The lattice theoretic background of the dimension theory of operator algebras
The lattice theoretic background of the dimension theory of operator algebras
复制标题
算子代数维数论的格论背景
DOI：
10.1090/memo/0018
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发表时间：
1955
期刊：
影响因子：
--
通讯作者：
L. H. Loomis
中科院分区：
文献类型：
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作者：
L. H. Loomis
冯·诺伊曼在他的连续几何理论[6]中和Kaplansky在他最近对AW*Banach代数的研究[1]中，都根据他给定的结构定义了一个等价关系，并且在每种情况下，等价是完全可加性的证明只是逐渐地演变，以一种必要的方式与维度理论本身的发展交织在一起。在这方面，我们的承诺要温和得多，因为我们一开始就假定完全相加等价性。这一假设对于应用于布尔代数是必要的。此外，它在第9节研究的算子代数和新的AW*-代数类中的应用也是合理的。最后，它允许该理论最自然、最直接的发展。虽然许多论点现在可以被认为是经典的，但他们仍然给出了一些细节，部分是为了完整性，部分是为了吸引还不熟悉维度理论的有趣读者。
Von Neumann, in his theory of continuous geometry [6], and Kaplansky, in his recent study of AW* Banach algebras [1], each defines an equivalence relation in terms of his given structure, and in each case the proof that equivalence is completely additive evolves only gradually, intertwining in a necessary way with the development of dimension theory itself. In this respect our undertaking is much more modest, since we assume complete additivity for equivalence at the outset. This assumption is necessary for the application to Boolean algebras. Moreover, it is justified in the application to operator algebras and the new class of AW*-algebras studied in section 9. Finally, it allows the theory its most natural, straightforward development. Although many of the arguments can by now be considered classical, they are nevertheless given in some detail, partly for completeness, and partly in the hope of interesting readers not yet acquainted with dimension theory.