Generalized B-Algebras
Generalized B-Algebras
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广义 B 代数
DOI：
10.1112/plms/s3-21.4.693
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发表时间：
1970
影响因子：
1.8
通讯作者：
P. Dixon
中科院分区：
文献类型：
--
作者：
P. Dixon
在(2)中，Allan构造了一类局部凸对合代数，称为GB*-代数，并证明了在交换情况下，GB*-代数与紧Hausdorff空间上的扩展复值连续函数代数在代数上同构。本文的主要目的是考虑非交换的情况。在对Allan的定义稍加修改后，我们将局部凸GB*-代数刻画为Hilbert空间上的一类闭算子代数。我们将首先展示如何将Allan的定义推广到非局部凸代数，例如，包括有限测度空间上的可测函数代数。这就是§2-4的主题。第5-7节发展了主要的表示定理（7.13）和相关的结果。最后一节讨论正泛函的连续性问题。
In (2), Allan denned a class of locally convex involution algebras called GB*-algebras, and proved that, in the commutative case, a GB*-algebra is (algebraically) isomorphic to an algebra of extended-complex-valued continuous functions on a compact Hausdorff space. The main object of the present paper is to consider the non-commutative case. After a slight modification of Allan's definition, we shall characterize locally convex GB*-algebras as a certain class of algebras of closed operators on a Hilbert space. We shall first show how Allan's definition may be extended to non-locally-convex algebras, to include, for example, the algebra of measurable functions on a finite measure space. This is the subject of §§ 2-4. Sections 5-7 develop the main representation theorem (7.13) and associated results. The final section is devoted to the question of the continuity of positive functionals.