Maps preserving peripheral spectrum of Jordan semi-triple products of operators

Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σ_π(A) of A is the set σ_π(A)={z∈σ(A):|z|=max_ω∈σ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:A→B is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σ_π(BAB)=σ_π(Φ(B)Φ(A)Φ(B)) for all A,B∈A if and only if there exists a scalar λ∈C with λ³=1 and either there exists an invertible operator T∈B(X,Y) such that Φ(A)=λTAT^-1 for every A∈A; or there exists an invertible operator T∈B(X^∗,Y) such that Φ(A)=λTA^∗T^-1 for every A∈A. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BA^∗B are also characterized. Such maps are of the form A?UAU^∗ or A?UA^tU^∗, where U∈B(H,K) is a unitary operator, A^t denotes the transpose of A in an arbitrary but fixed orthonormal basis of H.