Preduals and complementation of spaces of bounded linear operators

Abstract

For Banach spaces X and Y , we establish a natural bijection between preduals of Y and preduals of L(X, Y ) that respect the right L(X)module structure. If X is reflexive, it follows that there is a unique predual making L(X) into a dual Banach algebra. This removes the condition that X have the approximation property in a result of Daws.We further establish a natural bijection between projections that complement Y in its bidual and projections that complement L(X, Y ) in its bidual as a right L(X)-module. It follows that Y is complemented in its bidual if and only if L(X, Y ) is (either as a module or as a Banach space).Our results are new even in the well-studied case of isometric preduals.

Extending representations of Banach algebras to their biduals

“…Thus, ϕ * * : (A * * , ♦) → (L(X) * * , ♦) is multiplicative. The map α X is multiplicative for the right Arens product ♦ on L(X) * * , and we have γ X = α X • κ L(X) ; see Corollary 3.23 and Lemma 3.19 in [GT16a]. We thus obtain the following commutative diagram, where each map is a multiplicative operator:…”

“…Throughout this section, A denotes a Banach algebra, X denotes a Banach space, and ϕ : A → L(X) denotes a representation. Using the multiplicative operator α X : (L(X) * * , ♦) → L(X, X * * ) constructed in [GT16a], we extend ϕ to a multiplicative operatorφ : (A * * , ♦) → L(X, X * * ); see Paragraph 2.1 and Proposition 2.2. The main result of this section is Theorem 2.5, where we characterize when the image ofφ is contained in L(X) in terms of weak compactness of orbit maps A → X.…”

“…This gives L(X, X * * ) the structure of a unital Banach algebra, with unit κ X ; see [GT16a, Paragraph 3.9]. Note that L(X, X * * ) is anti-isomorphic to L(X * ) by [GT16a,Proposition 3.11] We let γ X : L(X) → L(X, X * * ) be given by γ X (a) = κ X • a for a ∈ L(X). Then γ X is an isometric, multiplicative operator; see [GT16a, Paragraph 3.11].…”

“…However, in Section 2 we obtain a complete answer if we require the extension to be the 'canonical' one, in the following sense. In [GT16a], we introduced a natural multiplication on L(X, X * * ) and a multiplicative operator α X : (L(X) * * , ♦) → L(X, X * * ); see Paragraph 2.1 for details.…”

Preduals for spaces of operators involving Hilbert spaces and trace-class operators

“…To obtain these results, we use that isometric preduals of Y naturally correspond to contractive projections L(X, Y ) * * → L(X, Y ) that are right L(X)-module maps and have weak * closed kernel; see [GT16,Theorem 5.7]. Hence, we are faced with: Problem 1.2.…”