We certainly tried to formulate Ricci curvature for von Neumann or matrix algebras, but it turns out not to work. In this talk I will review the connection between Wasserstein distance-entropy inequalities and concentration, and how to circumvent Ricci curvature for log-Sobolev inequalities. Order inequalities and Lipschitz norm replace (some of) the Ricci-curvature tools in a noncommutative Bakry-Emry program.
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