Study of noncommutative probability and noncommutative entropy

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Two major trends in recent development of operator algebras are subfactor theory initiated by V.F.R. Jones and free probability theory created by D. Voiculescu. Related with these we studied noncommutative probability theory and noncommutative entropy theory in the present research. Also, we studied norm inequalities and trace inequalities for operators and matrices. Results obtained are summarized in the following :1. When a group G is acting on an inclusion N ⊂ M of factors with finite index, the standard invariant of the crossed product N × G ⊂ M × M is compared with that of N ⊂ M. Moreover, in a joint work with Masaki Izumi (Kyoto Univ.), amenability and strong amenability of general fusion algebras are investigated on the model of the subfactor case.2. Large deviation principle for random matrices and free entropy were studied jointly with D. Petz (Hungary). We discussed maximization problems for one-variable free entropy under various constraints, and systematically investigated the large deviation principle for the empirical eigenvalue density of random matrices. Moreover, we established the relation among three types of free entropies defined for noncommutative random variables which are selfadjoint, non-selfadjoint and unitary, respectively, and applied it to the additivity and the maximization problems of free entropy.3. A series of investigations were made for norm inequalities and trace inequalities for Hilbert space operators (in particular matrices). In a joint work with Tsuyoshi Ando (Hokusei Gakuen Univ.), we investigated what is the matrix (or trace) Holder inequality. Jointly with Ando and Okubo (Hokkaido Univ. of Education) we discuss trace inequalities for multiple products of two matrices. Jointly with Hideki Kosaki (Kyushu Univ.), we obtained norm inequalities refining the arithmetic-geometric mean inequality for unitarily invariant norms.