Uniqueness of Rank-Metric Completions of Bratteli Systems

Abstract

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 ℳ_R,rk, the rank completion of the direct system Mat_2^k(R) with connecting maps x↦diag(x,x) and normalized ranks 2^−krk. 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 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.

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BibTeX RIS

BibTeX

@misc{jiang2026uniqueness,
  title = {{Uniqueness of Rank-Metric Completions of Bratteli Systems}},
  author = {Baojie Jiang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12924},
  eprint = {2609.12924},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12924},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Uniqueness of Rank-Metric Completions of Bratteli Systems
AU  - Baojie Jiang
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12924
UR  - https://arxiv.org/abs/2609.12924
AB  - 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 ℳ_R,rk, the rank completion of the direct system Mat_2^k(R) with connecting maps x↦diag(x,x) and normalized ranks 2^−krk. 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 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.
ER  -

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