Gottfried T. Rüttimann

Expectation functionals of observables and counters

Abstract A discussion of properties, counters and observables in the framework of a quantum logic is given. We prove the following theorem: Let ( P ,⩽,′) be a quantum logic with a strong property (convex subset of states) M . If every M -detectable property (exposed face of M ) is detected (exposed) by an expectational counter then every state belonging to M is completely additive. From this result we draw several important conclusions.

A geometric characterization of structural projections on a JBW∗-triple☆

A structural projection R on a Jordan∗-triple A is a linear projection such that, for all elements a, b and c in A, R{aRbc}={RabRc}. The L-orthogonal complement G◊ of a subset G of a complex Banach space E is the set of elements x in E such that, for all elements y in G, ||x±y||=||x||+||y||. A contractive projection P on E is said to be neutral if the condition that ||Px||=||x|| implies that the elements Px and x coincide, and is said to be a GL-projection if the L-orthogonal complement (PE)◊ of the range PE of P is contained in the kernel ker(P) of P. It is shown that, for a JBW∗-triple A, with predual A∗, a linear projection R on A is structural if and only if it is the adjoint of a neutral GL-projection on A∗, thereby giving a purely geometric characterization of structural projections.

On conditional probability in GL spaces

We investigate the notion of conditional probability and the quantum mechanical concept of state reduction in the context of GL spaces satisfying the Alfsen-Shultz condition.

Symmetries on quantum logics

Abstract We study conditions under which the group of symmetries of a quantum logic is isomorphic to the group of symmetries on certain subsets of the state space of the logic. The notions of Jordan–Hahn decomposition and ultrafulness of the set of states under consideration play a fundamental role in these investigations. They are used to establish a connection between the elements of the logic and the weak ∗ -exposed points or extreme points of the unit interval of the Banach dual of the signed state space. The results are then interpreted in the standard logic of quantum mechanics.

On blocks in quantum logics

Let L be a quantum logic, Ω(L) the convex set of states on L and M a property, i.e. a convex subset of Ω(L). For any P⊆L we define AM(P)={peL∣μ, veM and μ|P=v|P⇒μ(p)=v(p)}. The subset AM(P)⊆L is orthomodular and AM is a closure operator on the subsets of L. We call P⊆L M-dense, provided AM(P)=L. We show that a non-classical quantum logic satisfying the chain condition and having a full and unital property M has no block which is M-dense. We also prove that a quantum logic with a property M for which every counter is expectational and no block is M-dense necessarily has uncountably many blocks. In this setting we then discuss projection lattices of von Neumann algebras.