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On the embedding rigidity problem for uniformly locally finite coarse spaces

  • Teng Zhang
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Abstract

In this paper, we construct countable uniformly locally finite metric spaces X and Y such that Cu*(X) is isomorphic to a hereditary C*-subalgebra of Cu*(Y), but neither space coarsely embeds into the other. This gives a negative answer to the embedding rigidity problem for uniformly locally finite coarse spaces. On the positive side, we prove that a hereditary embedding Cu*(X)→ Cu*(Y) implies an injective coarse embedding X→ Y whenever Cu*(Y) contains no block-rank-one ghost projection. In particular, this strengthens the coarse embedding theorem of Braga, Farah, and Vignati by obtaining injectivity under the same hypothesis that every sparse subspace of Y yields only compact ghost projections.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026embedding,
  title = {{On the embedding rigidity problem for uniformly locally finite coarse spaces}},
  author = {Teng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.16949},
  eprint = {2607.16949},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.16949},
}
RIS
TY  - UNPB
TI  - On the embedding rigidity problem for uniformly locally finite coarse spaces
AU  - Teng Zhang
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2607.16949
UR  - https://arxiv.org/abs/2607.16949
AB  - In this paper, we construct countable uniformly locally finite metric spaces X and Y such that C_u^*(X) is isomorphic to a hereditary C^*-subalgebra of C_u^*(Y), but neither space coarsely embeds into the other. This gives a negative answer to the embedding rigidity problem for uniformly locally finite coarse spaces. On the positive side, we prove that a hereditary embedding C_u^*(X)→ C_u^*(Y) implies an injective coarse embedding X→ Y whenever C_u^*(Y) contains no block-rank-one ghost projection. In particular, this strengthens the coarse embedding theorem of Braga, Farah, and Vignati by obtaining injectivity under the same hypothesis that every sparse subspace of Y yields only compact ghost projections.
ER  -