  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) $K^r(E,F)=L^r(E,F)$.
