JAUCH PIRON STATES AND σ-ADDITIVITY L. J. BUNCE Department of Mathematics Reading University Whiteknights, Reading RG6 AX UK JAN HAMHALTER Czech Technical Univ. Faculty of Electrical Engineering Department of Mathematics Technicka 2, 166 27 Prague 6 Czech Republic E-mail: hamhalte@math.feld.cvut.cz 1991 Mathematics Subject Classification: 46L30, 46L50, 46L60 Received 6 May 1998 Revised 14 June 1999 We study σ-additivity of the physically plausible Jauch–Piron states on a von Neumann algebra M. Amongst other consequences we extend our earlier results [5] by showing that geometric conditions much weaker than pureness imply σ-additivity for a Jauch–Piron state and, further, that if M is properly infinite and the continuum hypothesis is assumed to be true then all Jauch–Piron factor states are σ-additive. 1. Introduction A state % on a von Neumann algebra which, in the context of quantum mechan- ics, satisfies the physically plausible condition that % vanishes on e ∨ f whenever it vanishes on projections e and f , is said to be Jauch–Piron state. Thus, given that projections are interpreted as yes-no questions of the ambient physical system, it has been suggested that a state should of necessity satisfy the Jauch–Piron condi- tion in order to qualify as a physical state. A convenient mathematical idealization of the Jauch–Piron condition is that of σ-additivity. However, whereas σ-additive states are always Jauch–Piron states, the converse is far from being true. So it seems both mathematically and physically desirable to determine conditions under which Jauch–Piron states are σ-additive. Initiated in [18], the study of mathematical implications of the Jauch–Piron condition has been pursued extensively for more general systems than are gone in to here (see [17]). For relevant discussion of quantum mechanical proposition system the reader is referred to [3, 11, 12, 17]. For Jauch–Piron states on von Neumann algebras key results were obtained by Amann [3]. In particular, for a von Neumann algebra with separable predual and without abelian part, Amann showed that all pure Jauch–Piron states are normal. 767 Reviews in Mathematical Physics, Vol. 12, No. 6 (2000) 767–777 c  World Scientific Publishing Company