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When is a bi-Jordan homomorphism bi-homomorphism? (English) Zbl 1483.47073

Summary: For Banach algebras \(\mathcal{A}\) and \(\mathcal{B}\), we show that, if \(\mathcal{U}=\mathcal{A}\times\mathcal{B}\) is commutative (weakly commutative), then each bi-Jordan homomorphism from \(\mathcal{U}\) into a semisimple commutative Banach algebra \(\mathcal{D}\) is a bi-homomorphism. We also prove the same result for 3-bi-Jordan homomorphism with the additional hypothesis that the Banach algebra \(\mathcal{U}\) is unital.

MSC:

47B48 Linear operators on Banach algebras
46L05 General theory of \(C^*\)-algebras
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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