An old result of Dixmier, Day and others states that every continuous bounded representation of an amenable group on Hilbert space is similar to a unitary representation. In similar vein, one can ask if amenable subalgebras of $B(H)$ are always similar to self-adjoint subalgebras. This problem was open for many years, but it was recently shown by Farah and Ozawa that in general the answer is negative; their approach goes via showing that the Dixmier--Day result is false when $B(H)$ is replaced by the Calkin algebra.



In this talk, I will give some of the background, and then outline a simplified and more explicit version of their construction; this is taken from joint work with Farah and Ozawa (2014) . It turns out that the key mechanism behind these negative results is the large supply of projections in $\ell_\infty / c_0$, rather than the complicated structure of $B(H)$.