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Transactions of the American Mathematical Society

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Local derivations on $C^*$-algebras are derivations


Author: B. E. Johnson
Journal: Trans. Amer. Math. Soc. 353 (2001), 313-325
MSC (2000): Primary 46L57, 46H40
DOI: https://doi.org/10.1090/S0002-9947-00-02688-X
Published electronically: September 18, 2000
MathSciNet review: 1783788
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Abstract:

Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any $C^*$-algebra $\mathfrak{A}$ into any Banach $\mathfrak{A}$-bimodule $\mathfrak{X}$. Most of the work is involved with establishing this result when $\mathfrak{A}$ is a commutative $C^*$-algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra $C^1[0,1]$ of continuously differentiable functions on $[0,1]$. We also give an automatic continuity result, that is, we show that local derivations on $C^*$-algebras are continuous even if not assumed a priori to be so.


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Additional Information

B. E. Johnson
Affiliation: Department of Mathematics, University of Newcastle, Newcastle upon Tyne, England NE1 7RU
Email: b.e.johnson@ncl.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-00-02688-X
Received by editor(s): June 24, 1999
Published electronically: September 18, 2000
Article copyright: © Copyright 2000 American Mathematical Society