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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: June 6, 2006
MathSciNet review: 2231917
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

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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
Email: moslehian@ferdowsi.um.ac.ir

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08376-6
PII: S 0002-9939(06)08376-6
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.