Automatic continuity for Banach algebras with finite-dimensional radical

Abstract

The paper [3] proved a necessary algebraic condition for a Banach algebra A with finite-dimensional radical R to have a unique complete (algebra) norm, and conjectured that this condition is also sufficient. We extend the above theorem. The conjecture is confirmed in the case where A is separable and A/R is commutative, but is shown to fail in general. Similar questions for derivations are discussed.2000 Mathematics subject classification: primary 46H40.

On discontinuity of derivations, inducing inequivalent complete metric topologies

“…In the Fréchet case, though there are a few results on the uniqueness of the Fréchet algebra topology (see [4,6,12,17,18]), a class of Fréchet algebras with finite dimensional radicals for this result appears to be treated for the first time in this paper. Of course, in the Banach case, we have results on the uniqueness of the complete norm topology of Banach algebras with finite dimensional radicals (see [9,21]); however, the hypotheses and results are very different.…”

On discontinuity of derivations, inducing non-unique complete metric topologies

“…In the Fréchet case, though there are a few results on the uniqueness of the Fréchet algebra topology (see [3,5,11,16,18]), a class of Fréchet algebras with finite-dimensional radicals for this result appears to be treated for the first time in this paper. Of course, in the Banach case, we have results on the uniqueness of the complete norm topology of Banach algebras with finite-dimensional radicals (see [8,19]); however, the hypotheses and results are very different.…”