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

DOI: 10.2140/pjm.2019.301.67

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 $L_p\left[0,1\right]{\hat{\otimes }}_{\epsilon }X$ 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 $L_p\left[0,1\right]{\hat{\otimes }}_{\pi }X$ 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