Abstract
The predual of a von Neumann algebra is shown to be a neutral strongly facially symmetric space, thereby suggesting an affine geometric approach to operator algebras and their non-associative analogues. Geometric proofs are obtained for the polar decompositions of normal functionals in ordered and non-ordered settings. A fundamental problem in the operator algebraic approach to quantum mechanics is to determine those algebraic structures in Banach spaces which are characterized by a set of geometrical axioms defining the quantum mechanical measuring process. This problem was solved in the context of ordered Banach spaces by Alfsen, Hanche-Olsen, and Shultz ([2], [1]) and led to a characterization of the state spaces of /2?*-algebras and C*-algebras. The main thrust of the present authors' recent research has been to find those algebraic structures induced on (unordered) Banach spaces in which such quantum mechanical axioms are satisfied. This project, which was initiated in [14] and [15] using the affine geometry of the dual unit ball, is used here to give a geometric proof of the Tomita-Sakai-Effros polar decomposition of a normal functional on a von Neumann algebra. Thus, the purpose of this partially expository paper is to show the richness and power of the affine geometric structure of the dual space of an operator algebra, by working in a purely geometric model. Indeed, since this geometry can be described in terms of the underlying real structure, it can be used to obtain new results in the real structure of operator algebras and in the structure of real operator algebras. For example, by using this approach, Dang ([5]) has shown that a raz/-linear isometry of a C*-algebra is the sum of a linear and a conjugate linear isometry, and hence is multiplicative, thereby obtaining a real analogue of Kadison's non-commutative extension of the BanachStone Theorem. The category of strongly facially symmetric (SFS) spaces (simply called facially symmetric spaces in [14] and [15]) has been shown to
Highlights
The predual of a von Neumann algebra is shown to be a neutral strongly facially symmetric space, thereby suggesting an affine geometric approach to operator algebras and their non-associative analogues
The purpose of this partially expository paper is to show the richness and power of the affine geometric structure of the dual space of an operator algebra, by working in a purely geometric model. Since this geometry can be described in terms of the underlying real structure, it can be used to obtain new results in the real structure of operator algebras and in the structure of real operator algebras
Facially symmetric spaces include the preduals of von Neumann algebras, and more generally of JBW*-triples, and can serve as a geometric order-free model in which to study operator algebras and their non-associative analogues
Summary
The predual of a von Neumann algebra is shown to be a neutral strongly facially symmetric space, thereby suggesting an affine geometric approach to operator algebras and their non-associative analogues. We will show that the predual of a von Neumann algebra is a neutral strongly facially symmetric space, and that the set of generalized tripotents coincides with the set of partial isometries in the von Neumann algebra These facts will be used to give a geometric proof of the Tomita-Sakai-Eίfros polar decomposition of a normal functional. Lemma 2.3 shows that the map u »- Fu from the set of partial isometries in a von Neumann algebra A to the set of norm exposed faces in the unit ball A*t\ of the predual A* is onto. We have the following consequence of Lemmas 2.8 and 2.4
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