Abstract
Let X, Y be real or complex Banach spaces with dimension greater than 2 and let A, B be standard operator algebras on X and Y, respectively. In this paper, we show that every map completely preserving idempotence from A onto B is either an isomorphism or (in the complex case) a conjugate isomorphism; every map completely preserving square-zero from A onto B is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism.
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This work is partially supported by National Natural Science Foundation of China (No. 10771157), Provincial Natural Science Foundation of Shanxi (2007011016) and Research Grant to Returned Scholars of Shanxi (2007-38).
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Hou, J., Huang, L. Maps completely preserving idempotents and maps completely preserving square-zero operators. Isr. J. Math. 176, 363–380 (2010). https://doi.org/10.1007/s11856-010-0032-y
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DOI: https://doi.org/10.1007/s11856-010-0032-y