The Isomorphic Structure of Banach and Operator Spaces

文摘：本文讨论了经典Banach空间及其量子化类似物--算子空间的同构结构问题。拟研究的问题包括以下四个方面：一、C*代数中理想的补问题。II.可分算子空间的某些扩张性质。核C~*-代数的可补子空间的结构。IV.非交换L p-空间的结构.具体问题包括：区域I：设J是A/J可分的C*-代数A中的理想(闭的，双边的).J·巴纳赫在《A》中得到了补充吗？一个重要的特例是其中J=K，它是可分无限维Hilbert空间上的紧算子的理想；这与经典Banach空间理论中的一致逼近性质有关。如果X是局部自反的，且在每个可分的局部自反超空间中是完全可补的，则称X有CSCP。区域III：可分核C*-代数的每个可补子空间是否全同构于一个核算子空间？反之亦然？第四区：设N是von Neumann代数，X是N的前对偶的子空间(无穷维的，线性的，且是闭的)，L^p是否对某个1或=p或=2嵌入INX？如果M是另一个von Neumann代数，且M和N的乘积是Banach同构的，那么M和N有相同的类型吗？对算符空间的研究涉及到作为量子力学基础的数学。重要的“量化”Banach空间包括C~*-代数，如Hilbert空间上的费米子代数和紧算子代数，以及von Neumann代数的乘积，即非交换的L^1-空间。关于这些空间的结构问题，我们将从经典的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，产生了紧算子空间的基本性质，以及有限和无限von Neumann代数的前项之间的Banach区别。

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.