Bijections on the Unit Ball of B(H) Preserving ∗ -Jordan Triple Product

Let B1 denote the closed unit ball of B(H), the von Neumann algebra of all bounded linear operators on a complex Hilbert space H with dim H ≥ 2. Suppose that φ is a bijection on B1 (with no linearity assumption) satisfying φ(AB∗A) = φ(A)φ(B) ∗ φ(A), (A, B ∈ B1). 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 conse- quence, 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).