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

Full-text PDF Free Access

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.

- 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; 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** - J. R. Ringrose,
*Automatic continuity of derivations of operator algebras*, J. London Math. Soc. (2)**5**(1972), 432–438. MR**374927**, DOI https://doi.org/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 https://doi.org/10.2748/tmj/1178244484

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46L57,
46L05,
47B47

Retrieve articles in all journals with MSC (2000): 46L57, 46L05, 47B47

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.