Basic decomposition of elements and Jauch–Piron effect algebras

Abstract

We show that every element of a complete atomic effect algebra E has a unique basic decomposition into a sum of a sharp element and unsharp multiples of isotropic atoms of E. Consequently, for such effect algebras we obtain “the Smearing Theorem for states” establishing that every order-continuous state existing on sharp elements of E can be extended to a state on E. For a σ -complete separable atomic effect algebra E we prove that E is a unital and Jauch–Piron effect algebra if and only if the set S ( E ) of all sharp elements of E is a unital Jauch–Piron orthomodular lattice and for finite E, S ( E ) is a Boolean algebra.