Studies in Operator Algebras and Operator Spaces

亲爱的乔，请在下面找到我们为国家科学基金会拨款的摘要。我们希望这就是您所想的。祝福新的一年，Corran和Roger拟议的研究涉及算子代数和算子空间的理论。Smith将专注于von Neumann代数理论，并继续他关于II1因子上同调群的Kadison-RingRose猜想的工作。他还将研究算子代数的Haagerup不变量，它解决了区分各种von Neumann代数和C*-代数的困难问题。Webster将继续他对局部算子空间的研究，局部凸向量空间的量子化版本，集中于C*-代数的光滑元的子代数和无界算子的代数的应用。他还将继续探索Krein-Milman定理在算子凸性理论中的应用，其特别目的是阐明边界表示的存在问题。在联合工作中，提出者将考虑算子代数和算子空间的各种逼近性质，这些性质将与矩阵集的Haagerup不变量和算子紧性联系在一起。算子论和算子代数是有限矩阵理论的无限维类比。冯·诺伊曼最先把它们作为量子力学的框架进行研究。随后，它们已经并继续在控制理论、量子群、非对易几何、非对易概率和纽结理论中有重要的应用。这项提案中的工作与数学物理和量子力学关系最密切。我们将考虑由海森堡测不准原理产生的无界算符的代数，以及在物理中模拟量子力学系统的时间演化的交叉积C*-代数。我们使用的技术在很大程度上依赖于算子空间理论，这是一个在过去十年中经历了相当大发展的话题。这些空间为物理中的自伴代数和控制理论中的非自伴代数提供了统一的框架。

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.