Topologically algebraic algebras (Q2568732)

Let \(A\) be a complex topological algebra and \(H(\mathbb{C})\) the algebra of all complex-valued entire functions. An element \(x\) in \(A\) is said to be topologically algebraic if all entire functions operate in \(A\) at the point \(x\) and if there exists a nonzero entire function \(f\) in \(H(\mathbb{C})\) such that \(f(x)=0\). The algebra \(A\) is said to be topologically algebraic if each element of \(A\) is topologically algebraic. In this paper, the authors give some spectral properties of topologically algebraic elements and establish that a locally convex algebra is topologically algebraic if and only if it is algebraic.