Operator Spaces and Locally Compact Quantum Groups

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RuanDuring the last few years, the PI has focused his research on the local theory of operator spaces and the applications of operator spaces to abstract harmonic analysis and locally compact quantum groups. In this proposal, he plans to continue his research in these directions and he proposes the following research projects. (1) Investigate some local properties of operator spaces. (2) Investigate some problems on locally compact quantum groups.(3) Investigate some problems related to Kadison-Singer problem and the q-deformation factors introduced by Bozejko and Speicher.The theory of operator space theory is a recently developed new research area in modern analysis. The PI, together with some other mathematicians, has made significant contributions to this area. The projects proposed in this proposal contain some important questions in operator spaces, and their applications to operator algebras, abstract harmonic analysis, and locally compact quantum groups. The progress on these problems will have significant impact to these areas. It will also have important impact to some other related areas such as quantum information theory, noncommutative geometry, and geometric group theory. Projects in this proposal also provide excellent resources and research problems for PI's Ph.D students. The PI will continue to organize learning seminars on recent research progress at the University of Illinois. This will encourage and attract his students to get involved in these research projects.The PI will also continue to organize Wabash Seminar and Miniconference, which has provided a very unique opportunity for his students, postdocs, and visitors to regularly meet and exchange ideas with some leading experts in the fields.