On the embedding rigidity problem for uniformly locally finite coarse spaces

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 $C_u^{\ast }(Y)$, while $X$ does not coarsely embed into$Y$. 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 $C_u^{\ast }(X)$ onto a hereditary $C^{\ast }$-subalgebra of $C_u^{\ast }(Y)$ induces an injective coarse embedding $X\to Y$. This strengthens a main result in \cite{BFV20} by upgrading coarse embeddability to injective coarse embeddability under the same hypothesis.