Integral Equations and Operator Theory

, Volume 89, Issue 1, pp 89–110 | Cite as

Inner Derivations and Weak-2-Local Derivations on the \(\hbox {C}^*\)-Algebra \(\varvec{C_0(L,A)}\)

Article

Abstract

Let L be a locally compact Hausdorff space. Suppose A is a \(\hbox {C}^*\)-algebra with the property that every weak-2-local derivation on A is a (linear) derivation. We prove that every weak-2-local derivation on \(C_0(L,A)\) is a (linear) derivation. Among the consequences we establish that if B is an atomic von Neumann algebra or a compact \(\hbox {C}^*\)-algebra, then every weak-2-local derivation on \(C_0(L,B)\) is a linear derivation. We further show that, for a general von Neumann algebra M, every 2-local derivation on \(C_0(L,M)\) is a linear derivation. We also prove several results representing derivations on \(C_0(L,B(H))\) and on \(C_0(L,K(H))\) as inner derivations determined by multipliers.

Keywords

Derivation 2-Local linear map 2-Local \(^*\)-derivation 2-Local derivation Weak-2-local mapping Weak-2-local derivation 

Mathematics Subject Classification

Primary 47B49 46L05 Secondary 46L40 46T20 47L99 

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Notes

Acknowledgements

We would like to thank our colleague Prof. J. Bonet for his useful suggestions during the preparation of this note. We also thank the anonymous referee for his/her constructive suggestions. A part of this work was done during the visit of the second author to Universitat Politécnica de Valencia in Alcoy and to the IUMPA in Valencia. He would like to thank the Department of Mathematics of the Universitat Politécnica de Valencia and the first author for the hospitality.

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Escuela Politécnica Superior de Alcoy, IUMPAUniversitat Politécnica de ValenciaAlcoySpain
  2. 2.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain

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