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On Centralizers of Generalized Uniform Subgroups of Locally Compact Groups * 1 )
| Content Provider | Semantic Scholar |
|---|---|
| Author | Lee, Deuk Ho |
| Copyright Year | 2010 |
| Abstract | Let G be a locally compact group and H a closed subgroup of G such that the homogeneous space G/H admits a finite invariant measure. Let Zq(H) be the centralizer of H in G. It is shown that if G is connected then Zq(H) modulo its center is compact. If G is only assumed to be locally connected it is shown that the commutator subgroup of Zq(H) has compact closure. Consequences of these results are found for special classes of groups, such as Lie groups. An example of a totally disconnected group G is given to show that the results for Zq{H) need not hold if G is not connected or locally connected. 0. Introduction. Let G be a locally compact group and H a closed subgroup of G such that G/H admits a finite invariant measure (we call such a subgroup a generalized uniform subgroup of G). In [1], Borel showed that the centralizer ZG(H) of H in G is equal to the center Z(G) of G when G is a semisimple connected Lie group without compact factors. Recently, Greenleaf-Moskowitz-Rothschild [3] extended Borel's result to those connected Lie groups G with the property that Z(G) = B(G) where B(G) is the set of bounded elements of G (an element x of G is bounded if the conjugacy class of x has compact closure). In [16] we prove a result analogous to that of Greenleaf-Moskowitz-Rothschild for linear algebraic groups. Here we investigate compactness conditions satisfied by ZG(H) (see ยง1 for all compactness conditions on locally compact groups). S. P. Wang [19] showed that ZG(H) is a [Z] group and therefore an [FD]~ group (see [5]) when G is a connected Lie group and H is discrete. In [11], using Borel's density theorem above, D. H. Lee showed that ZG(H) is an [FD] ~ group for Received by the editors June 18, 1973 and, in revised form, November 8, 1973. AMS (MOS) subject classifications (1970). Primary 22D05, 22E15, 22E40; Secondary 28A70. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.ams.org/journals/tran/1975-201-00/S0002-9947-1975-0354923-2/S0002-9947-1975-0354923-2.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
