Let V be a GL-space and let \(V^*\) be its dual unital GM-space. Suppose that V satisfies the condition that every norm exposed face of the base K of the cone \(V_+\) on V consisting of elements of norm one is projective in the sense of Alfsen and Shultz. By studying the mappings \(F\to F'\) and \(G\to G_{'}\) defined, for subsets F of the unit ball \(V_ 1\) of V and G of the unit ball \(V^*_ 1\) of \(V^*\), by \[ F'=\{a:\quad a\in V^*_ 1a(x)=1,\quad \forall x\in F\},\quad G_{'}=\{x:\quad x\in V_ 1,\quad a(x)=1,\quad a\in G\}. \] The following results are proved:
(i) The weak* semi-exposed faces of \(V^*_ 1\) are the order intervals [2p-e, 2q-e] where p and q are projective units in \(V^*\) and e is the order unit in \(V^*.\)
(ii) Every norm semi-exposed face of \(V_ 1\) is norm exposed and is of the form \(conv(F_ P\cup (-F_ 0))\) where P and Q are orthogonal P- projections on V with corresponding projective faces \(F_ P\) and \(F_ Q\) of K.
Moreover equivalent conditions are found to describe the situation in which every weak* semi-exposed face of \(V^*_ 1\) is weak* exposed.
These results are used to study the example in which V is the pre-dual of a JBW-algebra. In this case it is shown that every weak* closed face of \(V^*_ 1\) is weak* semi-exposed and every norm closed face of \(V_ 1\) is norm semi-exposed thereby completing the description of the facial structure unit balls in a JBW-algebra and its pre-dual.