Geometry of the Dual Ball of the Spin Factor

The main result of this paper is a geometric characterization of the unit ball of the dual of a complex spin factor. THEOREM. A strongly facially symmetric space of type I2 in which every proper norm closed face in the unit ball is norm exposed, and which satisfies ‘symmetry of transition probabilities’, is linearly isometric to the dual of a complex spin factor. This result is an important step in the authors' program of showing that the class of all strongly facially symmetric spaces satisfying certain natural and physically significant axioms is equivalent isometrically to the class of all predual spaces of JBW-triples. The result can be interpreted as a characterization of the non-ordered state space of ‘two state’ physical systems. A new tool for working with concrete spin factors, the so-called facial decomposition, is also developed.

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