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. © 1992 University of North Carolina Press.

A well-known result of Haagerup from 1983 states that every C*-algebra A is weakly amenable, that is, every (associative) derivation from A into its dual is inner. A Banach algebra B is said to be ternary weakly amenable if every continuous Jordan triple derivation from B into its dual is inner. We show that commutative C*-algebras are ternary weakly amenable, but that B(H) and K(H) are not, unless H is finite dimensional. More generally, we inaugurate the study of weak amenability for Jordan-Banach triples, focussing on commutative JB*-triples and some Cartan factors. © 2012 2012. Published by Oxford University Press. All rights reserved.

Given a JBW*-triple Z and a normal contractive projection P : Z → Z, we show that the (Murray-von Neumann) type of each summand of P(Z) is dominated by the type of Z.