Maps on essentially infinite idempotent operators

Let H be an infinite-dimensional real or complex Hilbert space and I∞(H) the set of all bounded linear idempotent operators on H with infinite-dimensional image and infinite-dimensional kernel. We characterize three types of maps on I∞(H), namely poset automorphisms, bijective maps preserving orthogonality in both directions, and bijective maps preserving commutativity in both directions.