Generalized additive mapping in Banach modules and isomorphisms between C ∗ -algebras ✩

By C. Baak, D. Boo, Themistocles M. Rassias. 2006

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Abstract Let X , Y be vector spaces. It is shown that if an odd mapping f : X → Y satisfies the functional equation (1) r f ( ∑ j = 1 d x j r ) + ∑ ι ( j ) = 0 , 1 ∑ j = 1 d ι ( j ) = l r f ( ∑ j = 1 d ( − 1 ) ι ( j ) x j r ) = ( C l d − 1 − C l − 1 d − 1 + 1 ) ∑ j = 1 d f ( x j ) then the odd mapping f : X → Y is additive, and we prove the Hyers–Ulam stability of the functional equation (1) in Banach modules over a unital C ∗ -algebra. 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 n r n u y ) = h ( 2 n r n u ) h ( y ) for all unitaries u ∈ A , all y ∈ A , and n = 0 , 1 , 2 , … .

Subfields: Algebra and Number Theory, Applied Mathematics

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