study of concentration inequality and the mixing time for product systems. Nevertheless, in the noncommutative case where the underlying function space is replaced by a non-commutative operator algebra, it is not known that log-Sobolev inequality in general tensorizes. In this talk, I will discuss an approach to tensor stable (modified) log-Sobolev inequalities via the gradient forms, and how this approach applies to semigroups on Matrix algebras and finite graphs. Based on joint works with Marius Junge and Nicholas LaRacuente.
