operators with finite propagation is called the translation algebra $\mathcal{A}^\infty(X)$. The uniform Roe algebra $C^*_\mathrm{u}(X)$ is the operator norm closure of $\mathcal{A}^\infty(X)$. A coarse geometric property of $X$ sometimes implies an operator algebraic property of $C^*_\mathrm{u}(X)$. The typical example is the following:

\[Theorem 5.3 in [@SkandalisTuYu]\] Let $X$ be a uniformly locally finite metric space. The space $X$ has property A if and only if $C^*_\mathrm{u}(X)$ is nuclear.

Local reflexivity
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In this paper, we consider the following properties for C$^*$-algebras: Nuclearity, exactness, and local reflexivity. Nuclearity implies exactness. Conclusions in Kirchberg [@KirchbergExactUHF] show that exactness implies local reflexivity. They are all related to minimal tensor products between C$^*$-algebras. The following is the definition of local reflexivity.

A C$^*$-algebra $B$ is said to be locally reflexive if for every finite dimensional operator system $V \subseteq B^{**}$, there exists a net of contractive completely positive maps $\Phi_j \colon V \rightarrow B$ which converges to $\mathrm{id}_V$ in the point-ultraweak topology.

Instead of the definition, we only use the following features of local reflexivity.

\[Theorem 3.2, Theorem 5.1, and Proposition 5.3 of [@EffrosHaagerup]\] \[PropositionLocalSplit\]

-   A C$^*$-subalgebra of a locally reflexive C$^*$-algebra is also locally reflexive.

-   If $B$ is locally reflexive and 