Let A be a closed, not necessarily self-adjoint, subalgebra of $\ell^\infty\otimes {\mathbb M}_d$. If A is isomorphic to a C*-algebra, then it must be amenable as a Banach algebra. The converse implication is known to fail: A can be amenable without being isomorphic to a C*-algebra. However, if we restrict ourselves to A which arise as subalgebras of $C(X;{\mathbb M}_d)$ for metrizable X, it remains an open question whether amenability implies being isomorphic to C*-algebra.

In this talk, I will sketch a description of the counterexample to the original question (joint work with I. Farah and N. Ozawa) and report on some partial positive results for the second question (joint work with R. Green). In both cases, an important role is played by studying which bounded subgroups of matrix corona algebras are similar to unitary subgroups.