Duality theory for locally compact groups with precompact conjugacy classes. II. The dual space

Sund, Terje

doi:10.1090/S0002-9947-1976-0439982-1

Duality theory for locally compact groups with precompact conjugacy classes. II. The dual space
HTML articles powered by AMS MathViewer

by Terje Sund PDF

Trans. Amer. Math. Soc. 224 (1976), 313-321 Request permission

Abstract:

The present paper is concerned with the dual space Ĝ consisting of all unitary equivalence classes of continuous irreducible unitary representations of separable ${[FC]^ - }$ groups (i.e. groups with precompact conjugacy classes). The main purpose of the paper is to extend certain results from the duality theory of abelian groups and [Z] groups to the larger class of ${[FC]^ - }$ groups. In addition, we deal briefly with square-integrability for representations of ${[FC]^ - }$ groups. Most of our results are proved for type I groups. Our key result is that Ĝ may be written as a disjoint union of abelian topological ${T_4}$ groups, which are open in Ĝ.

References

Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62 (1966), 199. MR 207910
Larry Baggett, A separable group having a discrete dual space is compact, J. Functional Analysis 10 (1972), 131–148. MR 0346090, DOI 10.1016/0022-1236(72)90045-6
Larry Baggett and Adam Kleppner, Multiplier representations of abelian groups, J. Functional Analysis 14 (1973), 299–324. MR 0364537, DOI 10.1016/0022-1236(73)90075-x
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
Ky Fan, On local connectedness of locally compact Abelian groups, Math. Ann. 187 (1970), 114–116. MR 262414, DOI 10.1007/BF01350176
J. M. G. Fell, A new proof that nilpotent groups are CCR, Proc. Amer. Math. Soc. 13 (1962), 93–99. MR 133404, DOI 10.1090/S0002-9939-1962-0133404-1
J. M. G. Fell, Weak containment and induced representations of groups, Canadian J. Math. 14 (1962), 237–268. MR 150241, DOI 10.4153/CJM-1962-016-6
James Glimm, Locally compact transformation groups, Trans. Amer. Math. Soc. 101 (1961), 124–138. MR 136681, DOI 10.1090/S0002-9947-1961-0136681-X
Siegfried Grosser, Richard Mosak, and Martin Moskowitz, Duality and harmonic analysis on central topological groups. I, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973), 65–77. MR 0340470, DOI 10.1016/1385-7258(73)90039-5
Siegfried Grosser and Martin Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1–40. MR 284541, DOI 10.1515/crll.1971.246.1
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, DOI 10.1007/978-1-4419-8638-2
Eberhard Kaniuth, Zur harmonischen Analyse klassenkompakter Gruppen, Math. Z. 110 (1969), 297–305 (German). MR 263992, DOI 10.1007/BF01110324
Eberhard Kaniuth and Günter Schlichting, Zur harmonischen Analyse klassenkompakter Gruppen. II, Invent. Math. 10 (1970), 332–345 (German). MR 316973, DOI 10.1007/BF01418779
Adam Kleppner and Ronald L. Lipsman, The Plancherel formula for group extensions. I, II, Ann. Sci. École Norm. Sup. (4) 5 (1972), 459–516; ibid. (4) 6 (1973), 103–132. MR 342641, DOI 10.24033/asens.1235
H. Leptin, Zur harmonischen Analyse klassenkompakter Gruppen, Invent. Math. 5 (1968), 249–254 (German). MR 233936, DOI 10.1007/BF01389775
John R. Liukkonen, Dual spaces of groups with precompact conjugacy classes, Trans. Amer. Math. Soc. 180 (1973), 85–108. MR 318390, DOI 10.1090/S0002-9947-1973-0318390-5
George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI 10.1007/BF02392428
Richard D. Mosak, The $L^{1}$- and $C^{\ast }$-algebras of $[FIA]^{-}_{B}$ groups, and their representations, Trans. Amer. Math. Soc. 163 (1972), 277–310. MR 293016, DOI 10.1090/S0002-9947-1972-0293016-7
Martin Moskowitz, Homological algebra in locally compact abelian groups, Trans. Amer. Math. Soc. 127 (1967), 361–404. MR 215016, DOI 10.1090/S0002-9947-1967-0215016-3
L. S. Pontryagin, Nepreryvnye gruppy, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1954 (Russian). 2d ed. MR 0071705
Lewis C. Robertson, A note on the structure of Moore groups, Bull. Amer. Math. Soc. 75 (1969), 594–599. MR 245721, DOI 10.1090/S0002-9904-1969-12252-4
I. Schochetman, Topology and the duals of certain locally compact groups, Trans. Amer. Math. Soc. 150 (1970), 477–489. MR 265513, DOI 10.1090/S0002-9947-1970-0265513-X
Terje Sund, Duality theory for locally compact groups with precompact conjugacy classes. I. The character space, Trans. Amer. Math. Soc. 211 (1975), 185–202. MR 387490, DOI 10.1090/S0002-9947-1975-0387490-8