  Suppose $\mathcal{A}$ is a separable unital ASH C*-algebra, $\mathcal{R}$ is
a sigma-finite II$_{\infty}$ factor von Neumann algebra, and $\pi,\rho
:\mathcal{A}\rightarrow\mathcal{R}$ are unital $\ast$-homomorphisms such that,
for every $a\in\mathcal{A}$, the range projections of $\pi\left( a\right) $ and
$\rho\left( a\right) $ are Murray von Neuman equivalent in $\mathcal{R}% $. We
prove that $\pi$ and $\rho$ are approximately unitarily equivalent modulo
$\mathcal{K}_{\mathcal{R}}$, where $\mathcal{K}_{\mathcal{R}}$ is the norm
closed ideal generated by the finite projections in $\mathcal{R}$. We also
prove a very general result concerning approximate equivalence in arbitrary
finite von Neumann algebras.
