Some conditions under which left derivations are zero

Abstract

In this study, we show that every continuous Jordan left derivation on a (commutative or noncommutative) prime UMV-Banach algebra with the identity element 1 is identically zero. Moreover, we prove that every continuous left derivation on a unital finite dimensional Banach algebra, under certain conditions, is identically zero. As another result in this regard, it is proved that if R is a 2-torsion free semiprime ring such that ann{[y,z]| y,z  R}= {0}, then every Jordan left derivation L : R → R is identically zero. In addition, we provide several other results in this regard.