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We construct a singly generated subalgebra of ${\mathcal K}({\mathcal H})$which is non-amenable, yet is boundedly approximately contractible. The exampleembeds into a homogeneous von Neumann algebra. We also observe that there aresingly generated, biflat subalgebras of finite Type I von Neumann algebras,which are not amenable (and hence are not isomorphic to C*-algebras). Such anexample can be used to show that a certain extension property for commutativeoperator algebras, which is shown in arXiv:1012.4259 to follow fromamenability, does not necessarily imply amenability.

Singly generated operator algebras satisfying weakened versions of amenability