On the embedding rigidity problem for uniformly locally finite coarse spaces

Abstract

In this paper, we construct countable uniformly locally finite metric spaces $X$ and $Y$ such that $C_u^{\ast }(X)$ is isomorphic to a hereditary $C^{\ast }$-subalgebra of $C_u^{\ast }(Y)$, 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 $C_u^{\ast }(X)\to C_u^{\ast }(Y)$ implies an injective coarse embedding $X\to Y$ whenever $C_u^{\ast }(Y)$ 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.

arXiv commentv2, 17 pages; v1, 23 pages