#### 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