AI 中文总结
研究一致局部有限粗空间的嵌入刚性问题，构造反例给出否定答案，又证明在特定条件下\(C_u^*(X)\)到\(C_u^*(Y)\)遗传子代数的同构可诱导\(X\)到\(Y\)的单射粗嵌入，加强了相关结果。
AI 中文摘要
本文构造了可数的一致局部有限度量空间\(X\)和\(Y\)，使得\(C_u^*(X)\)同构于\(C_u^*(Y)\)的一个遗传\(C^*\)-子代数，而\(X\)并不粗嵌入到\(Y\)中，这对一致局部有限粗空间的嵌入刚性问题给出了否定答案。正面结果是，若\(Y\)的每个稀疏子空间仅产生紧幽灵投影，则\(C_u^*(X)\)到\(C_u^*(Y)\)的遗传\(C^*\)-子代数的任何同构诱导一个从\(X\)到\(Y\)的单射粗嵌入，在相同假设下将粗可嵌入性提升为单射粗可嵌入性，加强了文献[BFV20]中的一个主要结果。
英文摘要
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 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 $C_u^*(X)$ onto a hereditary $C^*$-subalgebra of $C_u^*(Y)$ 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.
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