algebra $C^*_\mathrm{u} (X)$ and its unital subalgebra $B$ have amenable traces. Let $\Psi_m \colon C^*_\mathrm{u} (X) \rightarrow \mathbb{B}(\ell _2 X_m) = \mathbb{M}(\sharp(X_m), \mathbb{C})$ be the unital completely positive map by compression. Since $X_m$ is isolated in the coarse space $X$, $\Psi_m$ is a $*$-homomorphism. The composition $\theta_m = \mathrm{tr} \circ \Psi_m$ is a trace on $C^*_\mathrm{u} (X)$. Let $\theta$ be an accumulation point of $\{\theta_m\}$ in the state space of $C^*_\mathrm{u} (X)$. By Theorem \[TheoremAmenableTrace\] (\[TheoremAmenableTraceConditionNet\]), this is an amenable trace of $C^*_\mathrm{u} (X)$ and of its unital C$^*$-subalgebra $B$.

Construction of groupoid associated to label {#SectionGroupoid}
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Groupoid is an algebraical object equipped with partially defined associative product, a space of units, and inverse. We often consider additional structures such as measure and topology. Measured groupoids are related to constructions of von Neumann algebras and ergodic theory of group actions. Topological groupoids serve as fundamental tools for operator K-theory and coarse geometry. For the axioms of groupoids, the reader is referred to Renault’s book [@BookRenault Chapter 1].

Skandalis, Tu, and Yu described a coarse space $X$ in terms of topological groupoid in [@SkandalisTuYu Section 3]. Their groupoid was constructed from Stone–Čech compactification of $X$. For the proof of our 