Amplification of Completely Bounded Operators and Tomiyama’s Slice Maps

Let (M,N ) be a pair of von Neumann algebras, or of dual operator spaces with at least one of them having property Sσ, and let Φ be an arbitrary completely bounded mapping on M. We present an explicit construction of an amplification of Φ to a completely bounded mapping on M⊗N . Our approach is based on the concept of slice maps as introduced by Tomiyama, and makes use of the description of the predual of M⊗N given by Effros and Ruan in terms of the operator space projective tensor product (cf. [Eff–Rua 90], [Rua 92]). We further discuss several properties of an amplification in connection with the investigations made in [May–Neu–Wit 89], where the special case M = B(H) and N = B(K) has been considered (for Hilbert spaces H and K). We will then mainly focus on various applications, such as a remarkable purely algebraic characterization of w∗-continuity using amplifications, as well as a generalization of the so-called Ge–Kadison Lemma (in connection with the uniqueness problem of amplifications). Finally, our study will enable us to show that the essential assertion of the main result in [May–Neu–Wit 89] concerning completely bounded bimodule homomorphisms actually relies on a basic property of Tomiyama’s slice maps.