Let
S
S
be a commutative, locally compact semigroup and
A
A
an
L
L
-subalgebra of
M
(
S
)
M(S)
. We obtain conditions on
A
A
to ensure that
S
S
is an inverse semigroup, a group, or an idempotent semigroup. We assume an involution on
A
A
, or equivalently on
S
S
. The proofs come readily from known facts.