Automatic continuity of derivations on algebras
Authors:
Madjid Mirzavaziri and Mohammad Sal Moslehian
Journal:
Proc. Amer. Math. Soc. 134 (2006), 33193327
MSC (2000):
Primary 46L57; Secondary 46L05, 47B47
Published electronically:
June 6, 2006
MathSciNet review:
2231917
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: Let be a algebra acting on a Hilbert space , let be a linear mapping and let be a derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous derivation, then there exists a continuous linear mapping such that is a derivation. The continuity of the socalled  derivations is also discussed.
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 J. Hartwig, D. Larsson, S. D. Silvestrov, Deformations of Lie algebras using derivations, Preprints in Math. Sci. 2003:32, LUTFMA50362003 Centre for Math. Sci., Dept. of Math., Lund Inst. of Tech., Lund Univ., 2003.
 2.
 I. Kaplansky, Functional analysis, 1958, Some aspects of analysis and probability, pp. 134 Surveys in Applied Mathematics. Vol. 4 John Wiley Sons, Inc., New York; Chapman Hall, London. MR 0101475 (21:286)
 3.
 M. Mirzavaziri and M. S. Moslehian, derivations in Banach algebras, arXiv:math.FA/0505319.
 4.
 M. S. Moslehian, Approximate contractibility, to appear in Nonlinear Funct. Anal. Appl.
 5.
 J. G. Murphy, Operator Theory and algebras, Academic Press, Inc., Boston, MA, 1990. MR 1074574 (91m:46084)
 6.
 T. W. Palmer, Banach algebras and the general theory of algebras, Vol. I. Algebras and Banach algebras, Encyclopedia of Mathematics and its Applications 49, Cambridge University Press, Cambridge, 1994. MR 1270014 (95c:46002)
 7.
 J. R. Ringrose, Automatic continuity of derivations of operator algebras, J. London Math. Soc. (2) 5 (1972), 432438. MR 0374927 (51:11123)
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 S. Sakai, On a conjecture of Kaplansky, Tôhoku Math. J. (2) 12 (1960) 3133. MR 0112055 (22:2913)
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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/S0002993906083766
PII:
S 00029939(06)083766
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.
