On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms
On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms
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论群拓扑和拓扑群归纳极限的酉表示以及微分同胚群的情况
DOI：
10.1215/kjm/1250518067
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发表时间：
1998
影响因子：
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通讯作者：
T. Hirai
中科院分区：
文献类型：
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作者：
Nobuhiko Tatsuuma;Hiroaki Shimomura;T. Hirai
这篇论文的目的有两个。第一部分是讨论拓扑群的归纳极限上的各种群拓扑，以及在某种情况下归纳极限群的酉表示;第二部分是讨论群拓扑在同构群的情况下。与[1]或[5]中的肯定陈述相反，拓扑群的归纳系统的拓扑的归纳极限并不总是给出群拓扑，或者更确切地说，乘法不一定关于归纳极限拓扑（记为ri d）是连续的.在本文的第一部分中，我们用一个简单的例子在交换群的情况下证明了这一点，然后一般地讨论了在归纳极限群上，在弱于T Hid · W e的条件下，可以选择哪些群拓扑，特别是在归纳系统是可数的且本质上由局部紧群组成的情况下.（F或确切定义，见§2.3。这样的系统称为可数L-C-G归纳系统然后我们证明了归纳极限拓扑r，n d在这种情况下给出群拓扑（定理2.7），并且在一个温和的条件下它本质上是唯一的（定理5.6）。进一步，对于可数LCG归纳系统，在一定程度上讨论了归纳极限群G = lim G的酉表示和连续正定函数，证明了在与定理J5.6相同的条件下，存在足够多的酉表示和连续正定函数，使得G的点可分离（定理5.7）.由于G上一般不存在Haar测度，所以讨论的重点是从局部紧群GI到G的极限过程.在第二部分中，我们讨论了连通的非紧Cr-流形M，1 < r < co上具有紧支撑的群G =Diffo（M）的情形，并证明了归纳极限拓扑T_j_n_d永远不给伊韦斯（定理6.1）.
T he purpose o f th is paper is tw ofo ld . T he first one is to discuss various group topologies on inductive lim its o f topological groups, a n d unitary representations of inductive limit groups in a certain case, and the second one is to treat group topologies in the case of the group o f diffeomorphisms. Contrary to the affirmative statement in [1] o r in [5], the inductive lim it of topologies of an inductive system o f topological groups does not always give a group topology, or m ore exactly, the multiplication is not necessarily continuous with respect to the inductive limit topology (denoted by r i d ) . In P art I of this paper, we show this by a simple example in the case of abelian groups, and then discuss in general which kinds of group topologies can be chosen on an inductive limit group under the condition that they are weaker than T Hid • W e study in particular the case where inductive system is countable and essentially consists of locally compact g roups. (F or exact definition, see §2.3. and such a system is called a countable L C G inductive system in s h o r t ) . Then we prove that the inductive limit topology r,„d gives a group topology in this case (Theorem 2.7), and also that it is essentially a unique one under a mild condition (Theorem 5.6). Further, for a countable LCG inductive system, we discuss in a certain extent unitary representations and continuous positive definite functions of the inductive limit group G = lim G and prove that, under the same condition as for Theorem J 5.6, there exist sufficiently many of them so that the points of G can be separated (Theorem 5.7). Since there does not exist in general a Haar measure o n G, the important point of the discussion is the limiting process from the case of locally compact groups GI t o G. In Part II, we discuss the case of the group G =Diffo (M ) of diffeomorphisms with compact supports on a connected, non-compact, Cr-manifold M , 1 < r < co, a n d prove that the inductive limit topology T j n d never g ives a group topology (Theorem 6.1).