Operators on injective tensor products of L1-preduals

TL;DR

This paper investigates operators on injective tensor products of L1-preduals, establishing extension techniques that preserve operator properties and unify approaches to tensor product analysis.

Contribution

It introduces a method to extend operators from tensor products to function spaces, linking properties of the tensor product to those of the component spaces.

Findings

Unconditionally converging operators are strongly bounded.

Operators can be extended to continuous F-valued functions on dual unit balls.

The approach unifies property proofs for tensor products based on component properties.

Abstract

Let X be an L1-predual and E,F be Banach spaces. We use the fact that an unconditionally converging operator T from the injective tensor product of X and E to F is strongly bounded and extend T to an operator S on continuous F-valued functions on the dual unit ball of X with the preservation of properties of T. This procedure provides a unified approach for proving properties of the tensor product of X and E based on the properties of E.