  Uniformly finite homology is a coarse homology theory, defined via chains
that satisfy a uniform boundedness condition. By construction, uniformly finite
homology carries a canonical $\ell^\infty$-semi-norm. We show that, for
uniformly discrete spaces of bounded geometry, this semi-norm on uniformly
finite homology in degree 0 with integral coefficients allows for a new
formulation of Whyte's rigidity result. In contrast, we prove that this
semi-norm is trivial on uniformly finite homology in higher degrees with real
coefficients.
