  We refine a construction of Choi, Farah and Ozawa to build a nonseparable
amenable operator algebra $\mathcal A\subseteq\ell_\infty(M_2)$ whose
nonseparable subalgebras, including $\mathcal A$, are not isomorphic to a
$C^*$-algebra. This is done using a Luzin gap and a uniformly bounded group
representation.
  Next, we study additional properties of $\mathcal A$ and of its separable
subalgebras, related to the Kadison Kastler metric.
