The Gleason theorem for the field of rational numbers and residue fields (Q1290796)

The author considers functionals on the logic of all projectors \({\mathcal P}(F^n)\) where \(F\) is a field. The functionals considered are charges, i.e., mappings \(\mu:{\mathcal P}(F^n)\to F\) such that \(\mu(P_1+P_2)=\mu(P_1)+\mu(P_2)\) whenever \(P_1P_2=P_2P_1=0\). The main result is that if \(n\geq 3\) and \(F\) is either the field of rational numbers or a residue field, there exists a unique linear operator \(T\) such that \(\mu(P)=\text{tr}(TP)\) for all projectors \(P\in {\mathcal P}(F^n)\). This constitutes a Gleason-type representation analogous to that which follows from the celebrated theorem of Gleason, valid for charges on the logic of all orthogonal projections in a finite dimensional Hilbert space under a boundedness condition.