Comparative probability on von Neumann algebras

We continue here the study begun in earlier papers on implementation of comparative probability by states. Let $\mathcal{A}$ be a von Neumann algebra on a Hilbert space $\mathcal{H}$ and let $\mathcal{P}(\mathcal{A})$ denote the projections of $\mathcal{A}$. A comparative probability (CP) on $\mathcal{A}$ (or more correctly on $\mathcal{P} (\mathcal{A}))$ is a preorder $\preceq$ on $\mathcal{P}(\mathcal{A})$ satisfying:

$0\preceq P\forall P\in\mathcal{P}(\mathcal{A})$ with $Q\npreceq 0$ for some $Q\in\mathcal{P}(\mathcal{A})$.

If $P,Q\in\mathcal{P}(\mathcal{A})$, then either $P\preceq Q$ or $Q\preceq P$.

If $P$, $Q$ and $R$ are all in $\mathcal{P}(\mathcal{A})$ and $P\perp R$, $Q\perp R$, then $P\preceq Q\Leftrightarrow P+R\preceq Q+R$.

A state $\omega$ on $\mathcal{A}$ is said to implement a $\text{CP }\preceq$ on $\mathcal{A}$ if for $P,Q\in\mathcal{P}(\mathcal{A})$, $P\preceq Q\Leftrightarrow \omega(P)\le \omega(Q)$. In this paper, we examine the conditions for implementability of a CP on a general von Neumann algebra (as opposed to only type I factors). A crucial tool used here, as well as in earlier results, is the interval topology generated on $\mathcal{P}(\mathcal{A})$ by $\preceq$. A $\text{CP }\preceq$ will be termed continuous in a given topology on $\mathcal{A}$ if the interval topology generated by $\preceq$ is weaker than the topology induced on $\mathcal{P}(\mathcal{A})$ by the given topology. We show that uniform continuity of a comparative probability is necessary and sufficient if the von Neumann algebra has no finite direct summand. For implementation by normal states, weak continuity is sufficient and necessary if the von Neumann algebra has no finite direct summand of type I. We arrive at these results by constructing an appropriate additive measure from the CP.