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

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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.

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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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Authors and Affiliations

  1. Escuela Politécnica Superior de Alcoy, IUMPA, Universitat Politécnica de Valencia, Plaza Ferrándiz y Carbonell 1, 03801, Alcoy, Spain

    Enrique Jordá

  2. Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

    Antonio M. Peralta

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  1. Enrique Jordá
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  2. Antonio M. Peralta
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Correspondence to Antonio M. Peralta.

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First author partially supported by the Spanish Ministry of Economy and Competitiveness Project MTM2016-76647-P, project ACOMP/2015/186 (Spain) and GVA Project AICO/2016/054. Second author partially supported by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund Project No. MTM2014-58984-P and Junta de Andalucía grant FQM375.

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Jordá, E., Peralta, A.M. Inner Derivations and Weak-2-Local Derivations on the \(\hbox {C}^*\)-Algebra \(\varvec{C_0(L,A)}\) . Integr. Equ. Oper. Theory 89, 89–110 (2017). https://doi.org/10.1007/s00020-017-2390-x

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