The Isomorphic Structure of Banach and Operator Spaces

Banach空间与算子空间的同构结构

• 批准号: 0070547
• 负责人: Haskell Rosenthal
• 金额: $ 10.49万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2005-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070547&HistoricalAwards=false
• 关键词: Isomorphic; Structure; Banach; Operator; Spaces

ABSTRACT: This proposal deals with problems concerning the isomorphic structureof classical Banach spaces and their quantized analogues, operator spaces.The problems to be studied lie in the following four areas: I. The complementation problem for ideals in C* algebras. II. Certain extension properties for separable operator spaces. III. The structure of complemented subspaces of nuclear C*-algebras. IV. The structure of non-commutative L^p-spaces.Specific problems include the following:Area I: Let J be an ideal (closed, 2-sided) in a C*-algebra A with A/Jseparable. Is J Banach complemented in A? An important special case iswhere J = K, the ideal of compact operators on separable infinite-dimensionalHilbert space; this is related to the uniform approximation property inclassical Banach space theory.Area II: Does every separable operator space X with the CSCP completelyembed in K? X is said to have the CSCP if it is locally reflexive andcompletely complemented in every separable locally reflexive superspace.Area III: Is every complemented subspace of a separable nuclear C*-algebracompletely isomorphic to a nuclear operator space? Is the converse true?Area IV: Let N be a von Neumann algebra and X be a subspace (infinite-dimensional, linear, and closed) of the predual of N. Does l^p embed inX for some 1 or = p or = 2? If M is another von Neumann algebra, and thepreduals of M and N are Banach isomorphic, do M and N have the same type? The study of operator spaces involves mathematics underlying thefoundation of quantum mechanics. Important "quantized" Banach spaces includeC*-algebras such as the Fermion algebra and the algebra of compact operatorson Hilbert space, and the preduals of von Neumann algebras, i.e.,non-commutative L^1-spaces. The problems concerning the structure of thesespaces will be approached from the perspective of classical Banach spacetheory. This approach has already had considerable success, yielding basicproperties of the space of compact operators, and the Banach distinctionbetween the preduals of finite and infinite von Neumann algebras.

摘要：该提案涉及有关经典Banach空间的同构结构及其量化类似物（操作员空间）的问题。要研究的问题在于以下四个领域：I。C*代数中理想的互补问题。 ii。可分离操作员空间的某些扩展特性。 iii。核C* - 代数的互补子空间的结构。 iv。非交通级别的结构。特定问题包括以下内容：区域I：让J为具有A/JSE比程的C*-Algebra A中的理想（封闭，2侧）。 J Banach在A中得到补充吗？ 一个重要的特殊情况是J = K，这是可分离无限二维式赫伯特空间的紧凑型操作员的理想。这与包含Banach空间理论的均匀近似特性有关。区域II：每个可分离的操作员空间X是否具有CSCP在k中完全变为？ 据说如果CSCP具有局部反思性并且在每个可分离的局部反身级别空间中都具有局部反射性，则cscp是全面补充的：area iii：是否对核操作员空间的每个可分离核C* - 代理完全同构的互补子空间？ Converse是正确的吗？IV：让N为von Neumann代数，X是N的n。 如果m是另一个冯·诺伊曼（Von Neumann）代数，而m和n的原则是巴纳克同构的，那么m和n是否具有相同的类型？ 对操作空间的研究涉及量子力学基础的基础数学。 重要的“量化” Banach空间，包括诸如Fermion代数和紧凑型操作员空间的代数等代数，以及von Neumann代数的预期，即非交互性L^1个空间。 从古典Banach的空间的角度来看，有关这些问题结构的问题将解决。 这种方法已经取得了巨大的成功，从而产生了紧凑型操作员空间的基本证据，以及有限和无限的von Neumann代数之间的Banach区别。