Jordan derivations on triangular algebras and trivial extensions

Abstract

In this paper, we provide a necessary and sufficient condition for a class of trivial extensions so that each Jordan derivation on such algebras can be written as the sum of a derivation and an antiderivation. Among other things, we obtain some of the known results concerning the Jordan derivations on trivial extensions with fewer assumptions. Also, as a generalization of a well-known result on triangular matrix algebras, we show that every Jordan derivation from arbitrary triangular algebra into its certain bimodule is the sum of a derivation and an antiderivation. We also apply our results for upper triangular matrix algebra [Formula: see text], matrix algebra [Formula: see text] and the nest algebra [Formula: see text] to describe the Jordan derivations into their bimodules.