Under stated conditions on unital complex Banach algebras A and B, any surjective additive Φ: A→B with [Φ(x²), Φ(x)]=0 for all x is Φ(x)=λΨ(x)+ζ(x) where Ψ is a direct sum of additive homo- and anti-homomorphism, λ invertible in the center of B, and ζ additive into the center.
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Commutativity preserving mappings in Banach algebras
Under stated conditions on unital complex Banach algebras A and B, any surjective additive Φ: A→B with [Φ(x²), Φ(x)]=0 for all x is Φ(x)=λΨ(x)+ζ(x) where Ψ is a direct sum of additive homo- and anti-homomorphism, λ invertible in the center of B, and ζ additive into the center.