An algebra whose subalgebras are characterized by density

We refine a construction of Choi, Farah and Ozawa to build a nonseparable amenable operator algebra $\mathcal{A}\subseteq {\mathrm{\ell }}_{\mathrm{\infty }}(M_2)$ whose nonseparable subalgebras, including $\mathcal{A}$, are not isomorphic to a $C^{\ast }$-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.