STABILITY OF HOMOMORPHISMS IN BANACH MODULES OVER A C*-ALGEBRA ASSOCIATED WITH A GENERALIZED JENSEN TYPE MAPPING AND APPLICATIONS
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Let X and Y be vector spaces. It is shown that a mapping f : X rightarrow Y satisfies the functional equation (formula) if and only if the mapping f : X rightarrow Y is additive, and prove the Cauchy-Rassias stability of the functional equation {rm (ddag)} in Banach modules over a unital C*-algebra, and in Poisson Banach modules over a unital Poisson C*-algebra. Let mathcal A and Cal B be unital C*-algebras, Poisson C*-algebras, Poisson JC*-algebras or Lie JC*-algebras. As an application, we show that every almost homomorphism h : Cal A rightarrow Cal B of Cal A into Cal B is a homomorphism when h(dn u y) = h(dn u) h(y) or h(dn u circ y) = h(dn u) circ h(y) for all unitaries u inCal A, all y in Cal A, and n = 0, 1, 2, cdots. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in C*-algebras, Poisson C*-algebras, Poisson JC*-algebras or Lie JC*-algebras, and of Lie JC*-algebra derivations in Lie JC*-algebras.
Abstract
Let X and Y be vector spaces. It is shown that a mapping f : X rightarrow Y satisfies the functional equation (formula) if and only if the mapping f : X rightarrow Y is additive, and prove the Cauchy-Rassias stability of the functional equation {rm (ddag)} in Banach modules over a unital C*-algebra, and in Poisson Banach modules over a unital Poisson C*-algebra. Let mathcal A and Cal B be unital C*-algebras, Poisson C*-algebras, Poisson JC*-algebras or Lie JC*-algebras. As an application, we show that every almost homomorphism h : Cal A rightarrow Cal B of Cal A into Cal B is a homomorphism when h(dn u y) = h(dn u) h(y) or h(dn u circ y) = h(dn u) circ h(y) for all unitaries u inCal A, all y in Cal A, and n = 0, 1, 2, cdots. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in C*-algebras, Poisson C*-algebras, Poisson JC*-algebras or Lie JC*-algebras, and of Lie JC*-algebra derivations in Lie JC*-algebras.
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