Approximate identities for convolution measure algebras

Abstract

Let (A, *) be a commutative semisimple convolution measure algebra with structure semigroup Γ, It is proved that A has a weak bounded approximate identity if and only if Γ has a finite set of relative units; moreover, Γ has an identity if and only if some weak bounded approximate identity is of norm one. Considering now a commutative semigroup S, the existence of a bounded (norm) approximate identity in A=l1(S) is equivalent to the existence in S of a finite number of nets {up(i)}ρ(i)ϵfi, i=1, 2,···, n with the property that for every x∈S there exist j and ρ(j)x such that ρ(j)≥ρ(j)x implies xuρ(j)=x. © 1973 Pacific Journal of Mathematics.