#### Vol. 301, No. 1, 2019

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Complemented copies of $c_0(\tau)$ in tensor products of $L_p[0,1]$

### Vinícius Morelli Cortes, Elói Medina Galego and Christian Samuel

Vol. 301 (2019), No. 1, 67–88
##### Abstract

Let $X$ be a Banach space and $\tau$ an infinite cardinal. We show that if $\tau$ has uncountable cofinality, $p\in \left[1,\infty \right)$, and either the Lebesgue–Bochner space ${L}_{p}\left(\left[0,1\right],X\right)$ or the injective tensor product ${L}_{p}\left[0,1\right]{\stackrel{̂}{\otimes }}_{\epsilon }X$ contains a complemented copy of ${c}_{0}\left(\tau \right)$, then so does $X$. We show also that if $p\in \left(1,\infty \right)$ and the projective tensor product ${L}_{p}\left[0,1\right]{\stackrel{̂}{\otimes }}_{\pi }X$ contains a complemented copy of ${c}_{0}\left(\tau \right)$, then so does $X$.

##### Keywords
$c_0(\Gamma)$ spaces, injective tensor products, projective tensor products, Lebesgue–Bochner spaces $L_p([0,1],X)$, complemented subspaces
##### Mathematical Subject Classification 2010
Primary: 46B03, 46E15
Secondary: 46B25, 46E30, 46E40
##### Milestones
Received: 21 April 2018
Revised: 8 October 2018
Accepted: 18 October 2018
Published: 16 September 2019
##### Authors
 Vinícius Morelli Cortes Department of Mathematics University of São Paulo São Paulo Brazil Elói Medina Galego Department of Mathematics, IME University of São Paulo São Paulo Brazil Christian Samuel Aix Marseille Université, CNRS Marseille France