| 16:00〜 |
Alfons Van Daele
University of Leuven |
Locally compact quantum groups and the Tomita-Takesaki theory
Abstract： The operator algebra approach to quantum groups has in fact a longer history than the theory of quantum groups as it is understood nowadays. Indeed, it grew out of several attempts to generalize the famous Pontryagin duality theorem to non-abelian locally compact groups. The theory of Kac algebras, as developed in the early seventies, was at that time a great achievement as it was again a self-dual theory. However, due to the new developments in quantum groups in the late eighties, it became clear that Kac algebras could not be the final solution. The work of Masuda and Nakagami, later joint by Woronowicz, and independently the work of Kustermans and Vaes, resulted in a new and better (and simpler!) theory of what is now called locally compact quantum groups. The modular theory of Tomita and Takesaki on left Hilbert algebras has played a very crucial role in both the earlier theory of Kac algebras as well as in the theory of locally compact quantum groups as it was recently developed. In this talk, I will discuss some aspects of the use of the Tomita-Takesaki theory in a new and simplified approach to the theory of locally compact quantum groups. |