A noncompletely continuous operator on $L\sb{1}(G)$ whose random Fourier transform is in $c\sb{0}(\hat G)$

Let $T$ be a bounded linear operator from ${L_1}(G,\lambda )$ into ${L_1}(\Omega ,\mathcal{F},P)$, where $(G,\lambda )$ is a compact abelian metric group with its Haar measure, and $(\Omega ,\mathcal{F},P)$ is a probability space. Let $({\mu _\omega })$ be the random measure on $G$ associated to $T$; that is, $Tf(\omega ) = \int_G {f(t)d{\mu _\omega }(t)}$ for each $f$ in ${L_1}(G)$.

We show that, unlike the ideals of representable and Kalton operators, there is no subideal $B$ of $\mathcal{M}(G)$ such that $T$ is completely continuous if and only if ${\mu _\omega } \in B$ for almost $\omega$ in $\Omega$. We actually exhibit a noncompletely continuous operator $T$ such that ${\hat \mu _\omega } \in {l_{2 + \varepsilon }}(\hat G)$ for each $\varepsilon > 0$.