Precompact groups and property (T)

被引:1
|
作者
Ferrer, M. [1 ]
Hernandez, S. [2 ,3 ]
Uspenskij, V. [4 ]
机构
[1] Univ Jaume 1, Inst Matemat Castellon, Castellon de La Plana 12071, Spain
[2] Univ Jaume 1, INIT, Castellon de La Plana 12071, Spain
[3] Univ Jaume 1, Dept Matemat, Castellon de La Plana 12071, Spain
[4] Ohio Univ, Dept Math, Athens, OH 45701 USA
关键词
Compact group; Precompact group; Representation; Pontryagin-van Kampen duality; Compact-open topology; Fell dual space; Fell topology; Bohr compactification; Kazhdan property (T); Determined group; TOPOLOGIES;
D O I
10.1016/j.jmaa.2013.03.004
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
For a topological group G, the dual object (G) over cap is defined as the set of equivalence classes of irreducible unitary representations of G equipped with the Fell topology. It is well known that, if G is compact, (G) over cap is discrete. In this paper, we investigate to what extent this remains true for precompact groups, that is, dense subgroups of compact groups. We show that: (a) if G is a metrizable precompact group, then (G) over cap is discrete; (b) if G is a countable non-metrizable precompact group, then (G) over cap is not discrete; (c) every non-metrizable compact group contains a dense subgroup G for which (G) over cap is not discrete. This extends to the non-Abelian case what was known for Abelian groups. We also prove that, if G is a countable Abelian precompact group, then G does not have Kazhdan's property (T), although (G) over cap is discrete if G is metrizable. (C) 2013 Elsevier Inc. All rights reserved.
引用
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页码:221 / 230
页数:10
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