The most fundamental properties for coarse spaces are property A defined by Yu [@Yu:CoarseHilbert Definition 2.1] and coarse embeddability into a Hilbert space. Yu dealt with these two properties in the study of the coarse Baum–Connes conjecture. The conjecture states that the geometric K-theory of a metric space and the analytic K-theory are isomorphic. Property A implies coarse embeddability and coarse embeddability implies the conjecture.

An expander sequence does not coarsely embed into a Hilbert space, since its components are highly connected. In this paper, we introduce a generalization of expander sequence, which is called a sequence of weak expander spaces. It is proved that weak expander spaces do not have property A (Corollary \[CorollaryNotA\]). This means that connectivity of the spaces is high enough to negate property A.

We often analyze a uniformly locally finite coarse space $X$ by its uniform Roe algebra. The algebra is a C$^*$-algebra and can be regarded as a natural linear representation of the space $X$. Property A is equivalent to nuclearity of the uniform Roe algebra $C^*_\mathrm{u}(X)$ (Skandalis, Tu, and Yu [@SkandalisTuYu Theorem 5.3]). Nuclearity of C$^*$-algebras can be interpreted as a finite dimensional approximation property (Choi–Effros [@ChoiEffrosNuclearCPAP Theorem 3.1], Kirchberg [@KirchbergNuclear]). In 