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Sheaf theory and Paschke duality

Published online by Cambridge University Press:  28 August 2013

John Roe  and
Paul Siegel
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Abstract

Let X be a locally compact metrizable space. We show that the Paschke dual construction, which associates to a representation of C0(X) its commutant modulo locally compact operators, can be sheafified. We use this observation to simplify several constructions in analytic K-homology.

Keywords

K-homologyPaschke dualsheaf theoryexcision46L8019K3358J22
Type
Research Article
Information
Journal of K-Theory , Volume 12 , Issue 2 , October 2013 , pp. 213 - 234
Copyright
Copyright © ISOPP 2013 

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References

1.Ara, Pere and Mathieu, Martin, Sheaves of C*-algebras, Mathematische Nachrichten 283(1) (2010), 21–39 (en).CrossRefGoogle Scholar
2.Atiyah, M.F., Global Theory of Elliptic Operators, Proceedings of the International Symposium on Functional Analysis, Tokyo, University of Tokyo Press, 1969, 21–30.Google Scholar
3.Brown, L. G., Douglas, R. G., and Fillmore, P. A., Extensions of C*-algebras and K-Homology, Annals of Mathematics 105(2) (1977), 265–324.Google Scholar
4.Godement, Roger, Topologie algebrique et theorie des faisceaux (Actualites scientifiques et industrielles), Hermann, 1998.Google Scholar
5.Gromov, Mikhael and Lawson, H. Blaine, Positive scalar curvature and the Dirac operator on complete riemannian manifolds, Publications Mathématiques de l'Institut des Hautes Études Scientifiques 58(1) (1983), 83–196 (en).Google Scholar
6.Higson, Nigel and Roe, John, The Baum-Connes Conjecture in Coarse Geometry, Proceedings of the 1993 Oberwolfach Conference on the Novikov Conjecture (Ferry, S., Ranicki, A., and Rosenberg, J., eds.), LMS Lecture Notes 227, Cambridge University Press, Cambridge, 1995, pp. 227–254.Google Scholar
7.Higson, Nigel and Roe, John, Analytic K-homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000, Oxford Science Publications.Google Scholar
8.Higson, Nigel and Roe, John, Mapping surgery to analysis. I. Analytic signatures, K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences 33(4) (2005), 277–299.Google Scholar
9.Hofmann, Karl, Bundles and sheaves are equivalent in the category of Banach spaces, K-Theory and Operator Algebras (Morrel, Bernard and Singer, I., eds.), Lecture Notes in Mathematics 575, Springer Berlin/Heidelberg, 1977, pp. 53–69.CrossRefGoogle Scholar
10.Kasparov, G G, The operator K-functor and extensions of C*-algebras, Mathematics of the USSR-Izvestiya 16(3) (1981), 513–572.Google Scholar
11.Paschke, William L., K-theory for commutants in the Calkin algebra., Pacific Journal of Mathematics 95(2) (1981), 427–434.Google Scholar
12.Roe, John, A note on the relative index theorem, The Quarterly Journal of Mathematics. Oxford. Second Series 42(167) (1991), 365–373.CrossRefGoogle Scholar
13.Roe, John, Comparing analytic assembly maps, The Quarterly Journal of Mathematics 53(2) (2002), 241–248.Google Scholar
14.Roe, John, Lectures on coarse geometry, University Lecture Series 31, American Mathematical Society, Providence, RI, 2003.Google Scholar
15.Roe, John, Positive curvature, partial vanishing theorems, and coarse indices, arXiv:1210.6100 (2012).Google Scholar
16.Siegel, Paul, Homological calculations with the analytic structure group, PhD thesis, Penn State, 09 2012.Google Scholar
17.Taylor, M., Pseudodifferential Operators, Princeton University Press, Princeton, N.J., 1982.Google Scholar
18.Daele, Alfons Van, K-theory for graded Banach algebras I, Oxford Quarterly Journal of Mathematics 39 (1988), 185–199.Google Scholar
19.Williams, Dana P., Crossed Products of C*-Algebras, American Mathematical Soc., 03 2007 (en).CrossRefGoogle Scholar