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Automatic continuity of $\sigma$-derivations on $C^*$-algebras


Authors: Madjid Mirzavaziri and Mohammad Sal Moslehian
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
Published electronically: June 6, 2006
MathSciNet review: 2231917
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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.


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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

Keywords: $*$-$(\sigma ,\tau )$-derivation, $\sigma$-derivation, derivation, automatic continuity, $C^*$-algebra
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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.