$K_0$-invariance of the completely faithful property of Iwasawa modules

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 ${\mathrm{\Lambda }}_G$ that are also finitely generated as ${\mathrm{\Lambda }}_H$-modules, where $G = $\mathbb{Z}$_{p} \times H$\mathrm{.}Weshowthatiftheclassofamodule$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.