Additive idempotence preservers (Q1855369)

Let \(\mathcal A\) be a local matrix algebra over a field of characteristic different from 2, \(\mathcal B\) an arbitrary algebra, and \(X\), \(Y\) Banach spaces. Additive maps \(\Phi :{\mathcal A}\rightarrow \mathcal B\) which preserve idempotents are characterized. The result is then used in classifying additive surjections \(\Phi :{\mathcal B}(X)\rightarrow {\mathcal B}(Y)\) which preserve idempotents and do not annihilate finite-rank operators.