International Journal of Theoretical Physics, Vol. 43, Nos. 7/8, August 2004 ( C  2004) States and Structure of von Neumann Algebras Jan Hamhalter 1 We summarize and deepen recent results on the interplay between properties of states and the structure of von Neumann algebras. We treat Jauch–Piron states and the concept of independence in noncommutative probability theory. KEY WORDS: states on von Neumann algebras; Jauch–Piron states; independence of algebras. 1. JAUCH–PIRON STATES, σ-ADDITIVITY, AND FACIAL STRUCTURE In this part we analyze connection between the Jauch–Piron property, σ -additivity of states and facial structure of duals of von Neumann algebras. (For basic facts on von Neumann algebras we refer to Kadison and Ringrose, 1983.) Let M be a von Neumann algebra with the projection lattice P ( M ). A state  (i.e. a positive normalized functional) on M is called Jauch–Piron if (e ∨ f ) = 0 whenever e and f are projections in M with (e) = ( f ) = 0. (The symbol e ∨ f stands for the supremum of projections e, f .) In the physical interpretation, pro- jection e represents a random event of the system given by the algebra M . The linear functional  describes probability distributions of all observables. The value (e) is probability of event e in state  of the system. The Jauch–Piron property now postulates that the events with zero probability obey the same law as in clas- sical probability theory: if events e and f have both probability zero then the probability that event e or f occurs is again zero. Not all states on von Neumann algebras have this property (Amann, 1989; Bunce and Hamhalter, 1994). That is why the concept of Jauch–Piron state has received a great deal of attention. It was discovered and first studied by Jauch and Piron (Jauch, 1968; Jauch and Piron, 1965, 1969) in connection with propositional calculus of quantum mechanics and the theory of hidden variables. The first results on Jauch–Piron states on operator algebras were obtained by G. R¨ utimann (1977). A nice characterization of pure 1 Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University, 166 27 Prague 6, Czech Republic; e-mail: hamhalte@math.feld.cvut.cz 1561 0020-7748/04/0800-1561/0 C  2004 Springer Science+Business Media, Inc.