A weak*-topological dichotomy with applications in operator theory
A weak*-topological dichotomy with applications in operator theory
复制标题
弱*-拓扑二分法及其在算子理论中的应用
DOI：
10.1112/tlms/tlu001
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发表时间：
2014
影响因子：
0.8
通讯作者：
Kania T
中科院分区：
文献类型：
--
作者：
Kania T
表示由所有可数序数组成的局部紧Hausdorff空间，配备有序拓扑，并设为定义在上并最终消失的标值连续函数的Banach空间。我们证明了的对偶空间的弱紧子集或者是一致Eberlein紧的，或者它包含一个特定形式的序区间的同胚拷贝.这种二分法为Banach空间和作用在其上的有界线性算子的Banach代数的大多数现有研究提供了一种统一的方法，它导致了几个新的结果，以及已知结果的更强版本.特别地，我们推导出一个是的商的Banach空间可以嵌入到一个Hilbert生成的Banach空间中，或者它同构于的直和和一个Hilbert生成的Banach空间的子空间;并且我们得到了描述Loy-Willis理想的几个等价条件，这是的唯一极大理想，包括以下条件：一个算子属于的当且仅当它因子分解通过Banach空间。其中后果的这些特点是thathas一个有界左近似身份，这解决了一个问题左开放洛伊和威利斯。
Denote bythe locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and letbe the Banach space of scalar‐valued, continuous functions which are defined onand vanish eventually. We show that a weak‐compact subset of the dual space ofis either uniformly Eberlein compact, or it contains a homeomorphic copy of a particular form of the ordinal interval.This dichotomy yields a unifying approach to most of the existing studies of the Banach spaceand the Banach algebraof bounded, linear operators acting on it, and it leads to several new results, as well as to stronger versions of known ones. Specifically, we deduce that a Banach space which is a quotient ofcan either be embedded in a Hilbert‐generated Banach space, or it is isomorphic to the direct sum ofand a subspace of a Hilbert‐generated Banach space; and we obtain several equivalent conditions describing the Loy–Willis ideal, which is the unique maximal ideal of, including the following: an operator belongs toif and only if it factors through the Banach space. Among the consequences of these characterizations ofis thathas a bounded left approximate identity; this resolves a problem left open by Loy and Willis.