Abstract
Let \({\mathcal{A}}\) and \({\mathcal{B}}\) be unital semisimple complex Banach algebras, and let \({\varphi_1}\) and \({\varphi_2}\) be maps from \({\mathcal{A}}\) onto \({\mathcal{B}}\). We show that if the socle of \({\mathcal{A}}\) is an essential ideal of \({\mathcal{A}}\), and \({\varphi_1}\) and \({\varphi_2}\) satisfy
for all \({a,b\in \mathcal{A}}\), then \({\varphi_1\varphi_2(1)}\) and \({\varphi_1(1)\varphi_2}\) coincide and are Jordan isomorphisms. We also show that a map \({\varphi}\) from \({\mathcal{A}}\) onto \({\mathcal{B}}\) satisfies
for all \({a,b\in \mathcal{A}}\) if and only if \({\varphi(1)}\) is a central invertible element of \({\mathcal{B}}\) for which \({\varphi(1)^3=1}\) and \({\varphi(1)^2\varphi}\) is a Jordan isomorphism.
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Bourhim, A., Mashreghi, J. & Stepanyan, A. Maps between Banach algebras preserving the spectrum. Arch. Math. 107, 609–621 (2016). https://doi.org/10.1007/s00013-016-0960-9
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DOI: https://doi.org/10.1007/s00013-016-0960-9