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Introduction to the Baum-Connes Conjecture (Paperback, 2002 ed.)
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Introduction to the Baum-Connes Conjecture (Paperback, 2002 ed.)
Series: Lectures in Mathematics. ETH Zurich
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The Baum-Connes conjecture is part of A. Connes' non-commutative
geometry programme. It can be viewed as a conjectural
generalisation of the Atiyah-Singer index theorem, to the
equivariant setting (the ambient manifold is not compact, but some
compactness is restored by means of a proper, co-compact action of
a group "gamma"). Like the Atiyah-Singer theorem, the Baum-Connes
conjecture states that a purely topological object coincides with a
purely analytical one. For a given group "gamma," the topological
object is the equivariant K-homology of the classifying space for
proper actions of "gamma," while the analytical object is the
K-theory of the C*-algebra associated with "gamma" in its regular
representation. The Baum-Connes conjecture implies several other
classical conjectures, ranging from differential topology to pure
algebra. It has also strong connections with geometric group
theory, as the proof of the conjecture for a given group "gamma"
usually depends heavily on geometric properties of "gamma." This
book is intended for graduate students and researchers in geometry
(commutative or not), group theory, algebraic topology, harmonic
analysis, and operator algebras. It presents, for the first time in
book form, an introduction to the Baum-Connes conjecture. It starts
by defining carefully the objects in both sides of the conjecture,
then the assembly map which connects them. Thereafter it
illustrates the main tool to attack the conjecture (Kasparov's
theory), and it concludes with a rough sketch of V. Lafforgue's
proof of the conjecture for co-compact lattices in in Spn1, SL(3R),
and SL(3C).
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