Rigidity of isometries on idempotents

Wigner’s unitary-antiunitary theorem, which states that every surjective isometry on the Grassmann space of rank one projections is induced by a unitary or an antiunitary, characterizes the rigidity of isometries on projections. We introduce the concept of quasi-lines and characterize their forms. By using the geometrical characterizations of quasi-lines, we can distinguish projections from non-self-adjoint idempotents. Then it is shown that every surjective isometry on the set of rank one idempotents is a unitary or an antiunitary similarity transformation possibly composed with the adjoint operation. This leads to a rigidity result for isometries on all idempotents, which states that every surjective isometry on the set of all nontrivial idempotents acting on a separable Hilbert space is a unitary or an antiunitary similarity transformation possibly composed with the adjoint operation or the complement operation.