The completion of a continuous inverse algebra need not be a continuous inverse algebra

Abstract

In 2006 Neeb asked whether the Hausdorff completion of a continuous inverse algebra must again be a continuous inverse algebra. The noncommutative case remained open, while the commutative case was known to be true. We give a negative answer after twenty years. We construct a Hausdorff metrizable locally m-convex complex continuous inverse algebra whose completion is a Fréchet locally m-convex algebra, where inversion stays continuous but the set of invertible elements is not open. The construction uses finite-support sequences in a dense nil subalgebra of a Jacobson-semisimple Banach algebra. Finite support makes every element nilpotent, giving continuous inversion without a locally uniform bound on nilpotence indices. In the completion, shifting a fixed noninvertible element to later and later coordinates produces noninvertible elements converging to the identity. Thus completion destroys exactly the local spectral stability at the identity. The counterexample is necessarily noncommutative and marks the precise boundary of the commutative completion theorem.