An algebra whose subalgebras are characterized by density

Abstract: 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.