Fubini's Theorem for Ultraproducts of Noncommutative Lp -Spaces

Junge, Marius

Canadian journal of mathematics, Tome 56 (2004) no. 5, pp. 983-1021

Résumé

Let ${{({{\mathcal{M}}_{i}})}_{i}}{{_{\in }}_{I}},{{({{\mathcal{N}}_{j}})}_{j\in J}}$ be families of von Neumann algebras and $\mathcal{U},\,{\mathcal{U}}'$ be ultrafilters in $I$ , $J$ , respectively. Let $1\,\le \,p\,<\,\infty $ and $n\,\in \,\mathbb{N}$ . Let ${{x}_{1}},...,{{x}_{n}}\,\text{in}\,\prod {{L}_{p}}({{\mathcal{M}}_{i}})$ and ${{y}_{1}},...,{{y}_{n\,}}\text{in }\Pi \,{{L}_{p}}({{\mathcal{N}}_{j}})$ be bounded families. We show the following equality $$\underset{i,\mathcal{U}}{\mathop{\lim }}\,\,\underset{j,{\mathcal{U}}'}{\mathop{\lim }}\,\,{{\left\| \sum\limits_{k=1}^{n}{{{x}_{k}}(i)\,\otimes \,{{y}_{k}}(j)} \right\|}_{{{L}_{p}}({{\mathcal{M}}_{i}}\otimes {{\mathcal{N}}_{j}})}}=\,\underset{j,{\mathcal{U}}'}{\mathop{\lim }}\,\,\underset{i,\mathcal{U}}{\mathop{\lim }}\,\,{{\left\| \sum\limits_{k=1}^{n}{{{x}_{k}}(i)\,\,\otimes \,{{y}_{k}}(j)} \right\|}_{{{L}_{p}}({{\mathcal{M}}_{i}}\otimes {{\mathcal{N}}_{j}})}}$$ For $p\,=\,1$ this Fubini type result is related to the local reflexivity of duals of ${{C}^{*}}$ -algebras. This fails for $P=\infty $ .

Détail

DOI : 10.4153/CJM-2004-045-1

Mots-clés : 46L52, 46B08, 46L07, noncommutative, Lp -spaces, ultraproducts

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