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)}p(i)∈ℱi,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.
