Introduction to the BaumConnes Conjecture 

Author:
 Valette, Alain 
Series title:  Lectures in Mathematics. ETH Zürich Ser. 
ISBN:  9783764367060 
Publication Date:  Apr 2002 
Publisher:  Birkhäuser Boston

Book Format:  Paperback 
List Price:  USD $59.99 
Book Description:

A quick description of the conjecture The BaumConnes 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 BaumConnes conjecture identifies two objects associated with r, one analytical and one geometrical/topological. The righthand side of the conjecture, or analytical side, involves the K...
More DescriptionA quick description of the conjecture The BaumConnes 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 BaumConnes conjecture identifies two objects associated with r, one analytical and one geometrical/topological. The righthand 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 Ktheory used here, Ki(C;r) for i = 0, 1, is the usual topological Ktheory for Banach algebras, as described e.g. in [85]. The lefthand side of the conjecture, or geometrical/topological side RKf(Er) (i=O,I), is the requivariant Khomology with rcompact supports of the classifying space Er for proper actions of r. If r is torsionfree, this is the same as the Khomology (with compact supports) of the classifying space Br (or K(r,l) EilenbergMac Lane space). This can be defined purely homotopically.