Complemented Copies of $c_0$ in Positive Tensor Products of Banach Lattices

Abstract

A result of Cembranos states that a non-trivial injective tensor product of a $C$-space contains a complemented copy of $c_0$, and in particular fails the Grothendieck property. We establish a positive analogue of the Cembranos theorem for tensor products of Banach lattices. If $E$ and $F$ are infinite dimensional Banach lattices, with $E$ containing $c_0$ and $E^{\ast }$ or $F^{\ast }$ having the bounded positive approximation property, then the Wittstock tensor product $E{\tilde{\otimes }}_{|\epsilon |}F$ contains a complemented copy of $c_0$. If $F$ is in addition reflexive, then $E{\tilde{\otimes }}_{|\epsilon |}F$ fails the positive Grothendieck property.