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Valette A. — Introduction to the Baum-Connes Conjecture
Valette A. — Introduction to the Baum-Connes Conjecture

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Название: Introduction to the Baum-Connes Conjecture

Автор: Valette A.


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).

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 106

Добавлена в каталог: 22.11.2009

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Предметный указатель
$\Gamma$-Banach algebra      86
$\Gamma-C^{*}$-algebra      47
$\Gamma-C^{*}$-proper      79
Action, amenable      18
Action, proper      12 33
Assembly map      15 56 60 88
B-pair      85
Banach KK-theory group      86
Bott map      30
Chern character      97
Classifying space      15
Classifying space for proper actions      92
Clifford algebra      80
Cone      29
Conjecture, Baum — Connes      6 17
Conjecture, Baum — Connes with coefficients      20
Conjecture, Baum — Connes with commutative coefficients      20
Conjecture, Gromov — Lawson — Rosenberg      6 16
Conjecture, Idempotents      6
Conjecture, Novikov      6 16
Conjecture, Valette      72
Contractible (Banach algebra)      29
Convolution      11
Crossed product, Banach      88
Crossed product, reduced      19
CW-spectrum      95
Cycle, degenerate      40 48
Cycle, equivalent      40 48
Cycle, homotopic      40 48
Cycle, Kasparov      48
Cycle, Lafforgue      86
Cycle, non self-adjoint      43
Descent homomorphism      51 88
Dirac element      81
Dual-Dirac element      82
Elliptic differential operator      39
Equivariant K-homology      40
Equivariant K-homology with compact supports      42
Exponential      27
Fourier transform      14
Functions of rapid decay      72
G-CW-complex      94
Generalized elliptic $\Gamma$-operator      37
Grothendieck group      23
Group algebra      11
Group, a-T-menable      12
Group, amenable      70
Group, polynomial growth      70
Group, property (RD)      69
Group, property (T)      13 84
Hilbert C*-module      47
Holomorphic functional calculus      75
Idempotent      11 59
Join      95
Jordan's simple curve theorem      31
K-theory, degree 0      23
K-theory, degree 1      26
length function      69
Morphism of B-pairs      85
Operator, compact      48 85
Operator, finite rank      48 85
Operator, Fredholm      37
Operator, properly supported      53
Pontryagin dual      13
Projective finite type module      23
Rational injectivity      16
Reduced C*-algebra      13
Representation, covariant      19 48 86
Representation, essential      44
Representation, left regular      13
Representation, unitary      48
Retopologization (of a Hilbert space)      34
Rips complex      35
Semi-group      23
Six-term exact sequence      31
Spectral subalgebra      74
Suspension      29
Trace      25
Trace, canonical      26
Trace, faithful      26
Trace, positive      25
Twisted convolution      19
Unconditional completion      87
Universal proper space      33
Weighted $\ell^{2}$-norm      69
Word length      69
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