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Proceedings of the American Mathematical Society

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Local automorphisms and derivations on ${\mathcal {B}}(H)$

Author: Peter Semrl
Journal: Proc. Amer. Math. Soc. 125 (1997), 2677-2680
MSC (1991): Primary 47B47
MathSciNet review: 1415338
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Abstract: Let ${\mathcal {A}}$ be an algebra. A mapping $\theta :{\mathcal {A}}\longrightarrow {\mathcal {A}}$ is called a $2$-local automorphism if for every $a,b\in {\mathcal {A}}$ there is an automorphism $\theta _{a,b}:{\mathcal {A}}\longrightarrow {\mathcal {A}}$, depending on $a$ and $b$, such that $\theta _{a,b}(a)=\theta (a)$ and $\theta _{a,b}(b)=\theta (b)$ (no linearity, surjectivity or continuity of $\theta $ is assumed). Let $H$ be an infinite-dimensional separable Hilbert space, and let ${\mathcal {B}}(H)$ be the algebra of all linear bounded operators on $H$. Then every $2$-local automorphism $\theta :{\mathcal {B}}(H)\longrightarrow {\mathcal {B}}(H)$ is an automorphism. An analogous result is obtained for derivations.

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Additional Information

Peter Semrl
Affiliation: Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia

Received by editor(s): April 19, 1996
Additional Notes: This work was supported by a grant from the Ministry of Science of Slovenia
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society