发表机构
Chongqing Normal University（重庆师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在极端调和函数且块权重趋于零的条件下，Bratteli 系统的秩完备化唯一同构于对角嵌入矩阵系统的秩完备化，并构造了指定秩的角，展示了系数秩的影响及非正则非单情形。
AI 中文摘要
设 $R$ 为配备 Sylvester 矩阵秩函数 $\rk$ 的单位环。Bratteli 图 $B$ 上的调和函数 $\alpha$ 定义了相关代数直接极限 $A(B,R)$ 上的加权矩阵秩。我们证明，若 $\alpha$ 是极端的且任意固定有界大小的块的总权重趋于零，则 $A(B,R)$ 的秩完备化与 $\mathcal M_{R,\rk}$ 同构，其中 $\mathcal M_{R,\rk}$ 是直接系统 $\Mat_{2^k}(R)$（连接映射为 $x\mapsto\diag(x,x)$，归一化秩为 $2^{-k}\rk$）的秩完备化。该同构保持单位 $R$-代数结构以及所有矩形矩阵上的秩。系数环不必是正则的，指定的秩也不必由正则环诱导。我们恢复了因子序列的唯一性，并构造了每个指定秩在 $(0,1]$ 中的角，这些角与 $\mathcal M_{R,\rk}$ 及其归一化秩同构。例子表明系数秩可以影响同构类型，且完备化可以是非正则且非单的。对于复系数，由 $\alpha$ 确定的迹给出了相关 AF $C^*$-代数的秩完备化，该完备化典范同构于其 GNS 闭包的附属算子环。代数直接极限的秩完备化可以是该环的真子环。
英文摘要
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 tends to zero, then the rank completion of $A(B,R)$ is isomorphic to $\mathcal M_{R,\rk}$, the rank completion of the direct system $\Mat_{2^k}(R)$ with connecting maps $x\mapsto\diag(x,x)$ and normalized ranks $2^{-k}\rk$. 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 $\mathcal M_{R,\rk}$ 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 $α$ gives a rank completion of the associated AF $C^*$-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.
