Baum-Connes Conjecture Research Articles

Torsion-freeness for fusion rings and tensor C*-categories

Torsion-freeness for discrete quantum groups was introduced by R. Meyer in order to formulate a version of the Baum–Connes conjecture for discrete quantum groups. In this note, we introduce torsion-freeness for abstract fusion rings. We show that a discrete quantum group is torsion-free if its associated fusion ring is torsion-free. In the latter case, we say that the discrete quantum group is strongly torsion-free. As applications, we show that the discrete quantum group duals of the free unitary quantum groups are strongly torsion-free, and that torsion-freeness of discrete quantum groups is preserved under Cartesian and free products. We also discuss torsion-freeness in the more general setting of abstract rigid tensor C*-categories.

Crystallographic T-duality

We introduce the notion of crystallographic T-duality, inspired by the appearance of K-theory with graded equivariant twists in the study of topological crystalline materials. Besides giving a range of new topological T-dualities, it also unifies many previously known dualities, motivates generalisations of the Baum–Connes conjecture to graded groups, provides a powerful tool for computing topological phase classification groups, and facilitates the understanding of crystallographic bulk–boundary correspondences in physics.

A coarse Cartan–Hadamard theorem with application to the coarse Baum–Connes conjecture

We establish a coarse version of the Cartan–Hadamard theorem, which states that proper coarsely convex spaces are coarsely homotopy equivalent to the open cones of their ideal boundaries. As an application, we show that such spaces satisfy the coarse Baum–Connes conjecture. Combined with the result of Osajda–Przytycki, it implies that systolic groups and locally finite systolic complexes satisfy the coarse Baum–Connes conjecture.

Burghelea conjecture and asymptotic dimension of groups

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum–Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which together imply the Burghelea conjecture for such groups. We prove both conjectures for many classes of groups. It is known that the Burghelea conjecture does not hold for all groups, although no finitely presentable counterexample was known. We construct a finitely presentable (even type [Formula: see text]) counterexample based on Thompson’s group [Formula: see text]. We construct as well a finitely generated counterexample with finite decomposition complexity.

Equivariant K-homology for hyperbolic reflection groups

We compute the equivariant $K$-homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological $K$-theory of the reduced $C^\ast$-algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated $K$-theory groups are torsion-free. As a result we can promote previous rational computations to integral compu- tations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many other classes of groups.

A categorical perspective on the Atiyah–Segal completion theorem in KK-theory

We investigate the homological ideal \mathcal J _G^H , the kernel of the restriction functors in compact Lie group equivariant Kasparov categories. Applying the relative homological algebra developed by Meyer and Nest, we relate the Atiyah–Segal completion theorem with the comparison of \mathcal J_G^H with the augmentation ideal of the representation ring. In relation to it, we study on the Atiyah–Segal completion theorem for groupoid equivariant KK-theory, McClure's restriction map theorem and permanence property of the Baum–Connes conjecture under extensions of groups.

On coarse geometric aspects of Hilbert geometry

We begin a coarse geometric study of Hilbert geometry. Actually we give a necessary and sufficient condition for the natural boundary of a Hilbert geometry to be a corona, which is a nice boundary in coarse geometry. In addition, we show that any Hilbert geometry is uniformly contractible and with coarse bounded geometry. As a consequence of these we see that the coarse Novikov conjecture holds for a Hilbert geometry with a mild condition. Also we show that the asymptotic dimension of any two-dimensional Hilbert geometry is just two. This implies that the coarse Baum–Connes conjecture holds for any two-dimensional Hilbert geometry via Yu’s theorem.

An equivariant index for proper actions II: Properties and applications

In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In this paper, we investigate properties and applications of this index. We prove that it has an induction property that can be used to deduce various other properties of the index. In the case of compact orbit spaces, the index is a special case of Kasparov’s index of transversally elliptic operators. In that case, we show how it is related to the analytic assembly map from the Baum–Connes conjecture, and an index used by Mathai and Zhang. In the case of noncompact orbit spaces, we use the index to define a notion of K -homological Dirac induction, and show that, under conditions, it satisfies the quantisation commutes with reduction principle.

