Unbounded symmetric operators in $K$-homology and the Baum-Connes Conjecture

Details Unbounded symmetric operators in $K$-homology and the Baum-Connes Conjecture Unbounded symmetric operators in $K$-homology and the Baum-Connes Conjecture

2004-12-18

arxiv.org/abs/math/0412364v1

Mathematics

K-Theory and Homology

42 pages, 3 figures

Scientific paper

Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an ``addition formula'' for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs are provided, one using topology and the other one, surprisingly involved, sticking to analysis, on the basis of the previous result. As a second application, we construct, in a purely analytical language, various homomorphisms linking the homology of a group in low degree, the K-homology of its classifying space and the analytic K-theory of its C^*-algebra, in close connection with the Baum-Connes assembly map. For groups classified by a 2-complex, this allows to reformulate the Baum-Connes Conjecture.

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