Geometry of State Spaces of Operator Algebras

This monograph presents a complete and self-contained solution to the long-standing problem of giving a geometric description of state spaces of C -algebras and von Neumann algebras, and of their Jordan algebraic analogs (JB-algebras and JBW-algebras). The material, which previously has appeared only in research papers and required substantial prerequisites for a reader's understanding, is made accessible here to a broad mathematical audience. §Key features include: The properties used to describe state spaces are primarily of a geometric nature, but many can also be interpreted in terms of physics. There are numerous remarks discussing these connections A quick introduction to Jordan algebras is given; no previous knowledge is assumed and all necessary background on the subject is given A discussion of dynamical correspondences, which tie together Lie and Jordan structures, and relate the observables and the generators of time evolution in physics The connection with Connes' notions of orientation and homogeneity in cones is explained §Chapters conclude with notes placing the material in historical context Prerequisites are standard graduate courses in real and complex variables, measure theory, and functional analysis Excellent bibliography and index