Uniqueness of Rank-Metric Completions of Bratteli Systems

Let $R$ be a unital ring equipped with a Sylvester matrix rank function $\rk$. A harmonic function $α$ on a Bratteli diagram $B$ defines a weighted matrix rank on the associated algebraic direct limit $A(B,R)$. We prove that, if $α$ is extreme and the total weight of blocks of any fixed bounded size