George A. Willis

Contraction groups and scales of automorphisms of totally disconnected locally compact groups

We study contraction groups for automorphisms of totally disconnected locally compact groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale and this scale is shown to be the inverse value of the modular function on the closure of the contraction group at the automorphism. The closure of the contraction group is represented as acting on a homogenous tree and closed contraction groups are characterised.

Amenability and weak amenability of second conjugate Banach algebras

For a Banach algebra A, amenability of A∗∗ necessitates amenability of A, and similarly for weak amenability provided A is a left ideal in A∗∗. For G a locally compact group, indeed more generally, L1(G)∗∗ is amenable if and only if G is finite. If L1(G)∗∗ is weakly amenable, then M(G) is weakly amenable. 0. Introduction For a Banach algebra A,A∗∗ is a Banach algebra under two Arens products, of which we will always take the first, or left, product. For further details see the survey article [8]. This product can be characterized as the extension to A∗∗×A∗∗ of the bilinear map A×A→ A : (x, y) 7→ xy with the following continuity properties: for fixed y ∈ A∗∗, x 7→ xy is weak*-continuous on A∗∗; for fixed y ∈ A, x 7→ yx is weak*-continuous on A∗∗. Here, as elsewhere, we identify A with its canonical image in A∗∗. In terms of the asymmetry here, define the topological centre of A∗∗ by Zt(A ∗∗) = {y ∈ A∗∗ : x 7→ yx is weak∗-continuous}. Clearly, Zt(A ∗∗) contains the (algebraic) centre Z(A∗∗) of A∗∗; it also contains A. In the case that Zt(A ∗∗) = A∗∗, A∗∗ is said to be Arens regular . Any C∗-algebra is Arens regular [6, Theorem 7.1], but for a locally compact group G, L(G) is Arens regular if and only if G is finite [25]. A Banach algebra A is amenable if every derivation D : A → X∗ is inner, for every Banach A-bimodule X . If one only considers the bimodule X = A, one has the notion of weak amenability. There are many alternative formulations of the notion of amenability, of which we need the following two, first introduced in [18]. For further details see [3, 16, 7]. The Banach algebra A is amenable if and only if either, and hence both, of the following hold: (i) A has an approximate diagonal , that is, a bounded net (mi) ⊂ A⊗A such that for each x ∈ A, mix− xmi → 0, π(mi)x→ x; (ii) A has a virtual diagonal , that is, an element M ∈ (A⊗A)∗∗ such that for each x ∈ A, xM = Mx, (π∗∗M)x = x. Received by the editors June 27, 1994 and, in revised form, October 19, 1994. 1991 Mathematics Subject Classification. Primary 46H20; Secondary 43A20.

Amenability of Banach algebras of compact operators

In this paper we study conditions on a Banach spaceX that ensure that the Banach algebraК(X) of compact operators is amenable. We give a symmetrized approximation property ofX which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties ofX are implied by the amenability ofК(X).

Introduction to Banach Algebras, Operators, and Harmonic Analysis

A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Introduction to Banach algebras, operators, and harmonic analysis / H. Garth Dales. .. [et al.]. p. cm. – (London Mathematical Society student texts ; 57) Includes bibliographical references and index.

Probability measures on groups and some related ideals in group algebras

Abstract Let G be a locally compact group. Random walks on G , some factorization problems in L 1 ( G ) and the significance for these of the amenability of G are studied. These topics are linked in this paper via ideals in L 1 ( G ) of the form [ L 1 (G) ∗ (δ e − μ) ] − , where μ is a probability measure on G . Cohens factorization theorem and some ideas from ergodic theory play an important part.

Classification of the simple factors appearing in composition series of totally disconnected contraction groups

Abstract Let G be a totally disconnected, locally compact group admitting a contractive automorphism α. We prove a Jordan-Hölder theorem for series of α-stable closed subgroups of G, classify all possible composition factors and deduce consequences for the structure of G.

Locally normal subgroups of totally disconnected groups. Part II: Compactly generated simple groups

We use the structure lattice, introduced in Part I, to undertake a systematic study of the class

Totally disconnected, nilpotent, locally compact groups

\mathscr S

The nub of an automorphism of a totally disconnected, locally compact group

consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are non-discrete. Given

Scale functions and tree ends

G \in \mathscr S