SHARP UNIFORM CONVEXITY AND SMOOTHNESS INEQUALITIES FOR TRACE NORMS

We prove several sharp inequalities specifying the uniform convexity a nd uniform smoothness properties of the Schatten trace ideals C(p), wh ich are the analogs of the Lebesgue spaces L(p) in non-commutative int egration. The inequalities are all precise analogs of results which ha d been known in L(p), but were only known in C(p) for special values o f p. In the course of our treatment of uniform convexity and smoothnes s inequalities for C(p) we obtain new and simple proofs of the known i nequalities for L(p).