Automatic continuity of $\sigma$-derivations on $C^*$-algebras
HTML articles powered by AMS MathViewer
- 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
- J. Hartwig, D. Larsson, S. D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, Preprints in Math. Sci. 2003:32, LUTFMA-5036-2003 Centre for Math. Sci., Dept. of Math., Lund Inst. of Tech., Lund Univ., 2003.
- Irving Kaplansky, Functional analysis, Some aspects of analysis and probability, Surveys in Applied Mathematics. Vol. 4, John Wiley & Sons, Inc., New York, N.Y.; Chapman & Hall, Ltd., London, 1958, pp. 1–34. MR 0101475
- M. Mirzavaziri and M. S. Moslehian, $\sigma$-derivations in Banach algebras, arXiv:math.FA/0505319.
- M. S. Moslehian, Approximate $(\sigma -\tau )$-contractibility, to appear in Nonlinear Funct. Anal. Appl.
- Gerard J. Murphy, $C^*$-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990. MR 1074574
- Theodore W. Palmer, Banach algebras and the general theory of $^*$-algebras. Vol. I, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994. Algebras and Banach algebras. MR 1270014, DOI 10.1017/CBO9781107325777
- J. R. Ringrose, Automatic continuity of derivations of operator algebras, J. London Math. Soc. (2) 5 (1972), 432–438. MR 374927, DOI 10.1112/jlms/s2-5.3.432
- Shôichirô Sakai, On a conjecture of Kaplansky, Tohoku Math. J. (2) 12 (1960), 31–33. MR 112055, DOI 10.2748/tmj/1178244484
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