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).
Fang, Linear maps preserving the idempotency of Jordan products of operators, Electron
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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).