We study in this paper a Hilbert space HV associated with the coarse geometry of an infinite connected graph X(V, E) with vertex set V and edge set E. We show that X(V,E) is uniformly expanding if and only ifl2(V)can be continuously included in HV as a closed subspace,and that the inner product structure of HV is topologically invariant under uniform coarsening of the graph. We also discuss the functorial properties of these Hilbert spaces.
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