Inverse map and equicontinuity of power maps in locally convex algebras (Q1616330)

In [C. R. Acad. Sci., Paris, Sér. A 270, 1686--1689 (1970; Zbl 0195.13804)], \textit{P. Turpin} proved that every commutative locally convex algebra in which the sequence of power maps \((x \mapsto x^n)_n \) is equicontinuous at zero is \(m\)-convex. Thus, the inverse map \( x \mapsto x^{-1}\) is continuous. In the main result of the paper, the authors get continuity of the inverse map by dropping commutativity. They also show, by an example, that continuity of the inverse map does not imply equiconinuity at zero of the sequence of power maps. By employing the main result, they conclude that in any unitary \(B_0\)-algebra (viz., a locally convex metrizable and complete algebra) not necessarily commutative in which entire functions operate, the inverse map is continuous. The authors also justify, via examples, that the latter result does not hold without completeness or metrizability.