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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Local derivations on $C^*$-algebras are derivations
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by B. E. Johnson PDF
Trans. Amer. Math. Soc. 353 (2001), 313-325 Request permission

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
  • 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