The Baum-Connes Conjecture via Localization of Categories

By R. Meyer, R. Nest. 2003

240 citations recorded by the source corpus.

We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This interpretation is ideal for studying functorial properties and gives analogues of the Baum-Connes assembly map for all equivariant homology theories.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c0e9883c87bba56454b7e9390289f32836076914

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Citations in the corpus

Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions (2016, background): “ence for every action α: GyO∞. An analogous fact is even true for all strongly self-absorbing C∗-algebras in place of O∞. The key fact that we get from Baum-Connes is the following: Theorem 4.11 (cf. [47]). Let Gbe a countable, amenable and torsion-free group. If Ais a separable C∗-algebra with KK(A”
Homological algebra in bivariant K-theory and other triangulated categories. II (2008, unknown): “If L̃pq is the range of the map lim←−r G q(Ãr)→ Gq(Ãp) for all p and q, then the ABC spectral sequence converges towards lim←−r G q(Ãr) and induces on this limit the decreasing filtration by the subspaces ker ( lim←−r G q(Ãr)→ Gq(Ãp) ) for p ∈ N.”
A universal characterization of higher algebraic K-theory (2010, background): “The “motivic” idea of constructing universal invariants is not new and appears in several different subjects: for example, Cortiñas-Thom’s work [11] on bivariant algebraic K-theory, Higson’s work [27] on Kasparov’s bivariant Ktheory, Meyer-Nest’s work [43] on C-algebras, Morel-Voevodsky’s work [41] ”
Notes on Motives in Finite Characteristic (2007, background): “Nest introduced in [8] a non-commutative analog of the triangulated category of spectra.”
Bivariant algebraic K-theory (2006, background): “py category and derived categories. Over the last few years, several papers have appeared, in which triangulated categories of rings or algebras (in particular operator algebras) are considered (e.g. [27, 22, 29]). Here, we will show that kkis a triangulated category. The deﬁnition of triangulated c”
On the K-theory of crossed products by automorphic semigroup actions (2012, methodology): “It is taken from [13], but is based on earlier work in [5, 25, 12], and gives a more detailed formulation of the principle (DC) of the introduction:”
The structure of crossed products of irrational rotation algebras by finite subgroups of SL2(ℤ) (2006, result): “A much stronger result than the one formulated in the proposition below has been obtained by Meyer and Nest in [47], Proposition 8.”
Noncommutative Geometry, Quantum Fields and Motives (2007, background): “Moreover, a recent result of Meyer and Nest [250] shows that the additive category KK is in fact a triangulated category.”
The Baum–Connes conjecture for free orthogonal quantum groups (2009, methodology): “As already mentioned above, our approach is motivated from the work of Meyer and Nest [20].”
Higher K-theory via universal invariants (2007, background): “This result gives us a completely new way to think about algebraic K-theory and furnishes us for free the higher Chern characters from K-theory to cyclic homology [26].”