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Sharp uniform convexity and smoothness inequalities for trace norms. (English) Zbl 0803.47037
Summary: 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 $$L_ p$$.

##### MSC:
 47L20 Operator ideals 46B20 Geometry and structure of normed linear spaces 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras
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##### References:
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