Abstract
The author surveys Connes’ results on the longitudinal Laplace operator along a (regular) foliation and its spectrum, and discusses their generalization to any singular foliation on a compact manifold. Namely, it is proved that the Laplacian of a singular foliation is an essentially self-adjoint operator (unbounded) and has the same spectrum in every (faithful) representation, in particular, in L 2 of the manifold and L 2 of a leaf. The author also discusses briefly the relation of the Baum-Connes assembly map with the calculation of the spectrum.
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This work was supported by a Marie Curie Career Integration Grant (No. FP7-PEOPLE-2011-CIG, No. PCI09-GA-2011-290823) and the FCT (Portugal) with European Regional Development Fund (COMPETE) and national funds through the project PTDC/MAT/098770/2008.
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Androulidakis, I. Laplacians and spectrum for singular foliations. Chin. Ann. Math. Ser. B 35, 679–690 (2014). https://doi.org/10.1007/s11401-014-0858-4
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DOI: https://doi.org/10.1007/s11401-014-0858-4