SOME PROPERTY OF LIE BANACH∗ -ALGEBRA HOMOMORPHISMS

Let B be a Lie Banach∗-algebra. For elements (a, b) and (c, d) in B2 := B × B, by definitions (a, b)(c, d) = (ac, bd), ∥(a, b)∥ = ∥a∥ + ∥b∥ and (a, b) ∗ = (a∗, b∗) B2 can be considered as a Banach∗-algebra. This Banach∗-algebra is called a Lie Banach∗-algebra whenever it is equipped with the following definition of Lie product: [(a, b),(c, d)] = (ac−ca2,bd−db2) for all a, b, c, d in B. Also, if B is a Lie Banach∗-algebra, then D : B2 −→ B2 satisfying D([(a, b),(c, d)]) = [D(a, b),(c, d)] + [(a, b), D(c, d)] for all a, b, c, d ∈ B, is a Lie derivation on B2. Furthermore, if B is a Lie Banach∗-algebra, then D is called a Lie∗ derivation on B2 whenever D is a Lie derivation with D(a, b)∗ = D(a∗, b∗) for all a, b ∈ B. In this paper, we investigate the Hyers-Ulam stability of Lie Banach∗-algebra homomorphisms and Lie∗derivations on the Banach∗-algebra B2.