On the embedding rigidity problem for uniformly locally finite coarse spaces

2026/07/18 by Teng Zhang
#math.OA

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Abstract

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), while X does not coarsely embed intoY. This gives a negative answer to the embedding rigidity problem for uniformly locally finite coarse spaces. On the positive side, we prove that, if every sparse subspace of Y yields only compact ghost projections, then any isomorphism of C_u^*(X) onto a hereditary C^*-subalgebra of C_u^*(Y) induces an injective coarse embedding X→ Y. This strengthens a main result in \citeBFV20 by upgrading coarse embeddability to injective coarse embeddability under the same hypothesis.

Citations

• Isomorphism rigidity of uniform Roe algebras over arbitrary uniformly locally finite coarse spaces

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