Nonlinear left ∗-Lie triple mappings of standard operator algebras

Abstract

Let \(\mathcal{H}\) be an infinite dimensional complex Hilbert space and \(\mathcal{A}\) be a standard operator algebra on \(\mathcal{H}\) which is closed under the adjoint operation.  For A, B ∈ \(\mathcal{A}\), define by _∗[A, B] = AB − B^∗A the left ∗-Lie product of A and B.  In this paper, we prove that a mapping ϕ : \(\mathcal{A}\) \(\longrightarrow\) \(\mathcal{B}\)(\(\mathcal{H}\)) satisfies ϕ(_∗[A, _∗[B, C]]) = _∗[ϕ(A), _∗[B, C]] + _∗[A, _∗[ϕ(B), C]] + _∗[A, _∗[B, ϕ(C)]], for all A, B, C ∈ \(\mathcal{A}\) is automatically linear.  Moreover, ϕ is an inner ∗-derivation.