Tensor products of measurable Banach bundles

TL;DR

This paper constructs measurable Banach bundles representing tensor products of modules over a probability space, providing a fiberwise approach to understanding injective and projective tensor products in this setting.

Contribution

It introduces a method to realize tensor products of measurable Banach bundles via their sections, linking module tensor products to bundle tensor products fiberwise.

Findings

Constructed measurable Banach bundles for tensor products

Established isomorphisms between bundle sections and module tensor products

Provided fiberwise representation of tensor products of Banach modules

Abstract

We study injective and projective tensor products of measurable Banach bundles. More precisely, given two separable measurable Banach bundles ${\bf E}$, ${\bf F}$ defined over a probability space $({\rm X},\Sigma,\mathfrak m)$, we construct two measurable Banach bundles ${\bf E}\hat\otimes_\varepsilon{\bf F}$ and ${\bf E}\hat\otimes_\pi{\bf F}$ over $({\rm X},\Sigma,\mathfrak m)$ such that $\Gamma({\bf E}\hat\otimes_\varepsilon{\bf F})\cong\Gamma({\bf E})\hat\otimes_\varepsilon\Gamma({\bf F})$ and $\Gamma({\bf E}\hat\otimes_\pi{\bf F})\cong\Gamma({\bf E})\hat\otimes_\pi\Gamma({\bf F})$, where ${\bf G}\mapsto\Gamma({\bf G})$ is the map assigning to a measurable Banach bundle ${\bf G}$ its space of $L^\infty(\mathfrak m)$-sections, while $\Gamma({\bf E})\hat\otimes_\varepsilon\Gamma({\bf F})$ and $\Gamma({\bf E})\hat\otimes_\pi\Gamma({\bf F})$ denote the injective and projective tensor products, respectively, of $\Gamma({\bf E})$ and $\Gamma({\bf F})$ in the sense of $L^\infty(\mathfrak m)$-Banach $L^\infty(\mathfrak m)$-modules. In combination with previous results, this provides a fiberwise representation of the injective tensor product $\mathscr M\hat\otimes_\varepsilon\mathscr N$ and the projective tensor product $\mathscr M\hat\otimes_\pi\mathscr N$ of two countably-generated $L^\infty(\mathfrak m)$-Banach $L^\infty(\mathfrak m)$-modules $\mathscr M$, $\mathscr N$.