Haar measure for measure groupoids

Hahn, Peter

doi:10.1090/S0002-9947-1978-0496796-6

Haar measure for measure groupoids
HTML articles powered by AMS MathViewer

by Peter Hahn PDF

Trans. Amer. Math. Soc. 242 (1978), 1-33 Request permission

Abstract:

It is proved that Mackey’s measure groupoids possess an analogue of Haar measure for locally compact groups; and many properties of the group Haar measure generalize. Existence of Haar measure for groupoids permits solution of a question raised by Ramsay. Ergodic groupoids with finite Haar measure are characterized.

References

Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62 (1966), 199. MR 207910
J. Dixmier, Algèbres quasi-unitaires, Comment. Math. Helv. 26 (1952), 275–322 (French). MR 52697, DOI 10.1007/BF02564306
Edward G. Effros, Global structure in von Neumann algebras, Trans. Amer. Math. Soc. 121 (1966), 434–454. MR 192360, DOI 10.1090/S0002-9947-1966-0192360-9
James Glimm, Families of induced representations, Pacific J. Math. 12 (1962), 885–911. MR 146297, DOI 10.2140/pjm.1962.12.885

Haar measure and convolution algebras on ergodic groupoids

Peter Hahn, The regular representations of measure groupoids, Trans. Amer. Math. Soc. 242 (1978), 35–72. MR 496797, DOI 10.1090/S0002-9947-1978-0496797-8
Wolfgang Krieger, On constructing non-$^{\ast }$isomorphic hyperfinite factors of type III, J. Functional Analysis 6 (1970), 97–109. MR 0259624, DOI 10.1016/0022-1236(70)90049-2

Topologie

George W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134–165. MR 89999, DOI 10.1090/S0002-9947-1957-0089999-2
George W. Mackey, Ergodic theory, group theory, and differential geometry, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 1184–1191. MR 165034, DOI 10.1073/pnas.50.6.1184
George W. Mackey, Ergodic theory and virtual groups, Math. Ann. 166 (1966), 187–207. MR 201562, DOI 10.1007/BF01361167
F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, DOI 10.2307/1968693
Arlan Ramsay, Virtual groups and group actions, Advances in Math. 6 (1971), 253–322 (1971). MR 281876, DOI 10.1016/0001-8708(71)90018-1
Arlan Ramsay, Boolean duals of virtual groups, J. Functional Analysis 15 (1974), 56–101. MR 0374386, DOI 10.1016/0022-1236(74)90026-3
Manuel Samuelidès and Jean-Luc Sauvageot, Algèbre de Krieger d’un système dynamique, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), Ai, A709–A712. MR 419735
Anthony Karel Seda, Un concept de mesures invariantes pour les groupoïdes topologiques, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), no. 23, Aii, A1603–A1605. MR 396836
Joel J. Westman, Harmonic analysis on groupoids, Pacific J. Math. 27 (1968), 621–632. MR 244443, DOI 10.2140/pjm.1968.27.621

Nontransitive groupoid algebras

Ergodic groupoid algebras and their representations