Operator Topologies and Reflexive Representability
Operator Topologies and Reflexive Representability
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算子拓扑和自反表示性
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发表时间：
2000
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通讯作者：
M. Megrelishvili
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作者：
M. Megrelishvili
利用可破碎性的概念，我们证明了弱连续群表示经常是强连续的。我们证明了如果Banach（甚至Frechet）空间X具有Radon-Nikodym性质RNP，那么在GL(X)的每一个有界（分别是等连续的）子群G上弱算子和强算子拓扑重合。我们还加强了Shtern关于拓扑群的可表示性的结论。1. 众所周知，Jayne和Rogers[14,13,29,9]意义上的可碎性概念在Banach空间理论的各个方面都是非常强大的。在[22]中，我们使用可破碎性及其自然推广作为主要工具来处理线性半群表示的连续性问题。如22]中所述，我们说局部凸空间（简称l.c.s）的子集a。对于A的每个非空子集B和X的自然均匀结构的每个元素”，X是碎片化的；存在X的弱开子集W，使得B \ W非空且“-小”。如果X是一个Banach空间，那么我们得到Jayne和Rogers的原始定义[14]。在Namioka联合连续性定理的各种应用中[28]，我们记得Banach空间的每个相对弱紧化子集都是碎片化的[29]。我们需要下面的局部凸版本。引理1.1。如果X的每一个有界子集A都是碎片化的，则lc.s. X的每一个相对弱紧子集A在X中都是碎片化的，则lc.s. X是有界碎片化的（简称BF）。例如，根据引理1.1，每一个半可伸缩的l.c.s. X都是BF。为了形成一个更强的结果，我们需要一个局部凸版本的Rieeel的可登集的概念。如果对于每一个邻域V(0)，在A中存在一个点A，使得A = 2cl (con(A n(A + V)))，则称lcs X的非空子集A是可细化的；式中cl（con）表示闭合凸包。如果X的每一个有界子集都是可登的，那么空间X就是可登的。关于可缩进空间的范围，见第7节[2]。因此，具有Radon-Nikodym性质的Frechet空间(在
Using the concept of fragmentability, we show that weakly continuous group representations are frequently strongly continuous. We show that if a Banach (or, even, Frechet) space X has the Radon-Nikodym property RNP, then the weak and strong operator topologies coincide on every bounded (respectively , equicontinuous) subgroup G of GL(X). We also strengthen a result of Shtern on reeexive representability of topological groups. 1. Fragmentable subsets of locally convex spaces It is now well known that the concept of fragmentability in the sense of Jayne and Rogers 14, 13, 29, 9] is very powerful in various aspects of Banach space theory. In 22] we deal with continuity problems of linear semigroup representations using as a main tool fragmentability and its natural generalizations. As in 22], we say that a subset A of a locally convex space (in short, l.c.s.) X is fragmented if, for every non-empty subset B of A and every element " of the natural uniform structure of X; there is a weakly open subset W of X such that B \ W is non-empty and "-small. If X is a Banach space then we obtain the original deenition of Jayne and Rogers 14]. Among various applications of Namioka's joint continuity theorem 28], we recall that every relatively weakly compact subset of a Banach space is fragmented 29]. We need the following locally convex version. Lemma 1.1. Every relatively weakly compact subset A of an l.c.s. X is fragmented in X. We say that an l.c.s. X is bound-fragmented (in short, BF) if every bounded subset A of X is fragmented. For instance, by Lemma 1.1, every semireeexive l.c.s. X is BF. In order to formulate a stronger result, we need a locally convex version of Rieeel's concept of a dentable set. A non-empty subset A of an l.c.s. X is said to be dentable if for every neighborhood V of 0, there exists a point a in A such that a = 2 cl(con(A n (a + V)); where cl(con) denotes the closed convex hull. A space X is called dentable if every bounded subset of X is dentable. For the extent of the class of dentable spaces see 7, section 2]. It follows that Frechet spaces with the Radon-Nikodym property (in