Group-Valued Measures on the Lattice of Closed Subspaces of a Hilbert Space

Abstract

We show there are no non-trivial finite Abelian group-valued measures on the lattice of closed subspaces of an infinite-dimensional Hilbert space, and we use this to establish that the unigroup of the lattice of closed subspaces of an infinite-dimensional Hilbert space is divisible. The main technique is a combinatorial construction of a set of vectors in R 2n generalizing properties of those used in various treatments of the KochenSpecker theorem in R 4 .

Generalised Kochen–Specker Theorem in Three Dimensions

“…It was shown in [14], that there is no ℤ 2 -coloring of vectors in ℝ 4 and consequently no ℤ 2 -coloring in dimensions greater than four, see [11]. We show that there is no nonconstant ℤ 2 -coloring even for ℝ 3 , which was an open question, formulated, e.g., in [6,10]. The existence of a ℤ 2 -coloring in ℝ 2 is trivial; hence we answer the only remaining case.…”

“…It was proved in the sixties that they are impossible in dimensions more than two. One of the elegant proofs [2] allowed to prove also the non-existence of ℤ 2 -colorings in dimensions greater than three [6,11]. The case of dimension three remained an open problem.…”

Topos Quantum Theory with Short Posets

“…Remark 3.6. A question related to the existence of a 0,1-valued measure on the projection lattice P(H) was raised in [17]. Does there exist a non-constant Z 2 -valued state on P(H) when H has dimension three?…”