The Arens-Calderon theorem for commutative topological algebras

A theorem of Arens and Calderon states that if A is a commutative Banach algebra with Jacobson radical Rad(A), and if a0 , . . . , an∈ A with a0 ∈ Rad(A) and a1 an invertible element of k A, then there exists y ∈ Rad(A) such that Σ ak yk = 0. In this paper, we give extensions of this result to commutative non-normed topological algebras, as this is vital for extending an embedding theorem of Allan in [2] regarding the embedding of the formal power series algebra C[[X]] into a commutative Banach algebra.