The Isomorphic Structure of Banach and Operator Spaces

摘要： 本文讨论了经典Banach空间及其量子化类似空间--算子空间的同构结构问题，所要研究的问题包括以下四个方面： I. C* 代数中理想的补问题。 二.可分算子空间的某些扩张性质。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：区域I：设J是C*-代数A中的理想（闭的，双边的），且A/J可分。J在A中是可补的吗？ 一个重要的特殊情况是J = K，可分无限维Hilbert空间上紧算子的理想，这与经典Banach空间理论中的一致逼近性质有关。 称X具有CSCP，如果它在每个可分局部自反超空间中是局部自反的且完全可补的.区域III：可分核C*-代数的每个可补子空间是否完全同构于核算子空间？ 匡威亦然吗？第四部分：设N是一个冯诺依曼代数，X是N的前对偶的一个子空间（无限维、线性、闭）。 l^p是否嵌入X，对于某个1或= p或= 2？ 若M是另一个von Neumann代数，且M和N的预射是Banach同构的，那么M和N是否有相同的型？ 算符空间的研究涉及到量子力学基础的数学。 重要的“量子化”Banach空间包括C *-代数，如Fermion代数和Hilbert空间上的紧算子代数，以及von Neumann代数的前身，即，非交换L^1-空间 关于这些空间的结构问题将从经典Banach空间理论的角度来探讨。 这种方法已经取得了相当大的成功，产生了紧致算子空间的基本性质，以及有限和无限冯诺依曼代数之间的Banach区分。

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.