  We examine rigidity phenomena for representations of amenable operator
algebras which have an ideal of compact operators. We establish that a
generalized version of Kadison's conjecture on completely bounded homomorphisms
holds for the algebra if the associated quotient is abelian. We also prove that
injective completely bounded representations of the algebra are similar to
complete isometries. The main motivating example for these investigations is
the recent construction of Choi, Farah and Ozawa of an amenable operator
algebra that is not similar to a $C^*$-algebra, and we show how it fits into
our framework. All of our results hold in the presence of the total reduction
property, a property weaker than amenability.
