Sharp uniform convexity and smoothness inequalities for trace norms

Abstract

We prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the Schatten trace ideals Cp, which are the analogs of the Lebesgue spaces Lp in non-commutative integration. The inequalities are all precise analogs of results which had been known in Lp, but were only known in Cp for special values of p. In the course of our treatment of uniform convexity and smoothness inequalities for Cp we obtain new and simple proofs of the known inequalities for Lp. © 1993 Springer-Verlag.