Study of noncommutative analysis and free probability theory on operator algebras

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We studied operator theory and operator algebra theory on Hubert spaces with applications to noncommutative probability theory. In recent years, free probability theory initiated by D.Voiculescu has been extensively developed as a branch of noncommutative probability theory. We investigated large deviation theory for random matrices and free relative entropy related to free probability theory. We investigated automorphsims on (interpolated) free group factors by the method of free products. Moreover, we studied norm inequalities for means of operators and obtained a number of inequalities extending the arithmetic-geometric norm inequalities for operators. More concretely,(1)We obtained a new norm inequality for matrices involving operator monotone functions and unitarily invariant norms.(2)We investigated free relative entropy extending free entropy into two variables We gave its two definitions by double logarithmic integral and by matrix approximation, and proved their coincidence by making use of large deviation for the empirical eigenvalue distribution of relevant random matrices.(3)We showed that there are continuously many different aperiodic automorphisms on each (interpolated) free group factor by the method of free products and clarified the structure of their crossed-products.(4)Based on theory of double integral transformation and the method of Schur multipliers, we obtained norm inequalities for unitarily invariant nroms of operators, which strengthen the arithmetic-geometric mean inequality.(5)We gave an inequality for positive semidefinite matrices, comparing submultiplicativity with subadditivity for unitarily invariant norms.(6)We re-proved the free probabilistic analog of Talagrand transportation cost inequality for measures due to Biane and Voiculescu by means of random matrix approximation.