  Let $H$ be a compact $p$-adic analytic group without torsion element, whose
Lie algebra is split semisimple and $\mathfrak{N}_H(G)$ be the full subcategory
of the category of finitely generated modules over the Iwasawa algebra
$\Lambda_G$ that are also finitely generated as $\Lambda_H$-modules, where $G =
\mathbb{Z}_{p} \times H$. We show that if the class of a module $N$ in the
Grothendieck group of $\mathfrak{N}_H(G)$ equals to the class of a completely
faithful module, then $q(N)$ is also completely faithful, where $q(N)$ denotes
the image of $N$ via the quotient functor modulo the full subcategory of
pseudonull modules. We also generalize a Theorem of Konstantin Ardakov
characterizing the completely faithful property to the case of more general
$p$-adic Lie groups.
