Quantum logics and convex spaces (Q1303604)

A logic \(L\) is an orthomodular \(\sigma\)-lattice. A state on \(L\) is a \(\sigma\)-additive normed measure on \(L\). A set \(M\) of states on \(L\) is called rich if for every \(a,b\in L\) with \(a\not\leq b\) there exists an \(m\in M\) with \(m(a)=1\) and \(m(b)\neq 1\). \((L,M)\) is called a \(u\)-spectral logic if \(L\) is a logic and \(M\) is a rich \(\sigma\)-convex set of states on \(L\) such that for every affine functional \(f\) from \(M\) to \([0,1]\) there exists a unique real observable \(x\) on \(L\) such that for every \(m\in M\), \(f(m)\) is the expectation of \(x\) in \(m\). \(u\)-spectral logics are compared with the non-commutative spectral theory of Alfsen and Shultz. The situation when these two approaches correspond to each other is characterized.