Product commuting maps with the λ-Aluthge transform

Let H and K be two Hilbert spaces. The algebra of all bounded linear operators acting on H is denoted by B(H). The main purpose of this paper is to obtain a characterization of bijective maps Φ:B(H)→B(K) satisfying the following conditionΔλ(Φ(A)Φ(B))=Φ(Δλ(AB))for all A,B∈B(H), where Δλ(T) stands for the λ-Aluthge transform of the operator T∈B(H). More precisely, we prove that a bijective map Φ satisfies the above condition, if and only if, there exists a unitary operator U:H→K, such that Φ(A)=UAU⁎ for all A∈B(H).