Yasuo Watatani | Last modified date：2016.09.23 |

Graduate School

Undergraduate School

E-Mail

Phone

092-802-4442

Fax

092-802-4405

Academic Degree

Doctor of Engineering

Field of Specialization

Theory of Operator Algebras

Outline Activities

We study theory of operator algebras in functional analysis.

Jones constructed index theory for subfactors as an analogue of

Galois thoory for extension fields. We intruduced 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 finte index is a finite lattice, and we studied which finte 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 partucular 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 stuying a

relation between Dynkin dyagrams and relative positions of subspaces along

finte graphs.

In seminars, students study books on the fundamentals of functional

analysis and operator algebras. Then they introduce some papers in journals

and find research problems on operator algebras.

I visited a high school and gave several lectures.

Jones constructed index theory for subfactors as an analogue of

Galois thoory for extension fields. We intruduced 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 finte index is a finite lattice, and we studied which finte 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 partucular 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 stuying a

relation between Dynkin dyagrams and relative positions of subspaces along

finte graphs.

In seminars, students study books on the fundamentals of functional

analysis and operator algebras. Then they introduce some papers in journals

and find research problems on operator algebras.

I visited a high school and gave several lectures.

Research

**Research Interests**

- C*-algebras associated with complex dynamical systems

keyword : C*-algebras, rational function

2000.01We construct purely infinite simple C*-algebras for rational functions.. - Relative position of subspaces in a Hilbert space

keyword : HIlbert space, relative position

1998.01We study relative position of n subspaces in a Hilbert space.. - Index for C*-subalgebras

keyword : C*-algebras, index

1988.01～2003.01Jones index theory for simple C*-algebras.

**Academic Activities**

**Papers**

1. | Norio Nawata and Yasuo Watatani, Fundamental group of simple C*-algebras with unique trace, Advances in Math. , 225, 307-318, 2010.08. |

2. | Watatani, Yasuo, Ｉｎｄｅｃｏｍｐｏｓａｂｌｅ ｒｅｐｒｅｓｅｎｔａｔｉｏｎｓ of quivers on infinite-dimensional Hilbert spaces, J. Functional Analysis, (256) 959-991, 2009.02. |

3. | M. Izumi, T. Kajiwara and Y. Watatani, KMS states and branched points , Ergodic Th. & Dynam. Sys. , 27 巻、 1887-1918, 2007.12. |

4. | M.Enomoto and Y. Watatani, Relative position of four subspaces in a Hilbert space, Adv. Math., Vol 201, 263-317, 2006.05. |

5. | T. Kajiwara and Y. Watatani, C*-algebras associated with complex dynamical systems, Indiana Univ. Math. J., 54, 3, Vol. 54, 755-778, 2005.05. |

**Presentations**

1. | Singularities in operator algebras. |

2. | C*-algebras associated with complex dynamical systems. |

**Awards**

- Research of operator algebra from the multidirectional viewpoint and its applications

Educational

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