Abstract For Banach algebras A A and B B, we study zero product preserving linear maps Θ: C 1 (0, 1, A) → C 1 (0, 1, B): C^1 (0, 1, A) C^1 (0, 1, B). In the case that B B is either an operator algebra or an irreducible Banach algebra, and Θ is continuous and has a dense range, it has a description according to a pair of families of (jointly) zero product linear maps from A A to B B and, in particular, it is a separating map. The cases where A = B (E) A=B (E), B = B (F) B=B (F) for some Banach spaces E and F and Θ is bijective (not necessarily continuous) as well as the case where Θ is a continuous zero product preserving linear map into a Banach function algebra will also be considered. Similar results are valid for two-sided zero product preserving maps. An application of the results is given for surjective homomorphisms between Banach algebras of operator-valued continuously differentiable functions.
Khodaie et al. (2026) studied this question.