Abstract
Let π be a β-algebra, Ξ΄ : π β π be a linear map, and z β π be fixed. We consider the condition that Ξ΄ satisfies xΞ΄(y)β + Ξ΄(x)yβ = Ξ΄(z) (xβΞ΄(y) + Ξ΄(x)βy = Ξ΄(z)) whenever xyβ = z (xβy = z), and under several conditions on π, Ξ΄ and z we characterize the structure of Ξ΄. In particular, we prove that if π is a Banach β-algebra, Ξ΄ is a continuous linear map, and z is a left (right) separating point of π, then Ξ΄ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map Ξ΄ satisfying the above conditions with z = 0 on two classes of β-algebras: zero product determined algebras and standard operator algebras.
(Communicated by Emanuel Chetcuti)
Acknowledgement
The authors would like to express their sincere thanks to the referee(s) of this paper.
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