, Volume 176, Issue 1, pp 363-380
Date: 08 Jun 2010

Maps completely preserving idempotents and maps completely preserving square-zero operators

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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.

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).