Boolean representations of fuzzy quantum spaces (Q1278886)

A fuzzy quantum space on a set \(X\) is a set \(F \subset [0,1]^X\) satisfying (i) the constant zero function belongs to \(F\), (ii) if \(a\in F\), then \(1-a \in F\), (iii) if \(\{a_i\}\) is a sequence of elements of \(F\), then \(\sup_i a_i \in F\), and (iv) the constant function 1/2 is not an element of \(F\). A Boolean representation of \(F\) is a Boolean \(\sigma\)-algebra \(\overline F\) with a surjective mapping \(h: \overline F \to F\) with some special conditions on the observables and states on \(F\) and on \(\overline F\). In the paper under review it is shown that (1) such a representation exists, (2) there exists a minimal and maximal representation, and (3) a minimal representation provides the uniqueness property. These representations make it possible to transfer many known result from Boolean \(\sigma\)-algebras to fuzzy quantum spaces.