Continuous semigroups in locally convex algebras (Q1814472)

The paper under review belongs to a larger project, due to the same author, and devoted to the study of semigroups in locally convex algebras. The main example and the motivation for this investigation is the convolution algebra of distributions supported by a fixed cone. Let \(A\) be a separately continuous, locally convex algebra with unit. The principal result of the note states that, if the multiplication operation in \(A\) is bounded, then a weakly continuous semigroup \(x(t)\) in \(A\) which satisfies \(\lim_{t\to 0+}x(t)=1\) is necessarily continuous. As a consequence it follows that a continuous semigroup in \(A\) is differentiable.