We shall present a solution of a problem which has been open for over twenty years.

In 1992, C. Akemann and G.K. Pedersen described the structure of the norm-closed faces of the unit ball of a C*-algebra A in terms of the compact partial isometries in A**. Three years earlier, C.M. Edwards and G.T.
Ruttimann gave a complete description of the weak*-closed faces of the unit ball of a JBW*-triple, and in particular, in a von Neumann algebra. However, the question whether the norm-closed faces of the unit ball in a JB*-triple E are determined by the compact tripotents in E** has remained open.

We shall survey the positive answer established by
C.M. Edwards, F.J. Fernndez-Polo, C. Hoskin and the speaker in a recent paper.