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Hyers–Ulam–Rassias stability of a generalized Apollonius–Jensen type additive mapping and isomorphisms between C *‐algebras
Author(s) -
Park Choonkil
Publication year - 2008
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510611
Subject(s) - mathematics , unital , bijection , banach algebra , isomorphism (crystallography) , pure mathematics , type (biology) , cauchy distribution , stability (learning theory) , algebra over a field , combinatorics , banach space , mathematical analysis , chemistry , crystallography , crystal structure , ecology , machine learning , computer science , biology
Abstract Let X, Y be Banach modules over a C *‐algebra. We prove the Hyers–Ulam–Rassias stability of the following functional equation in Banach modules over a unital C *‐algebra:It is shown that a mapping f : X → Y satisfies the above functional equation and f (0) = 0 if and only if the mapping f : X → Y is Cauchy additive. As an application, we show that every almost linear bijection h : A → B of a unital C *‐algebra A onto a unital C *‐algebra B is a C *‐algebra isomorphism when h (2 d uy ) = h (2 d u ) h ( y ) for all unitaries u ∈ A , all y ∈ A , and all d ∈ Z . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)