Yasuo Watatani | Last modified date：2018.06.04 |

Graduate School

Undergraduate School

E-Mail

Phone

092-802-4442

Academic Degree

Doctor of Engineering

Country of degree conferring institution (Overseas)

No

Field of Specialization

Theory of Operator Algebras

Total Priod of education and research career in the foreign country

00years00months

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**

**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

**Educational Activities**

In the cources for freshmen, I gave lectures on calculus, linear algebras,

core (mathematical sciences and

informations) and indivisual particular topics. I gave homework on

fundamental subjects like calculus and linear algebras every week and

students do exercises in a classroom every week. In the

undergraduate cources, I gave lectures on topology, Lebesgue integrals and analysis.

In the graduated cources, I gave a lecture on spectral theory of

compact operators on Hilbert spaces. In a undergraduate seminar, students

read books on fundamentals of functional analysis. In a graduate semnar,

students read books on elementary operator algebras and they also

introduce some papers in journals.

core (mathematical sciences and

informations) and indivisual particular topics. I gave homework on

fundamental subjects like calculus and linear algebras every week and

students do exercises in a classroom every week. In the

undergraduate cources, I gave lectures on topology, Lebesgue integrals and analysis.

In the graduated cources, I gave a lecture on spectral theory of

compact operators on Hilbert spaces. In a undergraduate seminar, students

read books on fundamentals of functional analysis. In a graduate semnar,

students read books on elementary operator algebras and they also

introduce some papers in journals.

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