Commutativity of Banach Algebras with Derivations and Local Identities

Abstract

In this paper we prove a transfer principle for Banach algebras and use it to show that, for local identities of product type, the algebraic hypotheses of the commutativity theorems of Yood type are superfluous. Such theorems assume an algebraic identity only on a small part of the algebra and only with an exponent that may change from one point to the next. Completeness then restores both globality and uniformity. Every proof in this line opens with the same Baire category argument, and that argument uses nothing about the map to which the identity is applied. The transfer principle isolates that argument, and it holds for an arbitrary countable family of continuous maps of several variables subject to two elementary polynomial conditions. Its proof also supplies a selection step that has been used in the literature without proof. Combined with a linearisation at the identity element, the principle shows that continuity and additivity alone force the map to agree with the identity map modulo the prescribed closed subspace and force every commutator to lie in that subspace. For a closed two-sided ideal the result is best possible. The local hypothesis is then equivalent to the commutativity of the quotient algebra together with the requirement that the map induce the identity map on it. The theorems of Yood, of Ali and Khan and of Ashraf and Wani are recovered without their primeness, non-degeneracy and auxiliary hypotheses. The analogues for skew, b-generalised, Jordan and higher derivations follow at the same time, and no derivation can satisfy the hypothesis modulo a proper closed ideal. Identities of power, commutator, Jordan and multiplicative type are treated by the same method, and a unital C*-algebra satisfying the hypothesis is commutative.