Introduction to the Baum-Connes Conjecture

di Alain Valette (Autore)

Birkhauser Verlag AG, 2002

A quick description of the conjecture The Baum-Connes conjecture is part of Alain Connes'tantalizing "noncommuta- tive geometry" programme [18]. It is in some sense the most "commutative" part of this programme, since it bridges with classical geometry and topology. Let r be a countable group. The Baum-Connes conjecture identifies two objects associated with r, one analytical and one geometrical/topological. The right-hand side of the conjecture, or analytical side, involves the K- theory of the reduced C*-algebra c;r, which is the C*-algebra generated by r in 2 its left regular representation on the Hilbert space C(r). The K-theory used here, Ki(C;r) for i = 0, 1, is the usual topological K-theory for Banach algebras, as described e.g. in [85]. The left-hand side of the conjecture, or geometrical/topological side RKf(Er) (i=O,I), is the r-equivariant K-homology with r-compact supports of the classifying space Er for proper actions of r. If r is torsion-free, this is the same as the K-homology (with compact supports) of the classifying space Br (or K(r,l) Eilenberg-Mac Lane space). This can be defined purely homotopically.

• Autore: Alain Valette
• Editore: Birkhauser Verlag AG
• Collana: Lectures in Mathematics. ETH Zürich
• Anno: 2002
• Rilegatura: Paperback / softback
• Pagine: 104 p.
• Dimensioni: 244 x 170 mm
• EAN: 9783764367060

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