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DOI: 10.23671/VNC.2018.2.14715

Derivations on Banach \(*\)-Ideals in von Neumann Algebras

Ber A. F. , Chilin, V. I. , Sukochev F. A.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 2.
Abstract:
It is known that any derivation \(\delta: \mathcal M \to \mathcal M\) on the von Neumann algebra \(\mathcal M\) is an inner, i.e. \(\delta(x) := \delta_a(x) =[a, x] =ax -xa\), \(x \in \mathcal M\), for some \(a \in \mathcal M\). If \(H\) is a separable infinite-dimensional complex Hilbert space and \(\mathcal K(H)\) is a \(C^*\)-subalgebra of compact operators in \(C^*\)-algebra \(\mathcal B(H)\) of all bounded linear operators acting in \(H\), then any derivation \(\delta: \mathcal K(H) \to \mathcal K(H)\) is a spatial derivation, i.e. there exists an operator \( a \in \mathcal B(H)\) such that \(\delta(x) = [x, a]\) for all \(x \in K(H)\). In addition, it has recently been established by Ber A. F., Chilin V. I., Levitina G. B. and Sukochev F. A. (JMAA, 2013) that any derivation \(\delta: \mathcal{E}\to \mathcal{E}\) on Banach symmetric ideal of compact operators \(\mathcal{E} \subseteq \mathcal K(H)\) is a spatial derivation. We show that the same result is also true for an arbitrary Banach \(*\)-ideal in every von Neumann algebra \(\mathcal{M}\). More precisely: If \(\mathcal{M}\) is an arbitrary von Neumann algebra, \(\mathcal{E}\) be a Banach \(*\)-ideal in \(\mathcal{M}\) and \(\delta\colon \mathcal{E}\to \mathcal{E}\) is a derivation on \(\mathcal{E}\), then there exists an element \( a \in \mathcal{M}\) such that \(\delta(x) = [x, a]\) for all \(x \in \mathcal{E}\), i.e. \(\delta \) is a spatial derivation.
Keywords: von Neumann algebra, Banach \(*\)-ideal, derivation, spatial derivation
Language: English Download the full text  
For citation: Ber A. F.,† Chilin V. I., Sukochev F. A. Derivations on† Banach \(*\)-Ideals in von Neumann Algebras. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 1, pp. 23-28. DOI 10.23671/VNC.2018.2.14715
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