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 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. Effros and Ruan (Internat. J. Math. 1(2) (1990) 163; J. Operator Theory 27 (1992) 179)). We further discuss several properties of an amplification in connection with the investigations made in May et al. (Arch. Math. (Basel) 53(3) (1989) 283), 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 et al. (1989) concerning completely bounded bimodule homomorphisms actually relies on a basic property of Tomiyama's slice maps.