Ultrafilters and semigroup algebras

Abstract

The operation defined on a discrete semigroup S can be extended to the Stone- Cech compactification S of S so that for all a 2 S, the left translation S 3 x 7! ax 2 S is continuous, and for all q 2 S, the right translation S 3 x 7! xq 2 S is continuous. Because any compact right topological semigroup, S contains a smallest two-sided ideal K( S) which is a completely simple semigroup. We give an exposition of some basic results related to the semigroup S and to the semigroup algebra `1( S). In particular, we review the result that `1( N) is semisimple if and only if `1(K( N)) is semisimple. We also review the reduction of the question whether `1(K( N)) is semisimple to a question about K( N).