States on Partial Rings

ISSN： 0020-7748

Source： International Journal of Theoretical Physics, Vol.37, Iss.1, 1998-01, pp. : 609-621

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Abstract

Recently, the structure theory of JB*-triples has received considerable attention. The reason is that JB*-triples and those JB*-triples which are dual spaces, the JBW*-triples, not only form natural generalizations of Jordan C*-algebras and C*-algebras, and Jordan W*-algebras and W*-algebras, but also provide a context for the study of infinite-dimensional holomorphy and infinite-dimensional Lie algebras. In a JBW*-triple the tripotents play the role of the projections in a W*-algebra. In analogy to the projection lattice of a W*-algebra, we investigate the partial ring of tripotents of JBW*-triple. Unlike on W*-algebras, states, i.e., positive normalized homomorphisms from the partial ring of tripotents of a JBW*-triple into the partial ring of real numbers, have not yet been discussed in the literature. We show that the partial ring of tripotents of a JBW*-triple admits a unital set of Jauch-Piron states.