Let G be a locally compact group, and let WAP(G) denote the space of weakly almost periodic functions on G. We show that, if G is a [SIN]-group, but not compact, then the dual Banach algebra WAP(G)∗ does not have a normal, virtual diagonal. Consequently, whenever G is an amenable, non-compact [SIN]-group, WAP(G)∗ is an example of a Connes-amenable, dual Banach algebra without a normal, virtual ...

Our main result is that the simple Lie group G = Sp(n, 1) acts properly isometrically on Lp(G) if p > 4n+2. To prove this, we introduce property (BP0 ), for V be a Banach space: a locally compact group G has property (BP0 ) if every affine isometric action of G on V , such that the linear part is a C0-representation of G, either has a fixed point or is metrically proper. We prove that solvable ...

We prove several superrigidity results for isometric actions on Busemann non-positively curved uniformly convex metric spaces. In particular we generalize some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups, and we give a proof of an unpublished result on commensurability superrigidity due to G.A. Margulis. The proofs re...

Theorem 1 proves (among the others) that for a locally compact topological group X the following assertions are equivalent: (i) X is metrizable and s-compact. (ii) CpðXÞ is analytic. (iii) CpðXÞ is K-analytic. (iv) CpðXÞ is Lindelöf. (v) CcðX Þ is a separable metrizable and complete locally convex space. (vi) CcðX Þ is compactly dominated by irrationals. This result supplements earlier results ...

We show that Haefliger's cohomology forétale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (oř Cech) cohomology for topological simplicial spaces.

The concept of an n-dimensional local field was introduced by A.N. Parshin [Pa1] with the aim of generalizing the classical adelic formalism to (absolutely) ndimensional schemes. By definition, a 0-dimensional local field is just a finite field, and an n-dimensional local field, n > 0, is a complete discrete valued field whose residue field is (n − 1)-dimensional local. Thus for n = 1 we get lo...

This paper introduces a generalization of Pontryagin duality for locally compact Hausdorff abelian groups to locally compact Hausdorff abelian group bundles. First recall that a group bundle is just a groupoid where the range and source maps coincide. An abelian group bundle is a bundle where each fibre is an abelian group. When working with a group bundle G we will use X to denote the unit spa...

We show that Haefliger’s cohomology for étale groupoids, Moore’s cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Čech) cohomology for topological simplicial spaces.

Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the k-Schur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar’s nilHecke ring, work of Peterson on the homology of based loops on a compact group, and earlier work of ours on non-commutative k-Schur functions.