In this paper we study structural and uniqueness questions for Cartan subalgebras of uniform Roe algebras. We characterise when an inclusion $B\subseteq A$ of $C^*$-algebras is isomorphic to the canonical inclusion of $\ell^\infty(X)$ inside a uniform Roe algebra $C^*_u(X)$ associated to a metric space of bounded geometry. We obtain uniqueness results for “Roe Cartans” inside uniform Roe algebras up to automorphism when $X$ coarsely embeds into Hilbert space, and up to inner automorphism when $X$ has property A.

中文翻译：

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一致Roe代数中的Cartan子代数

在本文中，我们研究均匀Roe代数的Cartan子代数的结构和唯一性问题。我们表征了何时$ C ^ * $-代数的包含$ B \ subseteq A $与统一Roe代数$ C ^ * _ u（X）$内的$ \ ell ^ \ infty（X）$的规范包含同构。与有界几何的度量空间相关联。我们获得统一Roe代数中“ Roe Cartans”的唯一性结果，当$ X $粗糙地嵌入希尔伯特空间时，Roe Cartans达到自同构，当$ X $具有属性A时，达到内自同构。