Operator spaces with prescribed sets of completely bounded maps

Suppose A is a dual Banach algebra, and a representation π : A → B (ℓ₂) is unital, weak* continuous, and contractive. We use a "Hilbert-Schmidt version" of Arveson distance formula to construct an operator space X, isometric to ℓ₂ ⊗ ℓ₂, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A) ⊗ I_ℓ2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K₁ (C B (X)) contains ℤ₂ as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.