Maps on idempotents
Let $X$ be an infinite-dimensional real or complex Banach space, $B(X)$ the algebra of all bounded linear operators on $X$, and $P(X)\subset B(X)$ the subset of all idempotents. We characterize bijective maps on $P(X)$ preserving commutativity in both directions. This unifies and extends the characterizations of two types of automorphisms of $P(X)$, with respect to the orthogonality relation and with respect to the usual partial order; the latter have been previously characterized by Ovchinnikov. We also describe bijective orthogonality preserving maps on the set of idempotents of a fixed finite rank. As an application we present a nonlinear extension of the structural result for bijective linear biseparating maps on $B(X)$.