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 C p, which are the analogs of the Lebesgue spaces L p in non-commutative integration. The inequalities are all precise analogs of results which had been known in L p, but were only known in C p for special values of p. In the course of our treatment of uniform convexity and smoothness inequalities for C p we obtain new and simple proofs of the known inequalities for L p.