Let B_1 denote the closed unit ball of B(H), the von Neumann algebra of all bounded linear operators on a complex Hilbert space H with dimH ≥ 2. Suppose that ϕ is a bijection on B_1 (with no linearity assumption) satisfying ϕ(AB^∗A) = ϕ(A)ϕ(B)^∗ϕ(A), (A,B ∈ B_1). If I and T denote the identity operator on H and the unit circle in C, respectively, and if ϕ is continuous on {λI : λ ∈ T}, then we show that ϕ(I) is a unitary operator and ϕ(I)ϕ extends to a linear or conjugate linear Jordan ∗-automorphism on B(H). As a consequence, there is either a unitary or an antiunitary operator U on H such that ϕ(A) = ϕ(I)UAU^∗, (A ∈ B1) or ϕ(A) = ϕ(I)UA^∗U^∗, (A ∈ B1).

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