We study the character amenability of semigroup algebras. We work
on general semigroups and certain semigroups such as inverse semigroups
with a nite number of idempotents, inverse semigroups with uniformly
locally nite idempotent set, Brandt and Rees semigroup and study the
character amenability of the semigroup algebra l1(S) in relation to the
structures of the semigroup S. In particular, we show that for any semi-
group S, if ℓ 1(S) is character amenable, then S is amenable and regular.
We also show that the left character amenability of the semigroup alge-
bra ℓ 1(S) on a Brandt semigroup S over a group G with index set J is
equivalent to the amenability of G and J being nite. Finally, we show
that for a Rees semigroup S with a zero over the group G, the left char-
acter amenability of ℓ 1(S) is equivalent to its amenability, this is in turn
equivalent to G being amenable.