Year: 2026
Type: preprint
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 $C_u^*(Y)$, 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 $C_u^*(X)\to C_u^*(Y)$ implies an injective coarse embedding $X\to Y$ 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.
Source: arXiv (Cornell University)
Author: Zhang, Teng
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7169869616