Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T₁\ast \cdots \ast T_k$ on elements in ${}_i$, which covers the usual product $T₁\ast \cdots \ast T_k=T₁\cdots T_k$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying $\sigma \left(\Phi \left(A₁\right)\ast \cdots \ast \Phi \left(A_k\right)\right)=\sigma \left(A₁\ast \cdots \ast A_k\right)$ whenever any one of $A_i$’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied by a root...