In contrast with known results for amenable operator algebras, we construct a
singly generated subalgebra of ${\mathcal K}({\mathcal H})$ which is
non-amenable, yet is boundedly approximately contractible. The example also
embeds into a homogenous von Neumann algebra.

We revisit the old result that biflat Banach algebras have the same cyclic
cohomology as $\mathbb C$, and obtain a quantitative variant (which is needed
in forthcoming joint work of the author). Our approach does not rely on the
Connes-Tsygan exact sequence, but is motivated strongly by its construction as
found in [Connes,1985] and [Helemskii,1992].

Amenability of any of the algebras described in the title is known to force
them to be finite-dimensional. The analogous problems for \emph{approximate}
amenability have been open for some years now. In this article we give a
complete solution for the first two classes, using a new criterion for showing
that certain Banach algebras without bounded approximate identities cannot be
approximately amenable. The method also provides a unified approach to existing
non-approximate amenability results, and is applied to the study of certain
commutative Segal algebras.