Amplification of Completely Bounded Operators and Tomiyama's Slice Maps

Abstract

Let (M,N) be a pair of von Neumann algebras, or of dual operator spaces with at least one of them having property S_σ, and let T be an arbitrary completely bounded mapping on M. We present an explicit construction of an amplification of T to a completely bounded mapping on the w^*-spatial tensor product of M and N. Our approach makes essential use of the description of the predual of the latter given by Effros and Ruan in terms of the projective operator space tensor product and is further based on the concept of slice maps as introduced by Tomiyama. We discuss several properties of an amplification in connection with the investigations made by May-Neuhardt-Wittstock, 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 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). Our study will eventually enable us to show that the essential assertion of the main result obtained in the above-mentioned work of May-Neuhardt-Wittstock concerning completely bounded bimodule homomorphisms actually relies on a basic property of Tomiyama's slice maps.