Mahmoud Filali

Finite-dimensional left ideals in the duals of introverted spaces

We use representations of a Banach algebra A to completely characterize all finite-dimensional left ideals in the dual of introverted subspaces of A∗ and in particular in the double dual A∗∗. We give sufficient conditions under which such ideals always exist and are direct sums of one-dimensional left ideals.

Right cancellation in βS andUG

Let βS be the Stone-Ĉech compactification of an infinite discrete cancellative semigroupS. The set of points in the growth βS/S at which right cancellation holds in βS is shown to be dense in βS/S. It is then deduced that these type of points also form a dense subset ofUG/G, whenG is a non-compact locally compact abelian group andUG is its uniform compactification.

On the ideal structure of some algebras with an Arens product

Abstract Let G be a non-compact locally compact group, L 1 ( G ) be its group algebra, LUC ( G ) be the space of bounded functions on G which are uniformly continuous with respect to the right uniformity on G. Let L 1 ( G )** be the second conjugate of L 1 ( G ) with the first Arens product and LUC ( G )* be the conjugate of the space LUC ( G ) with the first Arens-type product. We see how the maximal ideals of LUC ( G )* are related to those of L 1 ( G ), and give examples of weak* – dense maximal ideals in LUC ( G )*. For a large class of locally compact groups, we compute the dimension of every right ideal in LUC ( G )* and the dimension of every right ideal in L 1 ( G )** which is not generated by a right annihilator of L 1 ( G )**. In particuar, we see that they are infinite dimensional; a result which was known earlier only for the radicals of these algebras. Finally, we construct new elements in the radicals of LUC ( G )* and L 1 ( G )**.

Weak P‐points and Cancellation in βS

Let S be a cancellative semigroup and let βS be the Stone‐Čech compactification of S. Then βS is a semigroup with an operation which extends that of S and which is continuous only in one variable. The points s of S are easily shown to be right (left) cancellative in βS, i.e., ys and zs (sy and sz) are different elements of βS whenever y and z are. It is known that such a property is not valid for all the elements of βS\S. However, we will see that the set of points in βS\S which are cancellative in βS is dense in βS\S. In particular, we will see that the (weak) p‐points of βS\S, which are contained in the closure of some countable subset of S, are (right) cancellative in βS.

A Note on Distal Functions

ABSTRACT: The function f(n)= exp(ip(n)) is known to be distal on the group of the integers whenever p is a real polynomial. We establish a simple criterion from which we deduce the following generalization:

On some special measures on βG

LetG be a discrete abelian group,Ĝ the character group ofG, andl∞(G)* the conjugate ofl∞(G) equipped with an Arens product. In many cases, we can find unitary functionsf such that χf is almost convergent to zero for all χ∈Ĝ. Some of these functions are then used to produce elements μ∈l∞(G)* such that γμ=0 whenever γ is an annihilator ofC0(G). Regarded as Borel measures on βG, these elements satisfyxμ=0 for allx∈βG/G. They belong to the radical ofl∞(G)*, and each of them generates a left ideal ofl∞(G)* that contains no minimal left ideal.

Algebraic structure of semigroup compactifications: Pym's and Veech's Theorems and strongly prime points

Abstract The spectrum of an admissible subalgebra A ( G ) of LUC ( G ) ( G ) , the algebra of right uniformly continuous functions on a locally compact group G, constitutes a semigroup compactification G A of G. In this paper we analyze the algebraic behaviour of those points of G A that lie in the closure of A ( G ) -sets, sets whose characteristic function can be approximated by functions in A ( G ) . This analysis provides a common ground for far reaching generalizations of Veechs property (the action of G on G LUC ( G ) is free) and Pyms Local Structure Theorem. This approach is linked to the concept of translation-compact set, recently developed by the authors, and leads to characterizations of strongly prime points in G A , points that do not belong to the closure of G ⁎ G ⁎ , where G ⁎ = G A ∖ G . All these results will be applied to show that, in many of the most important algebras, left invariant means of A ( G ) (when such means are present) are supported in the closure of G ⁎ G ⁎ .

Harmonic analysis and applications

Purpose – The purpose of this paper is to survey briefly how harmonic analyis started and developed throughout the centuries to reach its modern status and its surprisingly wide range of applications.Design/methodology/approach – The author traces applications of harmonic analysis back to Mesopotamia, ancient Egypt and the Indus Valley, showing how the Greeks have applied trigonometry and influenced its birth, then the important developments in India in the sixth century laying the first brick to modern trigonometry with the definition of the sinus, then medieval India founding modern mathematical analysis. Trigonometry was developed further by the Arabs until the fourteenth century, then by the Europeans. The eighteenth century in France was particularly important when Bernoulli solved, with an infinite trigonometric series, the vibrating string problem, then Fourier, who studied these series extensively. The author goes on to harmonic analysis on locally compact groups, and ends up with a quick personal...

The size of the quotient LUC(G)/UC(G)

Abstract We consider the Banach space LUC(G) (RUC(G)) of the bounded left (right) uniformly continuous functions on a locally compact group G, the Banach space UC(G) = LUC(G)∩RUC(G) of the bounded uniformly continuous functions on G and the quotient Banach space LUC(G)/UC(G). If G is a sin group, that is, the right and the left uniformities coincide, then the quotient LUC(G)/UC(G) is trivial, so we are concerned with non-sin groups. We bring up the topological structure of G to the Banach space structure of the quotient by showing first that when the compact covering number k (G) is greater than or equal to the local weight b(G), there is a linear isometric copy of in LUC(G)/UC(G). Then, using the generalised Kakutani–Kodaira theorem due to Zhiguo Hu, we deduce that in the case of a general non-sin group, there exists a compact normal subgroup N of G such that ℵ0 ≤ b(G/N) ≤ k (G) and LUC(G)/UC(G) contains a linear isometric copy of .