On a conjecture of Chou and applications by Mahmoud Filali

Electronic Version 24 Summer (1999)

On a conjecture of Chou and applications

Mahmoud Filali

Let S be an infinite, discrete, cancellative semigroup and let \betaS be the Stone- Cech compactification of S. Then \betaS is a semigroup with an operation which extends that of S and which is continuous only in one variable. We start with a subset V of S such that sV \cap tV is finite for all s,t \in S, s =/= t. We show that if x₁,x₂ \in [`V], || x₁ || = || x₂ || and x₁ =/= x₂ then \betaSx₁ \cap \betaSx₂ = \emptyset. This result is a partial positive answer to a conjecture given by C. Chou about thirty years ago, and it implies the known result that the number of these ideals is 2^{2^|S|}. We also show that any point in [`V] is right cancellative in \betaS. This result improves our earlier result where V was countable. With these two theorems, we deduce that the dimension of any non-zero right ideal of l_\infty(S)^* when equipped with the first Arens product is 2^{2^|S|}. This latter result was known only for the radical of l_\infty(S)^* when S is an amenable group.

Volume 24 Summer: table of contents, information on access
Topology Proceedings Electronic Version