Abstract:We give a self-contained exposition on generalized s-numbers of τ-nieasurable operators affiliated with a semi-finite von Neumann algebra.As applications, dominated convergence theorems for a gage and...We give a self-contained exposition on generalized s-numbers of τ-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.This work was completed during a stay of the first named author at the Mathematical Sciences Research Institute of Berkeley.The author is grateful to the Institute for its warm hospitalityRead More