Kyushu University Academic Staff Educational and Research Activities Database
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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.
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, 基本群とはMurrayとvon Neumann がRings of Operator Algebrasという題の連作の論文を書いて作用素環という分野を創設した時にすでに導入された, 正の実数の乗法的部分群を値としてとる、II_1型因子環の不変量です。しかし、Murrayとvon Neumann はhyperfinite II_1型因子環の基本群が正の時数全体になることを示しただけで、他のII_1型因子環に対しては何も計算できなかったといういわくつきの不変量でした。それをA.Connesが、property(T)を持つICC群の場合にはその群von Neumann環の基本群が可算群になることを示し突破口を開きました。それが近年Popaらにより剛性と絡んでどんな可算部分群も実現できることがわかり急展開しました。この論文では、世界で初めて唯一つのトレースを持つ単純C*-環に対してその基本群という不変量を導入することに成功し、基本的な性質を導きました。.
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., 10.1512/iumj.2005.54.2530, 54, 3, 755-778, 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
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.