Noncommutative Sobolev type inequalities

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Li, Haojian

Contributors dc:contributor

• Junge, Marius
• Boca, Florin
• Oikhberg, Timur
• Leditzky, Felix

• Sobolev inequalities
• noncommutative analysis
• open quantum system

Statement dc:rights

• Copyright 2021 Haojian Li

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/113210

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/113210