Operator Spaces and Applications to Related Areas

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The PI's research is mainly in the theory of operator spaces and its applications to operator algebras, non-commutative Lp-spaces and non-commutative harmonic analysis. During the last few years, the PI, together with some other mathematicians, has made some significant contributions to these areas. He has obtained a number of important results on the local (i.e. finite dimensional) operator space properties and related approximation properties of C*-algebras and non-commutative Lp-spaces. He has also obtained some interesting applications of operator spaces to non-commutative harmonic analysis. In this proposal, he plans to continue his research in these directions and proposes the following research projects: (1) investigate some problems on non-commutative Lp-spaces, (2)investigate some problems on group C*-algebras and Fourier algebras, (3)investigate the possible generalizations to Kac algebras and locally compact quantum groups; (4)investigate some related problems in non-commutative/free probability. The theory of operator spaces is a natural quantization of functional analysis, or more precisely, a natural quantization of Banach space theory. Operator spaces were first realized by William Arveson in 1969 and were abstractly characterized by the PI in his Ph.D thesis in 1987. Since then there have been some remarkable developments in operator spaces and the theory has been quickly developed into a very active research area in modern analysis. The projects proposed here contain some important questions in operator spaces, operator algebras, non-commutative/quantum harmonic analysis and non-commutative/free probability. The progress on these projects will have significant impact in these areas, as well as in some other related mathematics research areas such as quantum group theory, non-commutative geometry and geometric group theory. Projects in this proposal also provide the outstanding resources and research problems for the PI's Ph.D students and post-docs. The support for the Wabash Seminar and Miniconference is requested in this proposal. The Wabash Seminar, together with its annual Miniconference, is devoted to the stimulation and dissemination of significant contributions to analysis in the Midwest region. This has already provided (and will continuously provide) a unique opportunity for young researchers, visitors, post-docs and graduate students from the Midwest to meet regularly and exchange ideas with leading experts in the fields.