Functional analysis, Harmonic analysis
Continuity of the dual Haar measure
Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 415-419.

Given a continuous field of locally compact groups, we show that the field of the Plancherel weights of their C*-algebras is lower semi-continuous. As a corollary, we obtain that the dual Haar system of a continuous Haar system of a locally compact abelian group bundle is also continuous.

Étant donné un champ continu de groupes localement compacts, on montre que le champ des poids de Plancherel de leurs C*-algèbres est semi-continu inférieurement. On en déduit que, lorsque les groupes sont abéliens, le système de Haar dual d’un système de Haar continu est aussi continu.

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.183
Renault, Jean 1

1 Institut Denis Poisson (UMR 7013) Université d’Orléans et CNRS, 45067 Orléans Cedex 2, France.
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Renault, Jean. Continuity of the dual Haar measure. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 415-419. doi : 10.5802/crmath.183. http://www.numdam.org/articles/10.5802/crmath.183/

[1] Combes, François Poids sur une C * -algèbre, J. Math. Pures Appl., Volume 47 (1968), pp. 57-100 | MR | Zbl

[2] Combes, François Poids associé à une algèbre hilbertienne à gauche, Compos. Math., Volume 23 (1971) no. 1, pp. 49-77 | Numdam | Zbl

[3] Dixmier, Jacques Les algèbres de von Neumann, Cahiers scientifiques, 25, Gauthier-Villars, 1969 | Zbl

[4] Dixmier, Jacques Les C * -algèbres et leurs représentations, Cahiers scientifiques, 29, Gauthier-Villars, 1969 | Zbl

[5] Landsman, Nicolaas P.; Ramazan, Birant Quantization of Poisson algebras associated to Lie algebroids, Groupoids in analysis, geometry, and physics. AMS–IMS–SIAM joint summer research conference, University of Colorado, Boulder, CO, USA, June 20-24, 1999 (Contemporary Mathematics), Volume 282, American Mathematical Society (2001), pp. 159-192 | MR | Zbl

[6] Muhly, Paul S.; Renault, Jean N.; Williams, Dana P. Equivalence and isomorphism for groupoid C * -algebras, J. Oper. Theory, Volume 17 (1987), pp. 3-22 | MR | Zbl

[7] Muhly, Paul S.; Renault, Jean N.; Williams, Dana P. Continuous trace groupoid C * -algebras. III, Trans. Am. Math. Soc., Volume 348 (1996) no. 9, pp. 3621-3641 | DOI | MR | Zbl

[8] Pedersen, Gert C*-algebras and their automorphism groups, London Mathematical Society Monographs, 14, Academic Press Inc., 1979 | MR | Zbl

[9] Renault, Jean N. A groupoid approach to C * -algebras, Lecture Notes in Mathematics, 793, Springer, 1980 | MR | Zbl

[10] Strătilă, Şerban Modular theory in operator algebras, Abacus Press, 1981 | Zbl

[11] Takesaki, Masamichi Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, 128, Springer, 1970 | MR | Zbl

[12] Takesaki, Masamichi Theory of operator algebras. II., Encyclopaedia of Mathematical Sciences, 125, Springer, 2003 | MR | Zbl

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