Graph Hörmander Systems

Abstract

This paper extends the Bakry-Émery theorem connecting the Ricci curvature and log-Sobolev inequalities to the matrix-valued setting. Using tools from noncommuative geometry, it is shown that for a right invariant second order differential operator on a compact Lie group, a lower bound for a matrix-valued modified log-Sobolev inequality is equivalent to a uniform lower bound for all finite dimensional representations. Using combinatorial tools, we obtain computable lower bounds for matrix-valued log-Sobolev inequalities of graph-Hörmander systems using combinatorial methods.

References in corpus (4)

• Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps
• Poincaré type inequalities on the discrete cube and in the CAR algebra
• A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy
• On the Boltzmann Equation for Weakly Nonlinear Wave Equations

Cited by in corpus (5)

• Complete entropic inequalities for quantum Markov chains
• Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy
• Complete Sobolev Type Inequalities
• Ricci curvature of quantum channels on non-commutative transportation metric spaces
• Geometric Approach Towards Complete Logarithmic Sobolev Inequalities