Operator spaces with prescribed sets of completely bounded maps

Suppose A is a dual Banach algebra, and a representation : A → B( 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 2 ⊗ 2, such that the space of completely bounded maps on X consists of Hilbert–Schmidt perturbations of (A)⊗I 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 K1(CB(X)) contains Z2 as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace. © 2004 Elsevier Inc. All rights reserved