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 $\mathbf{E}$, $\mathbf{F}$ defined over a probability space $(\mathrm{X},\Sigma ,\mathfrak{m})$, we construct two measurable Banach bundles $\mathbf{E}{\hat{\otimes }}_{\epsilon }\mathbf{F}$ and $\mathbf{E}{\hat{\otimes }}_{\pi }\mathbf{F}$ over $(\mathrm{X},\Sigma ,\mathfrak{m})$ such that $\Gamma (\mathbf{E}{\hat{\otimes }}_{\epsilon }\mathbf{F})\cong \Gamma (\mathbf{E}){\hat{\otimes }}_{\epsilon }\Gamma (\mathbf{F})$ and $\Gamma (\mathbf{E}{\hat{\otimes }}_{\pi }\mathbf{F})\cong \Gamma (\mathbf{E}){\hat{\otimes }}_{\pi }\Gamma (\mathbf{F})$, where $\mathbf{G}\mapsto \Gamma (\mathbf{G})$ is the map assigning to a measurable Banach bundle $\mathbf{G}$ its space of $L^{\infty }(\mathfrak{m})$-sections, while $\Gamma (\mathbf{E}){\hat{\otimes }}_{\epsilon }\Gamma (\mathbf{F})$ and $\Gamma (\mathbf{E}){\hat{\otimes }}_{\pi }\Gamma (\mathbf{F})$ denote the injective and projective tensor products, respectively, of $\Gamma (\mathbf{E})$ and $\Gamma (\mathbf{F})$ 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 $\mathcal{M}{\hat{\otimes }}_{\epsilon }\mathcal{N}$ and the projective tensor product $\mathcal{M}{\hat{\otimes }}_{\pi }\mathcal{N}$ of two countably-generated $L^{\infty }(\mathfrak{m})$-Banach $L^{\infty }(\mathfrak{m})$-modules $\mathcal{M}$, $\mathcal{N}$.

30 pages