Bilinear ideals in operator spaces

Abstract

Abstract. We introduce a concept of bilinear ideal of jointly completely bounded mappings between operator spaces. In particular, we study the bilinear ideals N of completely nuclear, I of completely integral, E of completely extendible bilinear mappings, MB multiplicatively bounded and its symmetrization SMB. We prove some basic properties of them, one of which is the fact that I is naturally identified with the ideal of (linear) completely integral mappings on the injective operator space tensor product. Introduction and PreliminariesLet V, W and X be operator spaces. If we consider the underlying vector space structure, the relationshold through the two natural linear isomorphisms ν, ρ. In order for ν and ρ to induce natural morphisms in the operator space category, it is necessary to have appropriately defined an operator space tensor norm on V ⊗ W and specific classes of linear and bilinear mappings. This is the case, for instance, of the so called projective operator space tensor norm · ∧ , the completely bounded maps and the jointly completely bounded bilinear mappings, where ν and ρ induce the following completely bounded isometric isomorphisms:There are many possible ways to provide V ⊗ W with an operator space tensor norm and, of course, to define classes of mappings. Several authors, inspired by the success that the study of the relations between tensor products and mappings has had in the Banach space setting, have systematically study some analogous relations for operator spaces. This is the case, for instance, of the completely nuclear and completely integral linear mappings (see [7, Section III]).In this paper we follow this approach as well, but with the attention focused on the relations involving ν, the isomorphism in (1) which concerns bilinear mappings. In Section 2 we introduce the notion of an ideal of completely bounded bilinear mappings and study its general properties. In Section 3 we define the ideals of completely nuclear and completely integral bilinear mappings. The main result proved here is that the ideal of completely integral bilinear mappings is naturally identified with the ideal of completely integral linear mappings on the injective operator space tensor product, that is I(V × W, X) ∼ = L I (V ∨ ⊗ W, X) (see Theorem 3.8). This implies that, 2010 Mathematics Subject Classification. 47L25,47L22, 46M05.

Biduals of tensor products in operator spaces

“…Second, if N and I denote, respectively, the ideals of completely nuclear and completely integral bilinear mappings (see the definitions in [8]), then Theorem 3. If in the corollary it is further assumed that both dual spaces have CBAP, we get a stronger conclusion about the embedding of V * * ⊗W * * in (V ⊗W ) * * .…”

“…properly defined, we need to establish some properties of operator space tensor norms and to impose conditions on the spaces involved. We recall the notion of operator space tensor norm as defined in [8]: Definition 4.1. We say that α is an operator space tensor norm if α is an operator space matrix norm on each tensor product of operator spaces V ⊗ W that satisfies the following two conditions:…”

“…For an operator space tensor norm α, the set A α of bilinear mappings determined by the relation A α (V × W, X) ∼ = CB(V α ⊗ W, X) defines an operator space bilinear ideal [8,Proposition 2.4]. Thus, Corollary 4.5 also says that, under the hypothesis of local reflexivity, every bilinear form φ ∈ A α (V × W ) admits an extension φ ∈ A α (V * * × W * * ) with φ Aα = φ Aα .…”

“…We recall the duality and some definitions and notation related to tensor products and bilinear mappings in operator spaces. We refer to [11,20] for all the necessary background and notation from operator space theory, to [4] for tensor products of operator spaces and to [8] for some specific results on bilinear ideals on operator spaces.…”