Polar decomposition in e-rings

An e-ring is a generalization of the ring of bounded linear operators on a Hilbert space together with the subset consisting of all effect operators on that space. Associated with an e-ring is a partially ordered abelian group, called its directed group, that generalizes the additive group of bounded Hermitian operators on the Hilbert space. We prove that every element of the directed group of an e-ring has a polar decomposition if and only if every element has a carrier projection and is split by a projection into a positive and a negative part.