We use random walks to show that the Basilica group is amenable, answering an open question of Grigorchuk and Żuk. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups.

In this paper, we study approximate amenability of matrix algebras. We show that every derivation from Mn(A) into Mn(E) is the sum of an inner derivation and a derivation induced by a derivation from A into E(m), where A is a Banach algebra and E is a Banach A-bimodule. By using this, we provide many results in approximate and permanent weak amenability of these algebras.

Let A and B be Banach algebras with bounded approximate identities let Φ:A→B a surjective continuous linear map which preserves two-sided zero products (i.e., Φ(a)Φ(b)=Φ(b)Φ(a)=0 whenever ab=ba=0). We show that Φ is weighted Jordan homomorphism provided product determined weakly amenable. These conditions are in particular fulfilled when the group algebra L1(G) G any locally compact group. also...

LetG be a discrete group and letX be aG-finite, properG-CW-complex. We prove that Kasparov’s equivariant K-homology groups KK∗ (C0(X),C) are isomorphic to the geometric equivariant K-homology groups of X that are obtained by making the geometric K-homology theory of Baum and Douglas equivariant in the natural way. This reconciles the original and current formulations of the Baum-Connes conjectu...

We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have dis...

In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spinc geometry depending on whether the geometry is “real” or not. We attempt to flesh out the details of Connes’ ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudo-Riemann...

We study an equivariant co-assembly map that is dual to the usual Baum–Connes assembly map and closely related to coarse geometry, equivariant Kasparov theory, and the existence of dual Dirac morphisms. As applications, we prove the existence of dual Dirac morphisms for groups with suitable compactifications, that is, satisfying the Carlsson–Pedersen condition, and we study a K–theoretic counte...

We study in this paper the maximal version of the coarse BaumConnes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to the (maximal) Baum-Connes assembly map for the group and is an isomorphism for a class of expanders. We also introduce a quantitative Baum-Connes assembly ma...