In [7], the authors proposed the problem of giving a geometric characterization of those Banach spaces which admit an algebraic structure. Motivated by the geometry imposed by measuring processes on the set of observables of a quantum mechanical system, they introduced the categoryof facially symmetric spaces. A discrete spectral theorem for an arbitrary element in the dual of a reflexive facially symmetric space was obtained by using the basic notions of orthogonality, projective unit, norm exposed face, symmetric face, generalized tripotent and generalized Peirce projection, which were introduced and developed in this purely geometric setting. In this paper, we investigate geometric properties of these spaces. We analyze the generalized Peirce decomposition associated with a face and give a useful condition for two such decompositions to commute. A polar decomposition is proved for an arbitrary element and a characterization is given of semi-exposed faces in these spaces and their duals. The primary example of a facially symmetric space is a Banach space whose dual is a JB*-triple. In particular, this includes the preduals of von Neumann algebras, the duals of C*-algebras and JB*-algebras (=Jordan C*-algebras) as well as those of J*-algebras. The latter includes Hilbert spaces and Caftan factors as special cases. For an introduction to JB*-triples and JBW*-triples see [5, 6, 10]. Characterizations of the state spaces of C*-algebras and of JB*-algebras, based on physically significant axioms, are known and were constructed in a framework of ordered Banach spaces (cf. [1, 2]). On the other hand, the predual of a JBW*-triple enjoys analogues of essentially all the properties which are needed in these characterizations (cf. [5]). We expect that these properties, formulated in a facially symmetric space, will lead to an algebraic structure on its dual. Because of the lack of a global order structure however, this will be a triple product rather than a binary one (cf. the introduction in [7]). This will solve (and give precise meaning to) the problem stated in the first paragraph. One approach to solving this problem is via a spectral theorem, functional calculus, and polarization. However, obtaining a satisfactory continuous spectral theorem seems to be a difficult task, requiring a new version of Choquet theory. Moreover, even though spectral theory and functional calculus can be used to define cubes, it is non-trivial to show that the triple product defined via polarization is additive in each argument, since this requires a treatment of non-compatible elements, discussed below. © 1989, Cambridge Philosophical Society. All rights reserved.
