Extending classical theorems,
we obtain representations for bounded linear transformations from L-spaces to
Banach spaces with a separable predual. In the case of homomorphisms from a
convolution measure algebra to a Banach algebra, we obtain a generalization of
Šreĭder’s representation of the Gelfand spectrum via generalized characters. The
homomorphisms from the measure algebra on a LCA group, G, to that on the
circle are analyzed in detail. If the torsion subgroup of G is denumerable,
one consequence is the following necessary and sufficient condition that a
positive finite Borel measure on G be continuous: ∃γα→∞ in Ĝ such that
∀n≠0μ(γαn) → 0.
