Maps preserving two-sided zero products on Banach algebras

Let \(A\) and \(B\) be Banach algebras with bounded approximate identities and let \(\Phi:A\to B\) be a surjective continuous linear map which preserves two-sided zero products (i.e., \(\Phi(a)\Phi(b)=\Phi(b)\Phi(a)=0\) whenever \(ab=ba=0\)). We show that \(\Phi\) is a weighted Jordan homomorphism provided that \(A\) is zero product determined and weakly amenable. These conditions are in particular fulfilled when \(A\) is the group algebra \(L^1(G)\) with \(G\) any locally compact group. We also study a more general type of continuous linear maps \(\Phi:A\to B\) that satisfy \(\Phi(a)\Phi(b)+\Phi(b)\Phi(a)=0\) whenever \(ab=ba=0\). We show in particular that if \(\Phi\) is surjective and \(A\) is a \(C^*\)-algebra, then \(\Phi\) is a weighted Jordan homomorphism.