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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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