Controlled K-theory for groupoids & applications to Coarse Geometry

We develop a generalization of quantitative K-theory, which we call controlled K-theory. It is powerful enough to study the K-theory of crossed product of C⁎-algebras by action of étale groupoids and discrete quantum groups. In this article, we will use it to study groupoids crossed products. We define controlled assembly maps, which factorize the Baum–Connes assembly maps, and define the controlled Baum–Connes conjecture. We relate the controlled conjecture for groupoids to the classical conjecture, and to the coarse Baum–Connes conjecture. This allows to give applications to Coarse Geometry. In particular, we can prove that the maximal version of the controlled coarse Baum–Connes conjecture is satisfied for a coarse space which admits a fibred coarse embedding, which is a stronger version of a result of M. Finn-Sell.

The Minimal Exact Crossed Product

Given a locally compact group G , we study the smallest exact crossed-product functor (A,G,\alpha)\mapsto A\rtimes_{\mathcal E} G on the category of G - C^* -dynamical systems. As an outcome, we show that the smallest exact crossed-product functor is automatically Morita compatible, and hence coincides with the functor \rtimes_{\mathcal E} as introduced by P. Baum et al. in their reformulation of the Baum–Connes conjecture [Ann. K -Theory 1, No. 2, 155–208 (2016; Zbl 1331.46064)]. We show that the corresponding group algebra C_{\mathcal E}^*(G) always coincides with the reduced group algebra, thus showing that the new formulation of the Baum–Connes conjecture coincides with the classical one in the case of trivial coefficients.

Coaction functors, II

In further study of the application of crossed-product functors to the Baum-Connes Conjecture, Buss, Echterhoff, and Willett introduced various other properties that crossed-product functors may have. Here we introduce and study analogues of these properties for coaction functors, making sure that the properties are preserved when the coaction functors are composed with the full crossed product to make a crossed-product functor. The new properties for coaction functors studied here are functoriality for generalized homomorphisms and the correspondence property. We particularly study the connections with the ideal property. The study of functoriality for generalized homomorphisms requires a detailed development of the Fischer construction of maximalization of coactions with regard to possibly degenerate homomorphisms into multiplier algebras. We verify that all "KLQ" functors arising from large ideals of the Fourier-Stieltjes algebra $B(G)$ have all the properties we study, and at the opposite extreme we give an example of a coaction functor having none of the properties.

On the relation between K- and L-theory of $$C^*$$ C ∗ -algebras

We prove the existence of a map of spectra $$\tau _A:kA \rightarrow \ell A$$ between connective topological K-theory and connective algebraic L-theory of a complex $$C^*$$ -algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural equivalence $$KA\left[ \tfrac{1}{2}\right] \xrightarrow {\simeq } LA\left[ \tfrac{1}{2}\right] $$ of periodic K- and L-theory spectra after inverting 2. We show that this equivalence extends to K- and L-theory of real $$C^*$$ -algebras. Using this we give a comparison between the real Baum–Connes conjecture and the L-theoretic Farrell–Jones conjecture. We conclude that these conjectures are equivalent after inverting 2 if and only if a certain completion conjecture in L-theory is true.

K-THEORY FOR THE C*-ALGEBRAS OF THE SOLVABLE BAUMSLAG–SOLITAR GROUPS

AbstractWe provide a new computation of the K-theory of the group C*-algebra of the solvable Baumslag–Solitar group BS(1, n) (n ≠ 1); our computation is based on the Pimsner–Voiculescu 6-terms exact sequence, by viewing BS(1, n) as a semi-direct product ℤ[1/n] ⋊ ℤ. We deduce from it a new proof of the Baum–Connes conjecture with trivial coefficients for BS(1, n).

