On the Facial Structure of the Unit Balls in a JBW*-Triple and its Predual

The set of tripotents in a JBW*-triple U with its natural ordering and with a largest element adjoined is shown to be a complete lattice, order isomorphic to the lattice of norm closed faces in the unit ballU*1 of the predual U* of U and anti-order isomorphic to the lattice of weak* closed faces of the unit ball U1 in U. As a consequence, the set of partial isometries in a W*-algebra with its natural ordering and again with a largest element adjoined forms a complete lattice and every non-empty weak* closed face of its unit ball is of the form u+(1−uu*)U (1−u*u)1for some unique partial isometry