LIE TRIPLE DERIVATIONS ON FACTOR VON NEUMANN ALGEBRAS
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Let A be a factor von Neumann algebra with dimension greater than 1. We prove that if a linear map δ : A → A satisfies δ([[a, b], c]) = [[δ(a), b], c] + [[a, δ(b)], c] + [[a, b], δ(c)] for any a, b, c ∈ A with ab = 0 (resp. ab = P, where P is a fixed nontrivial projection of A), then there exist an operator T ∈ A and a linear map f : A → CI vanishing at every second commutator [[a, b], c] with ab = 0 (resp. ab = P) such that δ(a) = aT − Ta + f(a) for any a ∈ A.
Abstract
Let A be a factor von Neumann algebra with dimension greater than 1. We prove that if a linear map δ : A → A satisfies δ([[a, b], c]) = [[δ(a), b], c] + [[a, δ(b)], c] + [[a, b], δ(c)] for any a, b, c ∈ A with ab = 0 (resp. ab = P, where P is a fixed nontrivial projection of A), then there exist an operator T ∈ A and a linear map f : A → CI vanishing at every second commutator [[a, b], c] with ab = 0 (resp. ab = P) such that δ(a) = aT − Ta + f(a) for any a ∈ A.
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