Some results on weakly compact positive left multipliers of a certain group algebra

Abstract

In this paper, we first
characterize weakly compact positive left multipliers on
$L_0^\infty({\cal G})^*$, and prove that if $T$ is a weakly
compact left multiplier on $L_0^\infty({\cal G})^*$, then
$T=T^+-T^-$ for some weakly compact positive left multipliers
$T^+,T^-$ on $L_0^\infty({\cal G})^*$ if and only if the range of
$T$ contains in $L^1({\cal G})$. We then find conditions under
which the existence of a non-zero weakly compact left multipliers
on ideals of $L_0^\infty({\cal G})^*$ is equivalent to compactness
of ${\cal G}$.