Applying projective geometry to transformations on rank one idempotents

We present a short proof of Molnárʹs characterization of bijective transformations on the set of all rank one idempotent operators on a Banach space which preserve zero products in both directions. An improvement in the finite-dimensional case is given. We apply these results to describe automorphisms of standard operator semigroups and to improve Uhlhornʹs version of Wignerʹs theorem.