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：该提案涉及经典Banach空间及其量化类似物算子空间的同构结构问题。要研究的问题有以下四个方面： 一、C*代数中理想的补问题。 二.可分离运算符空间的某些扩展属性。三．核 C* 代数的补子空间的结构。 四．非交换L^p-空间的结构。具体问题包括以下内容： 区域I：设J 是A/J 可分离的C* 代数A 中的理想（封闭的、2 边的）。 A 中 J Banach 是否补足？ 一个重要的特例是其中 J = K，可分离无限维希尔伯特空间上的紧算子的理想；这与经典Banach空间理论中的一致近似性质有关。区域二：每个具有CSCP的可分离算子空间X是否完全嵌入在K中？ 如果 X 是局部自反的并且在每个可分离的局部自反超空间中完全互补，则称 X 具有 CSCP。 区域 III：可分离核 C* 代数的每个补子空间是否与核算子空间完全同构？ 反之亦然吗？区域 IV：设 N 为冯·诺依曼代数，X 为 N 预余数的子空间（无限维、线性且闭合）。对于某个 1 或 = p 或 = 2，l^p 是否嵌入 X 中？ 如果 M 是另一个冯·诺依曼代数，并且 M 和 N 的预理是 Banach 同构，那么 M 和 N 是否具有相同的类型？ 算子空间的研究涉及量子力学基础的数学。 重要的“量化”Banach 空间包括 C* 代数，例如费米恩代数和希尔伯特空间上的紧算子代数，以及冯诺依曼代数的预定义，即非交换的 L^1 空间。 有关这些空间结构的问题将从经典巴拿赫空间理论的角度来解决。 这种方法已经取得了相当大的成功，产生了紧算子空间的基本性质，以及有限和无限冯·诺依曼代数的预分值之间的巴纳赫区别。