Suppose A is a dual Banach algebra, and a representation π : A → B ( ℓ 2 ) is unital, weak * continuous, and contractive. We use a “Hilbert–Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ℓ 2 ⊗ ℓ 2 , such that the space of completely bounded maps on X consists of Hilbert–Schmidt perturbations of π ( A ) ⊗ I ℓ 2 . 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 K 1 ( CB ( X ) ) contains Z 2 as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.