Dimensionality and the duals of certain locally compact groups

Schochetman, I.

doi:10.1090/S0002-9939-1970-0265509-3

Dimensionality and the duals of certain locally compact groups

Author: I. Schochetman
Journal: Proc. Amer. Math. Soc. 26 (1970), 514-520
MSC: Primary 22.20
DOI: https://doi.org/10.1090/S0002-9939-1970-0265509-3
MathSciNet review: 0265509
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Abstract: Let $G$ be a separable locally compact group containing a closed normal subgroup which is type I and regularly embedded. The elements of the dual space $G$ are known to be induced from certain closed subgroups. In this article we first determine the kernel of an induced representation and then give necessary and sufficient conditions for $G$ to be maximally almost periodic. These results are then applied to the collection of compact extensions of abelian groups.

Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62 (1966), 199. MR 207910
J. M. G. Fell, A new proof that nilpotent groups are CCR, Proc. Amer. Math. Soc. 13 (1962), 93–99. MR 133404, DOI https://doi.org/10.1090/S0002-9939-1962-0133404-1
Siegfried Grosser and Martin Moskowitz, On central topological groups, Trans. Amer. Math. Soc. 127 (1967), 317–340. MR 209394, DOI https://doi.org/10.1090/S0002-9947-1967-0209394-9
Siegfried Grosser and Martin Moskowitz, Representation theory of central topological groups, Trans. Amer. Math. Soc. 129 (1967), 361–390. MR 229753, DOI https://doi.org/10.1090/S0002-9947-1967-0229753-8
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI https://doi.org/10.1007/BF02392428
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI https://doi.org/10.2307/1969423
Shingo Murakami, Remarks on the structure of maximally almost periodic groups, Osaka Math. J. 2 (1950), 119–129. MR 41860
Lewis C. Robertson, A note on the structure of Moore groups, Bull. Amer. Math. Soc. 75 (1969), 594–599. MR 245721, DOI https://doi.org/10.1090/S0002-9904-1969-12252-4
Theodore W. Wilcox, On the structure of maximally almost periodic groups, Math. Scand. 23 (1969), 221–232 (1969). MR 252568, DOI https://doi.org/10.7146/math.scand.a-10914