Stabilité des $C^{*}$-algèbres de feuilletages

Hilsum, Michel; Skandalis, Georges

Annales de l'Institut Fourier, Tome 33 (1983) no. 3, pp. 201-208.

Soit $A$ la $C^{*}$-algèbre, ou bien réduite ou bien maximale, associée à la variété feuilletée $(V,F)$, et $K$ la $C^{*}$-algèbre élémentaire des opérateurs compacts. Alors, si dim$F\ne 0$, on montre que $A$ est isomorphe à $A\otimes K$.

Let $A$ be either the reduced or the maximal $C^{*}$-algebra associated to a foliated manifold $V,F$, and let $K$ be the elementary $C^{*}$-algebra of compact operators. Then, it dim$F\ne 0$, it is shown that $A$ is isomorphic to the tensor product $A\otimes K$.

[1] A. Connes, Sur la théorie non commutative de l'intégration, Lect. Notes in Math., n° 725, Springer (1979), 19 à 143. | MR | Zbl

[2] A. Connes, Survey of foliations and operator algebras, Operator algebras and applications, Proc. of Symp. in Pure Math., vol 38, part 1, A.M.S., Providence 1982. | Zbl

[3] A. Connes, G. Skandalis, The longitudinal index theorem for foliations, Preprint I.H.E.S./M/82/24. | Zbl

[4] G.G. Kasparov, Hilbert C*-modules, Theorems of Stinespring and Voiculescu, Journal of Operator Theory, vol. 4 n° 1 (1980). | MR | Zbl

[5] J.F. Plante, Foliations with measure preserving holonomy, Ann. of Math., 102 (1975). | MR | Zbl

[6] J.N. Renault, A groupoid approach to C*-algebras, Lect. Notes in Math., n° 793, Springer (1980). | MR | Zbl

[7] M. Rieffel, Morita equivalence for C* and W* algebras, Journal of Pure and Applied Algebra, 5 (1974). | MR | Zbl