Generalized<i>s</i>-numbers of<i>τ</i>-measurable operators

We give a self-contained exposition on generalized s-numbers of -nieasurable operators affiliated with a semi-finite von Neumann algebra. As applications, dominated convergence theorems for a gage and convexity (or concavity) inequalities are investigated. In particular, relation between the classical //-norm inequalities and inequalities involving generalized s-numbers due to A. Grothendieck, J. von Neumann, H. Weyl and the first named author is clarified. Also, the Haagerup L pspaces (associated with a general von Neumann algebra) are considered. Haagerup L^-spaces "from scratch." Proofs are known, but it may not be without interest. In fact, some false proofs exist in the literature.

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