LIE DERIVATIONS AND HYER-ULAM STABILITY

Let A be a Lie Banach -algebra. For each elements (a; b) and (c; d) in A۲ := A A, by definitions(…)A۲ can be considered as a Banach -algebra. This Banach -algebra is called a Lie Banach -algebra wheneverit is equipped with the following definitions of Lie product:(...)for all a, b, c, d in A. Also, if A is a Lie Banach -algebra, then D : A۲ 􀀀! A۲ satisfying(...)for all a; b; c; d ۲ A, is a Lie derivation on A۲. Furthermore, if A is a Lie Banach -algebra, then D is calleda Lie derivation on A۲ whenever D is a Lie derivation with D(a; b) = D(a ; b ) for all a; b ۲ A. In thispaper, we investigate the Hyers-Ulam stability of Lie Banach -algebra homomorphisms and Lie derivationson the Banach -algebra A۲.