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Introduction to the Baum-Connes Conjecture by Alain Valette

Introduction to the Baum-Connes Conjecture

by Alain Valette
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Book cover type: Paperback
  • ISBN13: 9783764367060
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Springer
  • Publisher Imprint: Birkhauser
  • Publication Date:
  • Pages: 104
  • Original Price: EUR 49.99
  • Language: English
  • Edition: 2002
  • Item Weight: 236 grams
  • BISAC Subject(s): Geometry / Differential, Geometry / Algebraic, and Algebra / General

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.

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