Author/Authors :
Ding Y. نويسنده Department of Mathematics - East China University of Science and Technology, Shanghai , Li J. نويسنده Department of Mathematics - East China University of Science and Technology, Shanghai
Abstract :
Let A be a unital Banach algebra, M be a left A-module, and W in Z(A) be a left separating point of M. We show that if M is a unital left A-module and ? is a linear mapping from A into M, then the following four conditions are equivalent: (i) ? is a Jordan left derivation; (ii)? is left derivable at W; (iii) ? is Jordan left derivable at W; (iv)A?(B)+B?(A)=?(W) for each A,B in A with AB=BA=W.
Let A have property (B) (see Definition ???), M be a Banach left A-module, and ? be a continuous linear operator from A into M. Then ? is a generalized Jordan left derivation if and only if ? is Jordan left derivable at zero. In addition, if there exists an element C?Z(A) which is a left separating point of M, and RannM(A)={0}, then ? is a generalized left derivation if and only if ? is left derivable at zero.
