A version of Kalton's theorem for the space of regular operators

In this note we extend some recent results in the space of regular operators. In particular, we provide the following Banach lattice version of a classical result of Kalton: Let $E$ be an atomic Banach lattice with an order continuous norm and $F$ a Banach lattice. Then the following are equivalent: (i) $L^r(E,F)$ contains no copy of $\ell_\infty$, \,\, (ii) $L^r(E,F)$ contains no copy of $c_0$, \,\, (iii) $K^r(E,F)$ contains no copy of $c_0$, \,\, (iv) $K^r(E,F)$ is a (projection) band in $L^r(E,F)$, \,\, (v) $K^r(E,F)=L^r(E,F)$.