 
Summary: Helvetica Physica Acta, Vol. 52 (1979), Birkhäuser Verlag, Basel
A connection between propositional systems in
Hilbert spaces and von Neumann algebras
by Dirk Aerts1) and Ingrid Daubechies1)
Theoretische Natuurkunde, Vrije Universiteit Brüssel, Pleinlaan 2, B1050 Brüssel
(8. I. 1979)
Abstract. A theorem of Bade proves that for a complete Boolean sublattice S£ of 3>(3if) the
following holds:
SS {PeS£"; P is orthogonal projection operator}
We prove that this theorem does not hold for the physically interesting class of nonBoolean
propositional systems embedded in a 3>(3if); we derive however a necessary and sufficient condition
under which the theorem does hold. This condition is automatically satisfied if the propositional
system is Boolean.
1. Introduction
There exist several formalisms for the description of quantum phenomena.
One of these is the axiomatic approach of Jauch and Piron [1] where one starts
from some intuitive and physically very comprehensive ideas. Another one is the
algebraic approach. It is amusing to note that they were both initiated by von
Neumann [2], [3]. The well known Hilbert space approach can be considered as a
special case of the axiomatic approach [4] as well as of the algebraic approach.
