Tensor products of measurable Banach bundles

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 $\mathscr M$, $\mathscr N$.