On maximal ideals of codimension one in m-convex algebras

In Studia Math. 58:291–298, 1976, W. Żelazko gives a characterization of complex commutative complete unital m-convex algebras in which all maximal ideals are of codimension one. In these algebras every element has a bounded spectrum.

Here we present a similar result concerning sufficient conditions so that all maximal ideals of a complex commutative unital m-convex algebra to be of codimension one. Moreover, these conditions are proved to be equivalent to each other. We also give an example of a complex commutative unital advertibly complete m-convex algebra such that all its maximal ideals are of codimension one and has an element with unbounded spectrum.