Maps on B(X) or M_n preserving idempotency of the (scaled) Jordan product in both directions are signed similarity transformations (with field automorphisms and transposition in finite dimensions).
ˇSemrl, Non-linear commutativity preserving maps , Acta Sci
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Maps preserving the idempotency of Jordan products
Maps on B(X) or M_n preserving idempotency of the (scaled) Jordan product in both directions are signed similarity transformations (with field automorphisms and transposition in finite dimensions).