This paper may be
considered as another chapter in the theory of convolution algebras inaugurated by
Hewitt and Zuckerman. The interest here is in finding an L1 theory for locally
compact commutative topological semigroups which extends the known l1 theory for
discrete commutative semigroups and L1 theory for locally compact topological
groups. Let S be a locally compact commutative semigroup and m a nonnegative
regular Borel measure on S such that if x ∈ ∕S and E ⊂ S with m(E) = 0 then
m(Ex−1) = 0(Ex−1= [y : yx ∈ E]). When L1(S,m) is defined as the Banach space
of all bounded complex measures μ ∈ M(S) which are absolutely continuous with
respect to m, then L1(S,m) is a convolution algebra as a subalgebra of M(S). It is
shown that there is a one to one correspondence between the measurable
semicharacters on S and the multiplicative linear funclionals on L1(S,m)
analogus to the group situation. Extensions of the above results to those S
with a measure m satisfying the above condition in a local sense are also
obtained.
