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# Fubini's Theorem for Ultraproducts \\of Noncommutative $L_p$-Spaces

Published:2004-10-01
Printed: Oct 2004
• Marius Junge
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## Abstract

Let $(\M_i)_{i\in I}$, $(\N_j)_{j\in J}$ be families of von Neumann algebras and $\U$, $\U'$ be ultrafilters in $I$, $J$, respectively. Let $1\le p<\infty$ and $\nen$. Let $x_1$,\dots,$x_n$ in $\prod L_p(\M_i)$ and $y_1$,\dots,$y_n$ in $\prod L_p(\N_j)$ be bounded families. We show the following equality $$\lim_{i,\U} \lim_{j,\U'} \Big\| \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\|_{L_p(\M_i\otimes \N_j)} = \lim_{j,\U'} \lim_{i,\U} \Big\| \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\|_{L_p(\M_i\otimes \N_j)} .$$ For $p=1$ this Fubini type result is related to the local reflexivity of duals of $C^*$-algebras. This fails for $p=\infty$.
 Keywords: noncommutative $L_p$-spaces, ultraproducts
 MSC Classifications: 46L52 - Noncommutative function spaces 46B08 - Ultraproduct techniques in Banach space theory [See also 46M07] 46L07 - Operator spaces and completely bounded maps [See also 47L25]

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