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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Automatic continuity of $\sigma$-derivations on $C^*$-algebras
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by Madjid Mirzavaziri and Mohammad Sal Moslehian PDF
Proc. Amer. Math. Soc. 134 (2006), 3319-3327 Request permission

Abstract:

Let $\mathcal {A}$ be a $C^*$-algebra acting on a Hilbert space $\mathcal {H}$, let $\sigma :\mathcal {A}\to B(\mathcal {H})$ be a linear mapping and let $d:\mathcal {A}\to B(\mathcal {H})$ be a $\sigma$-derivation. Generalizing the celebrated theorem of Sakai, we prove that if $\sigma$ is a continuous $*$-mapping, then $d$ is automatically continuous. In addition, we show the converse is true in the sense that if $d$ is a continuous $*$-$\sigma$-derivation, then there exists a continuous linear mapping $\Sigma :\mathcal {A}\to B(\mathcal {H})$ such that $d$ is a $*$-$\Sigma$-derivation. The continuity of the so-called $*$-$(\sigma ,\tau )$-derivations is also discussed.
References
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Additional Information
  • Madjid Mirzavaziri
  • Affiliation: Department of Mathematics, Ferdowsi University, P.O. Box 1159, Mashhad 91775, Iran
  • Email: mirzavaziri@math.um.ac.ir
  • Mohammad Sal Moslehian
  • Affiliation: Department of Mathematics, Ferdowsi University, P.O. Box 1159, Mashhad 91775, Iran
  • MR Author ID: 620744
  • ORCID: 0000-0001-7905-528X
  • Email: moslehian@ferdowsi.um.ac.ir
  • Received by editor(s): May 26, 2005
  • Received by editor(s) in revised form: June 1, 2005
  • Published electronically: June 6, 2006
  • Communicated by: Joseph A. Ball
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 3319-3327
  • MSC (2000): Primary 46L57; Secondary 46L05, 47B47
  • DOI: https://doi.org/10.1090/S0002-9939-06-08376-6
  • MathSciNet review: 2231917