Geodesic
Laplacians on Smooth Distributions as $C^*$-Algebra Multipliers
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical physics, Tome 152 (2018), pp. 67-90.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold started in a previous paper. Under the assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distribution defines an unbounded, regular, self-adjoint operator in some Hilbert module over the $C^*$-algebra of the foliation.
Mots-clés : foliation, multiplier.
Keywords: Hilbert module, Laplacian, hypoelliptic operator, smooth distribution 
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Yu. A. Kordyukov. Laplacians on Smooth Distributions as $C^*$-Algebra Multipliers. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical physics, Tome 152 (2018), pp. 67-90. http://geodesic.mathdoc.fr/item/INTO_2018_152_a6/

[1] Kordyukov Yu. A., “Teoriya indeksa i nekommutativnaya geometriya na mnogoobraziyakh so sloeniem”, Usp. mat. nauk, 64:2 (386) (2009), 73–202 | DOI | MR | Zbl

[2] Kordyukov Yu. A., “Laplasiany na gladkikh raspredeleniyakh”, Mat. sb., 208:10 (2017), 91–112 | DOI | MR | Zbl

[3] Manuilov V. M., Troitskii E. V., $C^*$-Gilbertovy moduli, Faktorial Press, M., 2001

[4] Androulidakis I., “Laplacians and spectrum for singular foliations”, Chin. Ann. Math. Ser. B., 35:5 (2014), 679–690 | DOI | MR | Zbl

[5] Androulidakis I., Skandalis G., “The holonomy groupoid of a singular foliation”, J. Reine Angew. Math., 626 (2009), 1–37 | DOI | MR | Zbl

[6] Androulidakis I., Skandalis G., “Pseudodifferential calculus on a singular foliation”, J. Noncommut. Geom., 5:1 (2011), 125–152 | DOI | MR | Zbl

[7] Baaj S., Multiplicateurs non bornès, Ph.D. thesis, Univ. Pierre et Marie Curie, Paris, 1980

[8] Baaj S., Julg P., “Théorie bivariante de {K}asparov et opérateurs non bornés dans les $C^{\ast}$-modules hilbertiens”, C. R. Acad. Sci. Paris. Sér. I Math., 296:21 (1983), 875–878 | MR | Zbl

[9] Chernoff P., “Essential self-adjointness of powers of generators of hyperbolic equations”, J. Funct. Anal., 12 (1973), 401–414 | DOI | MR | Zbl

[10] Connes A., “Sur la théorie non commutative de l'intégration”, Lect. Notes Math., 725 (1979), 19–143 | DOI | MR | Zbl

[11] Connes A., A survey of foliations and operator algebras, Am. Math. Soc., Providence, Rhode Island, 1982 | MR | Zbl

[12] Fack T., Skandalis G., “Sur les représentations et idéaux de la $C^{\ast}$-algèbre d'un feuilletage”, J. Operator Theory, 8:1 (1982), 95–129 | MR | Zbl

[13] Folland G. B., “Subelliptic estimates and function spaces on nilpotent Lie groups”, Ark. Mat., 13:2 (1975), 161–207 | DOI | MR | Zbl

[14] Grigor'yan A., “Heat kernels on weighted manifolds and applications”, The Ubiquitous Heat Kernel, Contemp. Math., 398, Am. Math. Soc., Providence, Rhode Island, 2006, 93–191 | DOI | MR | Zbl

[15] Hörmander L., “Hypoelliptic second-order differential equations”, Acta Math., 119 (1967), 147–171 | DOI | MR

[16] Hörmander L., Melin A., “Free systems of vector fields”, Ark. Mat., 16:1 (1978), 83–88 | DOI | MR | Zbl

[17] Kordyukov Yu. A., “Functional calculus for tangentially elliptic operators on foliated manifolds”, Analysis and geometry in foliated manifolds, World Scientific, River Edge, New Jersey, 1995, 113–136 | MR | Zbl

[18] Kordyukov Yu. A., “Noncommutative geometry of foliations”, J. K-Theory, 2:2 (2008), 219–327 | DOI | MR | Zbl

[19] Lance E. C., Hilbert $C^*$-modules. A toolkit for operator algebraists, Cambridge Univ. Press, Cambridge, 1995 | MR

[20] Montgomery R., A tour of subriemannian geometries, their geodesics and applications, Am. Math. Soc., Providence, Rhode Island, 2002 | MR | Zbl

[21] Renault J., “A groupoid approach to $C\sp{\ast} $-algebras”, Lect. Notes Math., v. 793, Springer-Verlag, Berlin, 1980 | DOI | MR | Zbl

[22] Rothschild L. P., Stein E. M., “Hypoelliptic differential operators and nilpotent groups”, Acta Math., 137:3-4 (1976), 247–320 | DOI | MR

[23] Stefan P., “Accessible sets, orbits, and foliations with singularities”, Proc. London Math. Soc. (3), 29 (1974), 699–713 | DOI | MR | Zbl

[24] Sussmann H. J., “Orbits of families of vector fields and integrability of distributions”, Trans. Am. Math. Soc., 180 (1973), 171–188 | DOI | MR | Zbl

[25] Vassout S., “Unbounded pseudodifferential calculus on Lie groupoids”, J. Funct. Anal., 236:1 (2006), 161–200 | DOI | MR | Zbl