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Finite propagation speed and Connes' foliation algebra

Published online by Cambridge University Press:  24 October 2008

John Roe
John Roe
Affiliation:
Mathematical Institute, University of Oxford
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Extract

In [4], A. Connes has defined the convolution algebra associated to a foliation ℱ of the compact manifold M. Here is the graph or holonomy groupoid of the foliation ℱ (Winkelnkemper [15]). By forming the completion of in its regular representation, he obtains the C*-algebra C*{M, ℱ) associated to the foliation. The completeness of C*(M, ℱ) makes it easier to handle in some analytical contexts, but in others it seems to be too big, and it is necessary to consider instead some carefully selected dense subalgebra (cf. [6]). The purpose of this note is to show that certain spectral functions of leafwise elliptic operators, which might a priori be expected to belong to C*(M, ℱ), in fact belong to the more controllable dense subalgebra . We give a couple of applications, including a proof not passing through C*-algebras of Connes' index theorem for measured foliations [4]. It should be emphasized that the proof of that result offered here is essentially Connes' one, but the presentation may perhaps be more congenial to those who are not C*-algebra specialists.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 3 , November 1987 , pp. 459 - 466
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

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