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Staff Introduction

WATATANI, Yasuo / Professor

We study theory of operator algebras in functional analysis.

Jones constructed index theory for subfactors as an analogue of Galois theory for extension fields. We introduced Jones index theory for simple C*-algebras and developed the theory using bimodules and K-theory. Later we studied the structure of Hilbert C*-bimodules and the Pimsner algebras generated by them. We also showed that the set of intermediate subfactors for any irreducible subfactor of a type II_1 factor with finite index is a finite lattice, and we studied which finite lattices are realized as intermediate subfactor lattices. Now Kajiwara and I study C*-algebras associated with complex dynamical systems of the iterations of Rational functions. In particular we prove that the C*-algebras restricted on the Julia sets are purely infinite simple C*-algebras. By a similar method, we also show that the C*-algebras associated with proper contractions on the self-similar sets are purely infinite simple C*-algebras. Now we are studying a relation between the structure of singularity of rational functions and associated C*-algebras.

Following after Gelfand-Ponomarev study of relative positions of four subspaces in finite dimensional vector spaces, Enomoto and I study relative positions of subspaces in a infinite dimensional Hilbert space. We found uncountablly many indecomposable systems of four subspaces. We introduce a numerical invariant, called defect, using Fredholm index. We have determined the possible values of defect. Now we are studying a relation between Dynkin diagrams and relative positions of subspaces along finite graphs.

Keywords Theory of Operator Algebras, Operator Theory
Faculty , Department Faculty of Mathematics , Department of Mathematical Sciences