On compact normal semigroups

Abstract

A semigroup S is said to be normal if aS = Sa for each a in S. Thus the class of normal semigroups includes the class of groups and the class of Abelian semigroups. Given a compact semigroup S we write P(S) for the convolution semigroup of probability regular Borel measures on S. In (3), Theorem 7, Lin asserts that a compact semigroup S is normal if and only if P(S) is normal. We show in this paper that Lin's result is false. In fact, if S is the union of subsemigroups each of which has an identity element, we show that P(S) is normal if and only if S is Abelian. Thus any compact non-Abelian group contradicts Lin's result. What Lin's argument does establish is that if P(S) is normal then S is normal, and if S is normal then μP(S) = P(S)μ for each point mass measure μ.