Inverse map and equicontinuity of power maps in locally convex algebras

Abstract

We show that the inverse map $x\longmapsto x^{-1}$ is continuous in any unitary non commutative locally convex algebra in which the sequence of power maps $\left( x\longmapsto x^{n}\right) _{n}$ is equicontinuous at zero. As a consequence, we obtain that the inverse map is continuous in any unitary $B_{0}$-algebra not necessarily commutative in which entire functions operate.