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Zbl 1189.47035
Hou, Jinchuan; Huang, Li
(Hou, Jin-chuan)
Maps completely preserving idempotents and maps completely preserving square-zero operators.
(English)
[J] Isr. J. Math. 176, 363-380 (2010). ISSN 0021-2172; ISSN 1565-8511/e

Summary: Let $X,Y$ be real or complex Banach spaces with dimension greater than 2 and let $\mathcal A, \mathcal B$ be standard operator algebras on $X$ and $Y$, respectively. In this paper, we show that every map completely preserving idempotence from $\mathcal A$ onto $\mathcal B$ is either an isomorphism or (in the complex case) a conjugate isomorphism; every map completely preserving square-zero from $\mathcal A$ onto $\mathcal B$ is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism.
MSC 2000:
*47B49 Transformers (=operators on spaces of operators)

Keywords: idempotence preserving maps

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