K ‐homology and K ‐theory for the lamplighter groups of finite groups

Let F be a finite group. We consider the lamplighter group L = F ≀ Z over F. We prove that L has a classifying space for proper actions E ̲ L which is a complex of dimension 2. We use this to give an explicit proof of the Baum–Connes conjecture (without coefficients) that states that the assembly map μ i L : K i L ( E ̲ L ) → K i ( C ∗ L ) ( i = 0 , 1 ) is an isomorphism. Actually, K 0 ( C ∗ L ) is free abelian of countable rank, with an explicit basis consisting of projections in C ∗ L , while K 1 ( C ∗ L ) is infinite cyclic, generated by the unitary of C ∗ L implementing the shift. Finally we show that, for F abelian, the C ∗ -algebra C ∗ L is completely characterized by | F | up to isomorphism.

Kazhdan projections, random walks and ergodic theorems

Abstract In this paper we investigate generalizations of Kazhdan’s property (T) to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and that of Kazhdan projections. Our methods employ Markov operators associated to a random walk on the group, for which we provide new norm estimates and convergence results. This construction exhibits useful properties and flexibility, and allows to view Kazhdan projections in Banach spaces as natural objects associated to random walks on groups. We give a number of applications of these results. In particular, we address several open questions. We give a direct comparison of properties (TE) and FE with Lafforgue’s reinforced Banach property (T); we obtain shrinking target theorems for orbits of Kazhdan groups; finally, answering a question of Willett and Yu we construct non-compact ghost projections for warped cones. In this last case we conjecture that such warped cones provide counterexamples to the coarse Baum–Connes conjecture.

Local-Global Principle for the Baum–Connes Conjecture with Coefficients

We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group GL(2,F) where F is any global field.

Un analogue non archimédien d'un résultat de Haagerup et lien avec la propriété (T) renforcée

Inspired by a preprint of Haagerup and failed attempts to prove the Baum–Connes conjecture, we study the approximation of the trivial function on SL_3 or Sp_4 on a local non-archimedian field, by spherical functions with good Schur multiplier properties when restricted to the Borel subgroup, or its unipotent radical.

About the obstacle to proving the Baum–Connes conjecture without coefficient for a non-cocompact lattice in $Sp_4$ in a local field

We introduce property (T_\mathrm {Schur},G,K) and prove it for some non-cocompact lattice in Sp_4 in a local field of finite characteristic. We show that property (T_\mathrm {Schur},G,K) for a non-cocompact lattice \Gamma in a higher rank almost simple algebraic group in a local field is an obstacle to proving the Baum Connes conjecture without coefficient for \Gamma with known methods, and this is stronger than the well-known fact that \Gamma does not have the property of rapid decay (property (RD)). It is the first example (as announced in [7]) for which all known methods for proving the Baum–Connes conjecture without coefficient fail.

On an intermediate bivariant $K$-theory for $C^*$-algebras

We construct a new bivariant K -theory for C^* -algebras, that we call KE -theory. For each pair of separable graded C^* -algebras A and B , acted upon by a locally compact \sigma -compact group G , we define an abelian group KE_G(A,B) . We show that there is an associative product KE_G(A,D) \otimes KE_G(D,B) \rightarrow KE_G(A,B) . Various functoriality properties of the KE -theory groups and of the product are presented. The new theory is intermediate between the KK -theory of G.G. Kasparov, and the E -theory of A. Connes and N. Higson, in the sense that there are natural transformations KK_G \rightarrow KE_G and KE_G \rightarrow E_G preserving the products. The motivations that led to the construction of KE -theory were: (1) to give a concrete description of the map from KK -theory to E -theory, abstractly known to exist because of the universal characterization of KK -theory, (2) to construct a bivariant theory well adapted to dealing with elliptic operators, and in which the product is simpler to compute with than in KK -theory, and (3) to provide a different proof to the Baum–Connes conjecture for a-T-menable groups. This paper deals with the first two problems mentioned above; the third one will be treated somewhere else.

Coaction functors

A certain type of functor on a category of coactions of a locally compact group on C*-algebras is introduced and studied. These functors are intended to help in the study of the crossed-product functors that have been recently introduced in relation to the Baum-Connes conjecture. The most important coaction functors are the ones induced by large ideals of the Fourier-Stieltjes algebra. It is left as an open problem whether the "minimal exact and Morita compatible crossed-product functor" is induced by a large ideal.