2002 Fiscal Year Final Research Report Summary
Operator Algebraic Approach to Quantum Groups
| Project/Area Number |
12640216
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | JAPAN WOMEN'S UNIVERSITY |
|---|
Principal Investigator |
NAKAGAMI Yoshiomi Japan Women's Univ., Faculty of Science, Professor, 理学部, 教授 (70091246)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
YAMANOUCHI Takahiko Hokkaido Univ., Graduate School of Science, Professor, 理学研究科, 助教授 (30241293)
KUROSE Hideki Fukuoka Univ., Faculty of Science, Professor, 理学部, 教授 (00161795)
FUJII Kazuyuki Yokohama City Univ., Faculty of Science, Professor, 理学部, 教授 (00128084)
MINEMURA Katsuhiro Japan Women's Univ., Faculty of Science, Professor, 理学部, 教授 (20060684)
|
|---|
| Project Period (FY) |
2000 – 2002
|
| Keywords | operator algebra / von Neumann algebra / C^*-algebra / quantum group / locally compact quantum group / duality / multiplicative unitary / Tomita-Takesaki theory |
|---|
| Research Abstract |
More than 6 years has passed since we began to attack the problem of quantization of locally compact groups within the framework of C^*-algebras. While we could penetrate the global framework when we started our project and announced a part of our results under the name of "weighted Hopf C^*-algebra", we devoted enough time to solve technical problems occurring from the detail arguments and to pursuit the substantiality of the contents. During the time an analogous object as ours was announced by Kustermann and Vaes under the name of "locally compact quantum group" and we were obliged to play second fiddle to it However the originality of our arguments and the difference of outword looks of defining axioms made us convince that our results would be worth to be published and had to be continued. The most of our results are the C^*-versions of our previous results formulated in the framework of von Neumann algebras, under the name of "Woronowicz algebras", almost 10 years ago. These theory have a common serious problem that we must assume the existence of the Haar weight contrary to the classical case. Therefore our next main problem is to remove the existence of the Haar weights from our axioms.
On the way to the completion for the fundamental theory of the weighted Hopf C^*-algebras, the analysis of the dual object of the quantum groups were left open as one of the problems to be solved. While no deep results in this area have not been obtained even in the classical case, this area seems to be a core part of a noncommutative theory. To clarify the part corresponding to the part known at least in the classical case, we consider first the amenability for the weighted Hopf C^*-algebra. As a results we find that there exist two candidate for the definition of amenability contrary to the classical case and mat the relationship between discrete quantum groups and nuclear C^*-algebras are different from the case of Kac algebras.
|
