of weak expander spaces is a generalization in this context. A sequence of expander graphs can also be characterized by means of Poincar[' e]{} inequality. Ostrovskii [@Ostrovskii Theorem 2.4] and Tessera [@TesseraPoincare] independently characterized metric spaces which are not coarsely embeddable into Hilbert spaces and $L^1$-spaces. They made use of weaker versions of Poincar[' e]{} inequality. Tessera also studied non-embeddability into uniformly convex Banach spaces and ${\rm CAT}(0)$ spaces ([@TesseraPoincare Corollary 17]). Our generalization does not imply this kind of non-embeddability. Indeed, the box space $\square F_2$ of Arzhantseva, Guentner, and Špakula [@ArzhantsevaGuentnerSpakula] is a sequence of weak expanders and coarsely embeds into a Hilbert space.

Preliminary on operator algebra {#SectionLocalReflexivity}
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Translation algebra and uniform Roe algebra
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Let us recall the definition of a uniform Roe algebra. We regard the algebra as a natural linear representation of a coarse space. A bounded linear operator $a$ on $\ell_2 X$ is said to have finite propagation if its matrix coefficient is located on a controlled set. More precisely, there exists a controlled set $T_a \subseteq X \times X$ such that $\langle a \delta_y, \delta_x \rangle = 0, (x, y) \in (X \times X) \setminus T_a$. The collection of all the 