In the first part of this
paper the Arens multiplication on a space of bounded functions is used to simplify
and extend results by Day and Frey on amenability of subsemigroups and ideals of a
semigroup. For example it is shown that if S is a left amenable cancellation
semigroup then a subsemigroup A of S is left amenable if and only if each
two right ideals of A intersect. The remainder and major portion of this
paper is devoted to relations between Ieft invariant means on m(S) and
left idea1s of βS(=the Stone-Čech compactification of S). We find: If μ
is a left invariant mean on m(S) and if S has Ieft cancellation then 𝒮(μ),
the support of μ considered as a Borel measure on β(S), is a Iefl ideal of
β(S). An application is that if S is a left amenable semigroup and I is a Ieft
ideaI of βS, then K(I), the w∗-closed convex hull of If contains an extreme
left invariant mean; if in addition S has cancellation then K(I) contains
a left invariant mean which is the w∗-limit of a net of unweighted finite
averages.
