On the Facial Structure of the Unit Balls in a JBW∗ -Triple and its Predual
C. Martin Edwards, Gottfried T. Rüttimann · Journal of the London Mathematical Society · 1988
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 u.