  The goal of this paper is to solve the problem of existence of an $\ell^2$
relative eta morphism on the Higson-Roe structure group. Using the
Cheeger-Gromov $\ell^2$ eta invariant, we construct a group morphism from the
Higson-Roe maximal structure group constructed in [HiRo:10] to the reals. When
we apply this morphism to the structure class associated with the spin Dirac
operator for a metric of positive scalar curvature, we get the spin $\ell^2$
rho invariant. When we apply this morphism to the structure class associated
with an oriented homotopy equivalence, we get the difference of the $\ell^2$
rho invariants of the corresponding signature operators. We thus get new proofs
for the classical $\ell^2$ rigidity theorems of Keswani obtained in [Ke:00].
