标题： 可测巴拿赫丛的张量积 显示英文标题

标题： Tensor products of measurable Banach bundles

摘要： 我们研究可测巴拿赫丛的单射和投射张量积。 更准确地说，给定两个在概率空间$({\rm X},\Sigma,\mathfrak m)$上定义的可分可测巴拿赫丛${\bf E}$和${\bf F}$，我们构造两个在$({\rm X},\Sigma,\mathfrak m)$上的可测巴拿赫丛${\bf E}\hat\otimes_\varepsilon{\bf F}$和${\bf E}\hat\otimes_\pi{\bf F}$，使得$\Gamma({\bf E}\hat\otimes_\varepsilon{\bf F})\cong\Gamma({\bf E})\hat\otimes_\varepsilon\Gamma({\bf F})$和$\Gamma({\bf E}\hat\otimes_\pi{\bf F})\cong\Gamma({\bf E})\hat\otimes_\pi\Gamma({\bf F})$成立，其中${\bf G}\mapsto\Gamma({\bf G})$是将一个可测巴拿赫丛${\bf G}$映射到其$L^\infty(\mathfrak m)$-截面空间的映射，而$\Gamma({\bf E})\hat\otimes_\varepsilon\Gamma({\bf F})$和$\Gamma({\bf E})\hat\otimes_\pi\Gamma({\bf F})$分别表示在$L^\infty(\mathfrak m)$-巴拿赫$L^\infty(\mathfrak m)$-模的意义下，$\Gamma({\bf E})$和$\Gamma({\bf F})$的内射和张量积。 结合之前的结果，这提供了两个可数生成的$L^\infty(\mathfrak m)$-巴拿赫$L^\infty(\mathfrak m)$-模$\mathscr M$,$\mathscr N$的单射张量积$\mathscr M\hat\otimes_\varepsilon\mathscr N$和射影张量积$\mathscr M\hat\otimes_\pi\mathscr N$的纤维表示。 显示英文摘要

摘要： 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 $\mathscr M$, $\mathscr N$.