Preservers of λ-Aluthge transforms

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Let M and N be arbitrary von Neumann algebras. For any a in M or in N, let Δλ(a) denote the λ-Aluthge transform of a. Suppose that M has no abelian direct summand. We prove that every bijective map Φ:M→ N satisfying Φ(Δλ(a∘ b*))=Δλ(Φ(a) ∘ Φ(b)*), for all a,b∈ M, (for a fixed λ∈ [0,1]), maps the hermitian part of M onto the hermitian part of N (i.e. Φ(Msa) = Nsa) and its restriction Φ|Msa: Msa→ Nsa is a Jordan isomorphism. If we also assume that Φ(x +i y) = Φ(x) +Φ(i y) for all x,y∈ Msa, then there exists a central projection pc in M such that Φ|ₚc ₘ is a complex linear Jordan *-isomorphism and Φ|₍₁₋ₚc₎ ₘ is a conjugate linear Jordan *-isomorphism. Given two complex Hilbert spaces H and K with dim(H)≥ 2, we also establish that every bijection Φ: B(H)→ B(K) satisfying Φ(Δλ(a b*))=Δλ(Φ(a) Φ(b)*), for all a,b∈ B(H), must be a complex linear or a conjugate linear *-isomorphism.