On the continuity of Haar measure on topological groupoids
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- 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, Preprint, Saitama and Tokyo Metropolitan Universities, Japan, 1981. ${C^ * }$-News, No. 16. Moto O’uchi, Coverings of foliations and associated ${C^ * }$-algebras, Preprint, Ehime University, Japan. ${C^ * }$-News, No. 20.
- 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, C. R. Acad. Sci. Paris Sér. A 288 (1979), 21-24.
- 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 J. J. Westman, Nontransitive groupoid algebras, Univ. of California at Irvine, 1967.
- Joel J. Westman, Harmonic analysis on groupoids, Pacific J. Math. 27 (1968), 621–632. MR 244443 —, Groupoid theory in algebra, topology and analysis, Univ. of California at Irvine, 1971.
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 96 (1986), 115-120
- MSC: Primary 46L99; Secondary 22D40, 28C10, 43A05
- DOI: https://doi.org/10.1090/S0002-9939-1986-0813822-2
- MathSciNet review: 813822