A characterization of *-automorphism on ℬ(H)

Abstract

Let H be a Hilbert space and \( A \) be a standard *-subalgebra of ℬ(H). We show that a bijective map Φ: \( A \uparrow A \) preserves the Lie-skew product AB − BA* if and only if there is a unitary or conjugate unitary operator U ∈ ℬ(H) such that Φ(A) = UAU* for all A ∈ \( A \), that is, Φ is a linear *-isomorphism or a conjugate linear *-isomorphism.