Maps preserving two-sided zero products on Banach algebras (Q2154412)

Let \(A\) and \(B\) be algebras and \(\Psi:A\to B\) a map. The map \(\Psi\) is called a Jordan homomorphism if \(\Psi\) is linear and \(\Psi(ab+ba)=\Psi(a)\Psi(b)+\Psi(b)\Psi(a)\) for all \(a, b\in A\). A linear map \(W:B\rightarrow B\) is called a centralizer if \(W(cd)=W(c)d=cW(d)\) for all \(c, d\in B\). A map \(\Phi:A\rightarrow B\) is a weighted Jordan homomorphism if \(\Phi=W\Psi\) for some centralizer \(W\) and some Jordan homomorphism \(\Psi\). The main problem of the paper is the following: given Banach algebras \(A\) and \(B\) and a continuous linear map \(\Phi:A\rightarrow B\) for which from \(ab=ba=\theta_A\), with \(a, b\in A\), it follows that \(\Phi(a)\Phi(b)=\Phi(b)\Phi(a)=\theta_B\). Under which conditions does \(\Phi\) become a weighted Jordan homomorphism? The authors consider different sets of conditions on \(A\), \(B\) and \(\Phi\). They show that the following set of conditions is sufficient for \(\Phi\) to be a weighted Jordan homomorphism: \(A\) is either a) a \(C^*\)-algebra or b) the algebra of approximable operators on some Banach space \(X\) such that the dual \(X*\) of \(X\) has the bounded approximation property, \(B\) is a Banach algebra with bounded approximate identity, \(\Phi\) is surjective and satisfies, for all \(a, b\in A\), the condition: from \(ab=ba=\theta_A\) it follows that \(\Phi(a)\Phi(b)+\Phi(b)\Phi(a)=\theta_B\).