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

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	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 + References 1. Sakai S. \(C^\)-Algebras and* \(W^\)-Algebras, Berlin, Springer-Verlag, 1971. 2. Gohberg I., Krein M. Introduction to the Theory of Linear Nonselfadjoint Operators, Providence (R.I.), Amer. Math. Soc., 1969, Translat. of Math. Monogr., vol. 18. 3. Kalton N., Sukochev F. Symmetric Norms and Spaces of Operators. J. Reine Angew. Math., 2008, vol. 621, pp. 81-121. 4. Schatten R. Norm Ideals of Completely Continuous Operators. Second printing, Ergebnisse der Mathematik und ihrer Grenzgebiete, band 27, Berlin, Springer-Verlag, 1970, 98 p. 5. Simon B. Trace Ideals and Their Applications. Second edition, Math. Surveys and Monogr., vol. 120, Providence (R.I.): Amer. Math. Soc., 2005. 6. Ber A. F., Chilin V. I., Levitina G. B. and Sukochev F. A. Derivations with Values in Quasi-Normed Bimodules of Locally Measurable Operators. J. Math. Anal. Appl., 2013, vol. 397, no. 2, pp. 628-643. DOI: 10.101610/j.jmaa.2012.07.068. 7. Ber A. F., Chilin V. I. and Levitina G. B. Derivations with Values in Quasi-Normed Bimodules of Locally Measurable Operators. Sib. Adv. Math.,* 2015, vol. 25, no. 3, pp. 169-178. DOI: 10.3103/S1055134415030025. 8. Str\u{atil\u{a S., Zsido L. Lectures on von Neumann Algebras, Bucharest, Editura Academiei, 1979. 9. Takesaki M. Theory of Operator Algebras I, Berlin etc., Springer-Verlag, 1979. 10. Bratelli O., Robinson D. W. Operator Algebras and Quantum Statistical Mechanics 1, N.Y., Springer-Verlag, 1979. 11. Zsido L. The Norm of a Derivation in a \(W^\)-Algebra. Proc. Amer. Math. Soc., 1973, vol. 38, no. 1, pp. 147-150. 12. Chilin V. I., Levitina G. B. Derivations on Ideals in Commutative \(AW^\)-Algebras. Sib. Adv. Math., 2014, vol. 24, no. 1, pp. 26-42. DOI.10.3103/S1055134414010040. 13. Kusraev A. G. Automorphisms and Derivations on a Universally Complete Complex \(f\)-Algebra. Sib. Math. J., 2006, vol. 47, no. 1, pp. 77-85. DOI: 10.1007/s11202-006-0010-0. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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