Duality theory for locally compact groups with precompact conjugacy classes. I. The character space
HTML articles powered by AMS MathViewer
- by Terje Sund
- Trans. Amer. Math. Soc. 211 (1975), 185-202
- DOI: https://doi.org/10.1090/S0002-9947-1975-0387490-8
- PDF | Request permission
Abstract:
Let G be a locally compact group, and let $\mathcal {X}(G)$ consist of the nonzero extreme points of the set of continuous, G-invariant, positive definite functions f on G such that $f(e) \leq 1$. $\mathcal {X}(G)$ is called the character space, and is given the topology of uniform convergence on compacta. The purpose of the present paper is to extend the main results from the duality theory of abelian groups and [Z] groups to the class of ${[FC]^ - }$ groups (i.e., groups with precompact conjugacy classes), letting $\mathcal {X}(G)$ play the role of the character group in the abelian theory. Some of our theorems are only proved for the class ${[FD]^ - }\;( \subset {[FC]^ - })$. If $G \in {[FC]^ - }$ then $\mathcal {X}(G) \approx \mathcal {X}(H)$ where H is a certain ${[FIA]^ - }$ quotient group. Hence there is no loss of generality to study character spaces of ${[FIA]^ - }$ groups.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 note on groups with finite dual spaces, Pacific J. Math. 31 (1969), 569–572. MR 259015, DOI 10.2140/pjm.1969.31.569
- 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
- Jacques Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (Algèbres de von Neumann), Cahiers Scientifiques, Fasc. XXV, Gauthier-Villars, Paris, 1957 (French). MR 0094722
- 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
- F. P. Greenleaf and M. Moskowitz, Cyclic vectors for representations associated with positive definite measures: nonseparable groups, Pacific J. Math. 45 (1973), 165–186. MR 349896, DOI 10.2140/pjm.1973.45.165
- 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
- A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001
- 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
- J. Liukkonen and R. Mosak, Harmonic analysis and centers of group algebras, Trans. Amer. Math. Soc. 195 (1974), 147–163. MR 350322, DOI 10.1090/S0002-9947-1974-0350322-7
- John Liukkonen and Richard Mosak, The primitive dual space of $[FC]^{-}$ groups, J. Functional Analysis 15 (1974), 279–296. MR 0346088, DOI 10.1016/0022-1236(74)90036-6
- George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI 10.1007/BF02392428
- Calvin C. Moore and Joseph A. Wolf, Square integrable representations of nilpotent groups, Trans. Amer. Math. Soc. 185 (1973), 445–462 (1974). MR 338267, DOI 10.1090/S0002-9947-1973-0338267-9
- 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
- L. S. Pontryagin, Topological groups, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. Translated from the second Russian edition by Arlen Brown. MR 0201557
- 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
- A. I. Štern, Separable locally compact groups with discrete support for a regular representation, Dokl. Akad. Nauk SSSR 198 (1971), 1287–1290 (Russian). MR 0289722
Bibliographic Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 211 (1975), 185-202
- MSC: Primary 22D35; Secondary 22D10
- DOI: https://doi.org/10.1090/S0002-9947-1975-0387490-8
- MathSciNet review: 0387490