Functional representation of topological algebras
Peter Morris, Daniel E. Wulbert · Pacific Journal of Mathematics · 1967
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 such 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.