Jordan product maps commuting with the λ-Aluthge transform

Let B(H) be the algebra of all bounded linear operators acting on a Hilbert space H. The main purpose in this paper is to obtain a characterization of bijective maps Φ:B(H)→B(K), K Hilbert space, satisfying the following conditionΔλ(Φ(A)∘Φ(B))=Φ(Δλ(A∘B)) for all A,B∈B(H), where Δλ(T) stands the λ-Aluthge transform of the operator T∈B(H) and A∘B=12(AB+BA) is the Jordan product of A and B. We prove that a bijective map Φ satisfies the above condition, if and only if there exists a unitary linear bounded operator U:H→K, such that Φ has the form Φ(A)=UAU⁎ for all A∈B(H).