Singly generated operator algebras satisfying weakened versions of   amenability
  We construct a singly generated subalgebra of ${\mathcal K}({\mathcal H})$ which is non-amenable, yet is boundedly approximately contractible. The example embeds into a homogeneous von Neumann algebra. We also observe that there are singly generated, biflat subalgebras of finite Type I von Neumann algebras, which are not amenable (and hence are not isomorphic to C*-algebras). Such an example can be used to show that a certain extension property for commutative operator algebras, which is shown in arXiv:1012.4259 to follow from amenability, does not necessarily imply amenability. 
Yemon Choi
math.OA math.FA