Some Conditions under which Left Derivations are Zero

Abstract

The main purpose of this paper is to present some conditions

under which left derivations are identically zero on a given Banach algebras. As the main result, we prove the following:

Suppose that A is a unital UMV-Banach algebra such that either a = 0 or

b = 0 whenever ab = 0 in A. If δ: A → A is a left derivation, then the

following assertions are equivalent:

(i) δ is continuous;

(ii) δ(