PROJECTIONS IN BANACH ALGEBRAS
PROJECTIONS IN BANACH ALGEBRAS
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巴拿赫代数中的投影
DOI：
10.2307/1969540
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发表时间：
1951
期刊：
影响因子：
--
通讯作者：
I. Kaplansky
中科院分区：
文献类型：
--
作者：
I. Kaplansky
希尔伯特空间上的算子代数理论有两个主要贡献。在一系列五本回忆录中，默里和冯·诺伊曼向弱封闭案例的结构理论迈出了重要的一步。在俄罗斯学派开始的工作中，已经对更一般的情况进行了研究，在这种情况下，假设只有一致的闭合。在西格尔提出的术语中，我们分别称之为W*-代数和C*-代数。C*-情形的一个显著优点是，由于Gelfand和Neumark[2]的存在，存在一个优雅的内在公设系统；因此，人们可以并且确实以抽象的方式研究C*-代数，而不考虑任何特定的表示。TV*-代数的相应刻划目前尚不清楚，但已有几种替代方法被提出。在文[6]中，冯·诺伊曼从一开始就假设了第二个拓扑，它的行为类似于弱拓扑。Steen[8]假设了相对于正泛函诱导的拓扑的完备性。在这篇文章中，整个负担将放在一个更代数的，在某种意义上更基本的假设上；简而言之，我们的假设是在偏序投影集中的最小上界的假设--精确的公理在？2中给出。我们称所讨论的代数为ATV*代数(“A”表示“抽象”)。这项工作实质上是里卡特开始的研究的延续
There have been two main contributions to the theory of operator algebras on Hilbert space. In a series of five memoirs, Murray and von Neumann have made important strides toward the structure theory of the weakly closed case. In work begun by the Russian school, a study has been made of the more general case where merely uniform closure is assumed. In terminology suggested by Segal, we call these W*-algebras and C*-algebras respectively. A notable advantage of the C*-case is the existence of an elegant system of intrinsic postulates due to Gelfand and Neumark [2]; so one can, and does, study C*-algebras in an abstract fashion that pays no attention to any particular representation. A corresponding characterization of TV*-algebras is not known, but nevertheless several substitutes have been suggested. In [6] von Neumann postulated from the start a second topology behaving like the weak topology. Steen [8] assumed completeness relative to a topology induced by positive functionals. In the present paper the entire burden will be thrown upon a more algebraic, and in some sense more elementary assumption; briefly put, our postulate is the assumption of least upper bounds in the partially ordered set of projections-the precise axioms are given in ?2. We call the algebras in question ATV*algebras (the "A" suggesting "abstract"). This work is in essence a continuation of the study that was begun by Rickart