Introduction to the Baum-Connes Conjecture

Alain Valette (Autor)

Buch | Softcover

X, 104 Seiten

2002 | 2002
Springer Basel (Verlag)
978-3-7643-6706-0 (ISBN)

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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.

1 Idempotents in Group Algebras.- 2 The Baum-Connes Conjecture.- 3K-theory for (Group) C*-algebras.- 4 Classifying Spaces andK-homology.- 5 EquivariantKK-theory.- 6 The Analytical Assembly Map.- 7 Some Examples of the Assembly Map.- 8 Property (RD).- 9 The Dirac-dual Dirac Method.- 10 Lafforgue'sKKBanTheory.- G. Mislin: On the Classifying Space for Proper Actions.- A.1 The topologist's model.- A.2 The analyst's model.- A.4 Spectra.

"Overall, the book is a very valuable addition to the literature on the Baum-Connes conjecture. It is highly recommended reading for anyone interested in learning more about the conjecture, or who does research in areas related to it. Of course, the reader who wants to be an expert will eventually have to consult the original literature, but such is inevitable in a book of this size (around 100 pages) and not necessarily a bad thing."

--Mathematical Reviews

Erscheint lt. Verlag	1.4.2002
Reihe/Serie	Lectures in Mathematics. ETH Zürich
Zusatzinfo	X, 104 p.
Verlagsort	Basel
Sprache	englisch
Maße	170 x 244 mm
Gewicht	240 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Schlagworte	Algebra • algebraic topology • group algebras • group theory • Homology • K-theory • Non-Commutative Geometry
ISBN-10	3-7643-6706-7 / 3764367067
ISBN-13	978-3-7643-6706-0 / 9783764367060
Zustand	Neuware