Abstract

Let L be a quantum logic, here an orthoalgebra, and let Δ be a convex set of states on L. Then Δ generates a base-normed space, and the dual-order unit-normed space contains a canonically constructed homomorphic copy of L, denoted by eΔ(L). A convex set Δ of states on L is said to be ample provided that every state on L is obtained by restricting an element of the base of the bi-dual order unit-normed space to eΔ(L). For a quantum logic L we show that a convex set of states Δ is ample if and only if Δ is weakly dense in the convex set of all states on L. The notion of ampleness is then discussed in the context of Gleason-type theorems for W* algebras and JBW algebras and also in the context of classical logics