In this paper, we connect the rigidity problem and the coarse Baum-Connes conjecture for Roe algebras. In particular, we show that if $X$ and $Y$ are two uniformly locally finite metric spaces such that their Roe algebras are $*$-isomorphic, then $X$ and $Y$ are coarsely equivalent provided either $X$ or $Y$ satisfies the coarse Baum-Connes conjecture with coefficients. It is well-known that coarse embeddability into a Hilbert space implies the coarse Baum-Connes conjecture with coefficients. On the other hand, we provide a new example of a finitely generated group satisfying the coarse Baum-Connes conjecture with coefficients but which does not coarsely embed into a Hilbert space.

中文翻译：

---

Roe 代数的粗糙 Baum-Connes 猜想和刚性

在本文中，我们将刚性问题和 Roe 代数的粗糙 Baum-Connes 猜想联系起来。特别地，我们证明如果 $X$ 和 $Y$ 是两个一致局部有限度量空间，使得它们的 Roe 代数是 $*$-同构的，那么 $X$ 和 $Y$ 粗略等价，只要 $X$ 或$Y$ 满足带系数的粗鲍姆-康纳猜想。众所周知，希尔伯特空间的粗嵌入性意味着具有系数的粗鲍姆-康纳猜想。另一方面，我们提供了一个新的有限生成群的例子，该群满足带系数的粗鲍姆-康纳猜想，但没有粗略地嵌入到希尔伯特空间中。