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Maps completely preserving idempotents and maps completely preserving square-zero operators. (English) Zbl 1189.47035

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:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
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