Uniqueness of Rank-Metric Completions of Bratteli Systems

Let $R$ be a unital ring equipped with a Sylvester matrix rank function $\text{rk}$. A harmonic function $\alpha$ on a Bratteli diagram $B$ defines a weighted matrix rank on the associated algebraic direct limit $A(B,R)$. We prove that, if $\alpha$ is extreme and the total weight of blocks of any fixed bounded size tends to zero, then the rank completion of $A(B,R)$ is isomorphic to ${\mathcal{M}}_{R,\text{rk}}$, the rank completion of the direct system ${\text{Mat}}_{2^k}(R)$ with connecting maps $x\mapsto \text{diag}(x,x)$ and normalized ranks $2^{-k}\text{rk}$. The isomorphism preserves the unital $R$-algebra structure and the ranks on all rectangular matrices. The coefficient ring need not be regular, and the specified rank need not be induced from a regular ring. We recover factor-sequence uniqueness and construct corners of every prescribed rank in $(0,1]$ that are isomorphic to ${\mathcal{M}}_{R,\text{rk}}$ with their normalized ranks. Examples show that the coefficient rank can affect the isomorphism type and that the completion can be non-regular and non-simple. For complex coefficients, the trace determined by $\alpha$ gives a rank completion of the associated AF $C^{\ast }$-algebra canonically isomorphic to the affiliated-operator ring of its GNS closure. The rank completion of the algebraic direct limit can be a proper subring of this ring.