Linear Maps Which Are Homomorphisms or Jordan Homomorphisms at a Point

We study linear maps \(\theta:A\longrightarrow B\) between unital Banach algebras which are homomorphisms at zero or identity products. We show under special hypotheses that every such linear map is a homomorphism multiplied by a central invertible element. Zero and identity Jordan products preserving linear maps, and a more restrictive version of zero product preservers are also discussed.