A characterization of derivations on uniformly mean value Banach algebras

Abstract

In this paper, a uniformly mean value Banach algebra (briefly UMV-Banach algebra) is defined as a new class of Banach algebras, and we characterize derivations on this class of Banach algebras. Indeed, it is proved that if A is a unital UMV-Banach algebra such that either a = 0 or b = 0 whenever ab = 0 in A , and if δ : A → A is a derivation such that aδ(a) = δ(a)a for all a ∈ A , then the following assertions are equivalent: (i) δ is continuous; (ii) δ(e a) = e a δ(a) for all a ∈ A ; (iii) δ is identically zero.

The image of Jordan left derivations on algebras

“…A number of authors have presented many non-commutative versions of the Singer-Wermer theorem (see, e.g., [2,12,13]). Moreover, the question under which conditions all derivations are zero on a given Banach algebra has attracted much attention of authors (see, e.g., [5,6,9,14,15]). In the following, some significant works on the range of left derivations are reviewed.…”