Surjective Lp-isometries on rank 1 idempotents

We have studied surjective Lp-isometries on rank 1 idempotents acting on a Hilbert space H for p>1. We show that if φ is a surjective Lp-isometry on the set of rank 1 idempotents, then it leaves the set of rank 1 projections invariant. By applying the classical Wigner's theorem and some further calculations, we prove that either φ or φ⁎ is induced by a unitary or an anti-unitary.