On the continuity of Haar measure on topological groupoids

Seda, Anthony Karel

doi:10.1090/S0002-9939-1986-0813822-2

On the continuity of Haar measure on topological groupoids
HTML articles powered by AMS MathViewer

by Anthony Karel Seda PDF

Proc. Amer. Math. Soc. 96 (1986), 115-120 Request permission

Abstract:

It is shown that continuity of a family of invariant (Haar) measures on a topological groupoid $G$ is equivalent to the continuity of the implied convolution product $f * g$ for all pairs of functions $f$ and $g$. An example is given of a groupoid which admits no (continuous) Haar measure. It results, therefore, that the usual ${C^ * }$-algebra associated with a Haar measure on $G$ cannot, in general, be constructed. Some remarks are included concerning the construction of Haar measures on the holonomy groupoid of a foliated manifold.

References

Alain Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 19–143 (French). MR 548112
T. Fack and G. Skandalis, Sur les représentations et idéaux de la $C^{\ast }$-algèbre d’un feuilletage, J. Operator Theory 8 (1982), no. 1, 95–129 (French). MR 670180
Paul S. Muhly and Jean N. Renault, $C^{\ast }$-algebras of multivariable Wiener-Hopf operators, Trans. Amer. Math. Soc. 274 (1982), no. 1, 1–44. MR 670916, DOI 10.1090/S0002-9947-1982-0670916-3
Hiroshi Takai and Toshikazu Natsume, A. Connes’ noncommutative differential geometry, Sūgaku 35 (1983), no. 2, 97–112 (Japanese). MR 732432

Connes algebras associated to foliated bundles

Coverings of foliations and associated

-algebras

Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266
Jean N. Renault, $C^{\ast }$-algebras of groupoids and foliations, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 339–350. MR 679714
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
Anthony Karel Seda, A continuity property of Haar systems of measures, Ann. Soc. Sci. Bruxelles Sér. I 89 (1975), no. 4, 429–433. MR 402741
A. K. Seda, Haar measures for groupoids, Proc. Roy. Irish Acad. Sect. A 76 (1976), no. 5, 25–36. MR 427598

Quelques résultats dans la Catégorie des groupoids d’opérateurs

288

Anthony Karel Seda, Banach bundles and a theorem of J. M. G. Fell, Proc. Amer. Math. Soc. 83 (1981), no. 4, 812–816. MR 630060, DOI 10.1090/S0002-9939-1981-0630060-2
Anthony Karel Seda, Banach bundles of continuous functions and an integral representation theorem, Trans. Amer. Math. Soc. 270 (1982), no. 1, 327–332. MR 642344, DOI 10.1090/S0002-9947-1982-0642344-8
Hiroshi Takai, $C^\ast$-algebras of Anosov foliations, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 509–516. MR 799590, DOI 10.1007/BFb0074906

Nontransitive groupoid algebras

Joel J. Westman, Harmonic analysis on groupoids, Pacific J. Math. 27 (1968), 621–632. MR 244443

Groupoid theory in algebra, topology and analysis