[@ArzhantsevaGuentnerSpakula]). This is a conclusion of the argument of [@OzawaBook Proposition 3.7.11].

-   For an appropriate choice of finite index normal subgroups of the free group $F_2$, the box space $\square F_2$ coarsely embeds into a Hilbert Space, although it does not have property A (Arzhantseva, Guentner, and Špakula [@ArzhantsevaGuentnerSpakula]).

-   Let $G$ be a residually finite group with Kazhdan’s property (T). The box space $\square G$ is an expander sequence (see e.g., [@BekkaDelaHarpeValette Section 6.1]).

The construction of box spaces relies on discrete groups. In this paper, we generalize the notion of a box space in the context of coarse geometry and investigate its operator algebraic feature.

\[DefinitionGBS\] A uniformly locally finite coarse space $(X, \mathcal{C})$ is called a generalized box space, if there exists a disjoint decomposition $X = \bigsqcup_{m = 1}^\infty X_m$ satisfying the following conditions:

-   Every subset $X_m$ is finite and not empty,

-   Every controlled set $T \in \mathcal{C}$ is a subset of $\bigsqcup_{m=1}^\infty X_m \times X_m$.

The second condition means that the components $\{X_m\}$ are mutually disjoint in the sense of coarse geometry. This notion allows us to discuss box spaces and expander sequences in the same framework.

Weak expander spaces {#SubsectionWeakExpander}
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