Noncommutative Sobolev type inequalities

Abstract

In this thesis, we study Sobolev type inequalities in the noncommutative (quantum) settings. We establish the abstract Bakry-\'Emery criterion for the operator-valued $f$-Sobolev inequalities on the derivation triple. We recapture the celebrated Bakry-\'Emery theorem for operator-valued functions on Riemannian manifolds. We discuss examples including noncommutative $f$-Sobolev inequalities on the intervals, Lindblad operators of finite dimensional matrix algebras, and discrete graphs. By generalizing the monotone metrics in the space of quantum states, we develop a deeper understanding of $f$-Sobolev inequalities in the noncommutative setting.