Vol. 22, No. 2, 1967

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Functional representation of topological algebras

Peter Don Morris and Daniel Eliot Wulbert

Vol. 22 (1967), No. 2, 323–337

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.

Mathematical Subject Classification
Primary: 46.55
Received: 22 August 1966
Published: 1 August 1967
Peter Don Morris
Daniel Eliot Wulbert