RIGHT CANCELLATION IN THE ${\cal L}{\cal U}{\cal C}$ -COMPACTIFICATION OF A LOCALLY COMPACT GROUP

Recent Progress in the Topological Theory of Semigroups and the Algebra of βS

Ideals, idempotents and right cancelable elements in the uniform compactification

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 )**

On the ideal structure of some algebras with an Arens product

The LUG-compactification UG of a locally com­ pact group is a semigroup with an operation which extends that of G and which is continuous (only) in one variable. When G is discrete, UG and the Stone-Cech compactification {3G are identi­ cal. Some algebraic properties, such as the num­ ber of left ideals and cancellation, are known to hold in the semigroup {3N where N is the additive semigroup of the integers. We show that these properties are also true in UG for a large class of locally compact groups. The method used is to transfer the information from {3N to (3G where G is an infinite discrete group (or a cancellative commutative semigroup), and then to UG where G is not necessarily discrete.

M. Filali

Papers

On some semigroup compactifications