Studies in Operator Algebras and Operator Spaces

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Dear Joe, Please find below our abstract for the NSF grant. We hope it is what you had in mind. Best wishes for the new year, Corran and Roger The proposed research concerns the theory of operator algebras and operator spaces. Smith will concentrate on the theory of von Neumann algebras, and continue his work on the Kadison-Ringrose conjecture on the cohomology groups of type II_1 factors. He will also investigate the Haagerup invariant for operator algebras, which addresses the difficult problem of distinguishing between various von Neumann algebras and C*-algebras. Webster will continue his investigation of local operator spaces, the quantized version of locally convex vector spaces, concentrating on applications to subalgebras of smooth elements of C*-algebras, and algebras of unbounded operators. He will also pursue applications of the Krein-Milman theorem in operator convexity theory, with the particular aim of shedding light on the problem of the existence of boundary representations. In joint work the proposers will consider various approximation properties of operator algebras and operator spaces and these will be connected to both the Haagerup invariant and operator compactness of matrix sets. Operator theory and operator algebras are the infinite dimensional analogues of finite matrix theory. They were first studied by von Neumann as a framework for quantum mechanics. Subsequently they have had, and continue to have, important applications to control theory, quantum groups, non-commutative geometry, non-commutative probability and knot theory. The work in this proposal is most closely related to mathematical physics and quantum mechanics. We will consider algebras of unbounded operators which arise from the Heisenberg Uncertainty Principle, as well as crossed product C*-algebras which model the time evolution of quantum mechanical systems in physics. The techniques we use rely heavily on the theory of operator spaces , a topic which has undergone considerable development in the last ten years. These spaces provide a unifying framework for the self-adjoint algebras of physics and the non-self-adjoint algebras of control theory.