| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 6, 1999 |
| Latest Amendment Date: | July 6, 1999 |
| Award Number: | 9970369 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Joe W. Jenkins
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 15, 1999 |
| End Date: | December 31, 2001 (Estimated) |
| Total Intended Award Amount: | $62,364.00 |
| Total Awarded Amount to Date: | $62,364.00 |
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| Recipient Sponsored Research Office: |
110 INNER CAMPUS DR AUSTIN TX US 78712-1139 (512)471-6424 |
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| Primary Place of Performance: |
110 INNER CAMPUS DR AUSTIN TX US 78712-1139 |
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Performance Congressional District: |
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| NSF Program(s): | ANALYSIS PROGRAM |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Abstract
Oikhberg
The PI plans to continue his work on several aspects of geometry of operator spaces. One line of work deals with separable extension properties. Here the most intriguing problem is whether K (the space of compact operators on a separable Hilbert space) is complemented in every separable operator space containing it (this is a "non-commutative" version of a classical result of Sobczyk). A related issue is giving a complete description of separable locally reflexive operator spaces which are completely complemented in every separable locally reflexive operator superspace (an operator space counterpart of Zippin's characterization of the space of convergent sequences). Another direction of research is the study of maximal operator spaces (quotients of duals of commutative C*-algebras). The PI plans to determine the cardinality of the set of n-dimensional subspaces of maximal spaces, as well as the structure of n-dimensional subspaces of the dual of a 2n-dimensional commutative C*-algebra (in the spirit of Kashin's work). Possible applications of the proposed research include better understanding of the structure of function spaces, of the space of operators on a Hilbert space, and Banach space geometry in general (extensions of local reflexivity).
One of the key mathematical tools of quantum mechanics is replacing scalars (numbers) by operators on a Hilbert space (they can be thought of as infinite matrices). Further exploration in this direction led to the development of the theory of C*-algebras, and, later, to the introduction of the notion of operator space (also called "non-commutative" or "quantized Banach space"). One can view an operator space as a Banach space with additional structure, induced by its embedding into the C*-algebra B(H) of bounded linear operators on a Hilbert space H. The investigation of operator spaces has advanced very rapidly over the last ten years, combining ideas and techniques from Banach space theory and the theory of C*-algebras. It has already produced answers to some long-standing problems of Operator Theory. The proposed research deals with two themes: the existence of non-commutative analogs of classical Banach space results; and the use of Banach space methods in the operator space case. If successful, this research will advance our understanding of connections between Banach and operator spaces, and potentially, enhance our knowledge of physical phenomena.
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