Jordan isomorphisms and maps preserving spectra of certain operator products

Let A(1), A(2) be (not necessarily unital or closed) standard operator algebras on locally convex spaces X-1, X-2, respectively. For k >= 2, consider different products T-k *......* T-k on elements in A(i), which covers the usual product T-1 *...* T-k = T-1...T-k and the Jordan triple product T-1 * T-2 = T2T1T2. Let Phi : A(1) -> A(2) be a (not necessarily linear) map satisfying or (sigma(A(1)) *...* Phi(A(k))) = sigma(A(1) *...* A(k)) whenever any one of A(i)'s has rank at most one. It is shown that if the range of 45 contains all rank one and rank two operators then Phi must be a Jordan isomorphism multiplied by a root of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.