The strict topology for double centralizer algebras
The strict topology for double centralizer algebras
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双中心化代数的严格拓扑
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发表时间：
1970
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通讯作者：
D. C. Taylor
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作者：
D. C. Taylor
给出了严格拓扑下双中心化代数是Mackey空间的充分条件。 0. 简介。令 C(S) 为局部紧豪斯多夫空间 S 上所有有界复值连续函数的 B* 代数；令 CO(S) 为 C(S) 中无穷大消失的所有函数的代数，并令 C(S) 表示 : 或严格拓扑下的 C(S)。 1958年，R. C. Buck [3]证明了强拓扑下的C(S)严格对偶与Co(S)的范数对偶等距同构，并提出了以下问题：严格拓扑/是否真的与Mackey拓扑一致？ 1967 年，J. B. Conway [6] 大部分回答了这个问题。他证明，如果 S 是仿紧拓扑，那么严格拓扑确实是麦基拓扑，并且他还给出了局部紧空间 S 的示例，其中 C(S) 的严格拓扑不是麦基拓扑。最近，R. C. Busby [4] 在他对 B* 代数的双中心化器的研究中引入了严格拓扑的广义概念。具体来说，如果 A 是 B* 代数，M(A) 是其双中心化代数，则严格拓扑 ：对于 M(A) 被定义为由半范数 (Aa)aeA 和 (Pa)aeA5 生成的局部凸拓扑，其中 Aa(X) = ||ax 和 pa(x) = |xa |，我们让 M(A),6 表示严格拓扑下的 M(A)。尽管 Busby 研究了这种情况下严格拓扑的一些性质，但没有提及 M(A) 的严格对偶。因此，需要考虑的问题如下：（1）强拓扑下M(A)的严格对偶是否是与A的范数对偶等距同构的Banach空间？ (2) M(A)的严格拓扑是Mackey拓扑的充分条件是什么？问题（1）的答案是肯定的，为了回答问题（2），我们证明以下两个定理： 定理 I。设 {AA: A E A} 是 B* 代数族，并设 A= (> AA)O。则 M(A)2i 是 Mackey 空间当且仅当对于每个 A E A，M(AA)2 是 Mackey 空间。编辑于 1969 年 10 月 14 日收到。AMS 主题分类。小学 4650、4665；中学4601。
Sufficient conditions are given for a double centralizer algebra under the strict topology to be a Mackey space. 0. Introduction. Let C(S) be the B*-algebra of all bounded complex valued continuous functions on a locally compact Hausdorff space S; let CO(S) be the algebra of all functions in C(S) that vanish at infinity, and let C(S), denote C(S) under the : or strict topology. In 1958, R. C. Buck [3] proved that the strict dual of C(S) under the strong topology is isometrically isomorphic to the norm dual of Co(S) and then raised the following question: Is it in fact true that the strict topology / coincides with the Mackey topology? In 1967, J. B. Conway [6] answered this question for the most part. He showed that if S is paracompact, then indeed the strict topology is the Mackey topology and he also gave examples of locally compact spaces S where the strict topology for C(S) is not the Mackey topology. More recently, R. C. Busby [4] in his study of double centralizers of B*-algebras introduced a generalized notion of the strict topology. Specifically, if A is a B*algebra and M(A) is its double centralizer algebra, then the strict topology : for M(A) is defined to be that locally convex topology generated by the seminorms (Aa)aeA and (Pa)aeA5 where Aa(X) = ||ax and pa(x) = |xa |, and we let M(A),6 denote M(A) under the strict topology. Although Busby investigated some of the properties of the strict topology in this setting, no mention was made of the strict dual of M(A). Thus, the questions under consideration are the following: (1) Is the strict dual of M(A) under the strong topology a Banach space that is isometrically isomorphic to the norm dual of A ? (2) What are some sufficient conditions for the strict topology for M(A) to be the Mackey topology? The answer to question (1) is yes and to answer question (2) we prove the following two theorems: THEOREM I. Let {AA: A E A} be a family of B*-algebras and let A= (> AA)O. Then M(A)2i is a Mackey space if, and only if, for each A E A, M(AA)2 is a Mackey space. Received by the editors October 14, 1969. AMS Subject Classifications. Primary 4650, 4665; Secondary 4601.