We consider the following problem: suppose π : A → B(E) is a unital contractive representation (here, A is a Banach algebra, and E is a Banach space). When can we equip E with an operator space structure X in such a way that CB(X) consists of 2-summing perturbations of π(A)? We obtain a positive answer in some special cases. In particular, we show that, for any dual Banach algebra A with a separable predual, there exists a separable operator space X and a unital isometric representation π : A → B(X) s.t. CB(X) = π(A) + Π2(X). The key tool in the construction is the notion of “hyperreflexivity with respect to an operator ideal.”

The technique of representing Banach algebras in a way described above allows us to construct operator spaces with “pathological” properties. Time permitting, we will exhibit some examples of such spaces.