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.