Basic decomposition of elements and Jauch-Piron effect algebras

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.