It is well known that a complex-valued function ø, analytic on some open set Ω, extends to any commutative Banach algebra B so that the action of ø on B commutes with the action of the Gelfand transformation. In this paper, it is shown that if B is a homogeneous convolution Banach algebra over any compact group and if 0 ∈ Ω is a fixed point of ø, then a similar result holds, with the Gelfand transformation replaced by the Fourier-Stieltjes transformation. Care is required, in that discussion of this relation usually requires simultaneous consideration of the extension of ø to B and to certain operator algebras.

Let G be any compact group, connected or disconnected, with dual object Ĝ. We define a family of local Sidon subsets of Ĝ in terms of allowable images of the representations. Using this family we develop a straightforward criterion whereby the existence of infinite Sidon subsets of Ĝ may be decided.

If G is a compact connected Lie group every infinite subset of Ĝ contains an infinite central Λ(p) set, for p < 2 + 2 rank G/(dim G - rank G). A subset R of Ĝ is of type central Λ(2) if and only if the associated set of characters on the maximal torus is of type Λ(2). The dual of a compact connected semisimple Lie group contains infinite sets which are central p-Sidon for all p > 1. Every infinite subset of the dual of Su(2) contains such a set.

Various criteria, in terms of forward differences and related operations on coefficients, are shown to imply that certain series on bounded Vilenkin groups represent integrable functions. These results include analogues of known integrability theorems for trigonometric series. The method of proof is to pass from the given series to a derived series, and to deduce the integrability of the original series from smoothness properties of the latter.

Let G be a locally compact group, and let D(G) be a dense subalgebra of the convolution algebra L1(G). Suppose that π is a unitary representation of G and that, for each u in D(G), π(u)) is a trace-class operator. Then the linear functional u → tr(π(u)) (the trace of π(u)) is called the D-character of π. We give a simple proof that the D-character of such a representation determines the representation up to unitary equivalence. As an application, we give an easy proof of the result of Harish-Chandra that the K-finite characters of unitary representations of semisimple Lie groups determine the representations.

Let Gbe a primitive permutation group on a finite set Ω. We investigate the subconstitutents of G, that is the permutation groups induced by a point stabilizer on its orbits in Ω, in the cases where Ghas a diagonal action or a product action on Ω. In particular we show in these cases that no subconstituent is doubly transitive. Thus if G has a doubly transitive subconstituent we show that G has a unique minimal normal subgroup N and either N is a nonabelian simple group or N acts regularly on Ω: we investigate further the case where N is regular on Ω.

Let N be a nilpotent simply connected Lie group, and A a commutative connected d-dimensional Lie group of automorphisms of N which correspond to semisimple endomorphisms of the Lie algebra of N with positive eigenvalues. Form the split extension S = N × A ≅ N × a, a being the Lie algebra of A. We consider a family of “rectangles” Br in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f → Mf relative to left translates of the family {Br}. We prove that M is of weak type (1, 1). This complements a result of J.-O. Strömberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

The Lebesgue measure, λ (E + F), of the algebraic sum of two Borel sets, E, F of the classical “middle-thirds’ Cantor set on the circle can be estimated by evaluating the Cantor meaure, μ of the summands. For example log λ (E + F) exceeds a fixed scalar multiple of log μ (E)+ log μ (F). Several numerical inequalities which are required to prove this and related results look tantalizingly simple and basic. Here we isolate them from the measure theory and present a common format and proof.

There are some well-known laws that the commutator satisfies in groups, and that go by some or all of the names Jacobi, Witt, Hall; and there are also some lesser-known laws. This is an attempt at an axiomatic study of the interdependence and independence of these laws.

We study negative definite functions on a Hilbert space and use their properties to give a proof of the Lévy-Khinchin formula for an infinitely divisible probability distribution on .

Let X be either the d-dimensional sphere or a compact, simply connected, simple, connected Lie group. We define a mean-value operator analogous to the spherical mean-value operator acting on integrable functions on Euclidean space. The value of this operator will be written as ℳ f (x, a), where x ∈ X and a varies over a torus A in the group of isometries of X. For each of these cases there is an interval pO < p ≦ 2, where the p0 depends on the geometry of X, such that if f is in Lp (X) then there is a set full measure in X and if x lies in this set, the function a ↦ℳ f(x, a) has some Hölder continuity on compact subsets of the regular elements of A.

Recently M. Benedicks showed that if a function f ∈ L2(Rd) and its Fourier transform both have supports of finite measure, then f = 0 almot everywhere. In this paper we give a version of this result for all noncompact semisimple connected Lie groups with finite centres.

Riesz products are employed to give a construction of quasi-invariant ergodic measures under the irrational rotation of T. By suitable choice of the parameters such measures may be required to have Fourier-Stieltjes coefficients vanishing at infinity. We show further that these are the unique quasi-invariant measures on T with their associated Radon-Nikodym derivative.