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Nonlinear \(\ast\)-Lie derivations on factor von Neumann algebras. (English) Zbl 1263.46058

Let \(A\) be an associative complex algebra with an involution \(\ast\). A map \(\delta : A\to A\) is said to be an additive \(\ast\)-derivation if it is an additive derivation (i.e., \(\delta (a b) = \delta(a) b + a \delta(b)\) for every \(a,b\in A\)) and a symmetric mapping (i.e., \(\delta(a^*) = \delta(a)^*\) for all \(a\in A\)). A not necessarily additive mapping \(\phi : A\to A\) is called a nonlinear \(\ast\)-Lie derivation if \(\phi([a,b]_*)=[\phi(a),b]_* + [a,f(b)]_*\) for all \(a,b \in A\), where \([a,b]_* :=a b -b a^*\). The product \([\cdot,\cdot]_*\) has appeared, under a different notation, in [P. Šemrl, Colloq. Math. 59, No. 2, 241–251 (1990; Zbl 0723.46044); Stud. Math. 97, No. 3, 157–165 (1991; Zbl 0761.46047); Proc. Am. Math. Soc. 119, No. 4, 1105–1113 (1993; Zbl 0803.15024)], [L. Molnár, Linear Algebra Appl. 235, 229–234 (1996; Zbl 0852.46021)] and [M. Brešar and M. Fošner, Publ. Math. 57, No. 1–2, 121–134 (2000; Zbl 0969.16013)].
Continuous linear Lie derivations from a C\(^*\)-algebra \(A\) to a Banach \(A\)-bimodule have been described by B. E. Johnson [Math. Proc. Camb. Philos. Soc. 120, No. 3, 455–473 (1996; Zbl 0888.46024)]. Not necessarily continuous Lie derivations on a C\(^*\)-algebra were studied by M. Mathieu and A. R. Villena [J. Funct. Anal. 202, No. 2, 504–525 (2003; Zbl 1032.46086)]. Not necessarily linear Lie derivations have recently been explored, for example, by L. Chen and J.-H. Zhang [Linear Multilinear Algebra 56, No. 6, 725–730 (2008; Zbl 1166.16016)] who studied nonlinear Lie derivations on the class of upper triangular matrix algebras. More recently, the authors of this note showed that every nonlinear Lie derivation of a triangular algebra is the sum of an additive derivation and a map into its center that sends commutators to zero [Linear Algebra Appl. 432, No. 11, 2953–2960 (2010; Zbl 1193.16030)]. In [Acta Math. Sin., Chin. Ser. 54, No. 5, 791–802 (2011; Zbl 1249.47013)], F.-J. Zhang et al.proved that every nonlinear Lie derivation on a factor von Neumann algebra \(\mathcal{M}\) is of the form \(a\mapsto \varphi(a)+h(a)I\), where \(\varphi: \mathcal{M}\to \mathcal{M}\) is an additive derivation and \(h: \mathcal{M}\to \mathbb C\) is a nonlinear map with \(h(ab-ba)=0\) for all \(a, b\in \mathcal{M}\).
In the paper under review, the authors study nonlinear \(^*\)-Lie derivations on factor von Neumann algebras. The main result (Theorem 2.1) says that, if \(M\) is a factor von Neumann algebra acting on a complex Hilbert space \(H\) with dimension at least two, then every nonlinear \(^*\)-Lie derivation \(\phi:M \to M\) is an additive \(^*\)-derivation. As a consequence, the authors show that, if \(H\) is an infinite-dimensional complex Hilbert space and \(\phi :B(H) \to B(H)\) is a nonlinear \(^*\)-Lie derivation, then there exists \(s\in B(H)\) such that \(\phi (a)=as - sa\) for all \(a\in B(H)\).

MSC:

46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
47B49 Transformers, preservers (linear operators on spaces of linear operators)
16W25 Derivations, actions of Lie algebras
46L10 General theory of von Neumann algebras
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References:

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