Local derivations on $C^*$-algebras are derivations
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- by B. E. Johnson
- Trans. Amer. Math. Soc. 353 (2001), 313-325
- DOI: https://doi.org/10.1090/S0002-9947-00-02688-X
- Published electronically: September 18, 2000
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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.References
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Bibliographic Information
- B. E. Johnson
- Affiliation: Department of Mathematics, University of Newcastle, Newcastle upon Tyne, England NE1 7RU
- Email: b.e.johnson@ncl.ac.uk
- Received by editor(s): June 24, 1999
- Published electronically: September 18, 2000
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1783788