A topological algebra E is an
algebra over the real or complex numbers together with a topology such
that E is a topological vector space and sach that multiplication in E is
jointly continuous. For a topological space X,C(X) denotes the algebra of
all continuous, complex-valued functions on X with the usual pointwise
operations. Unless otherwise stated, C(X) is assumed to have the compact-open
topology. Our principal concern is with representing (both topologically and
algebraically) a commutative (complex) topological algebra, with identity, E
as a subalgebra of some C(X),X a completely regular Hausdorff space.
We obtain several characterizations of topological algebras which can be so
represented. The most interesting of these is that the topology on E be
generated by a family of semi-norms each of which behaves, with respect
to the multiplication in the algebra, like the norm in a (Banach) function
algebra.
