Relative norm averaging property and selflessness of almost periodic actions

Let $D$ be a unital separable $C^*$-algebra with a faithful state $σ$ and $G$ be a discrete group. We introduce a property called relative norm averaging property (RNA) for almost periodic $σ$-preserving actions by $G$ on $(D,σ)$. We prove that if an almost periodic $σ$-preserving action $α$ satisfies that $\overline{α(G)} \subseteq \text{Aut}(D,σ)$ is connected and nontrivial, then $α$ satisfies (RNA). As an application, we prove that if $G$ has property PHP of Ozawa and $α:G\curvearrowright (D,σ)$ is an ergodic (i.e. $D^G=\mathbb{C}$) almost periodic $σ$-preserving action with $\overline{α(G)}$ being connected, then $D\rtimes_r G$ is selfless. This includes certain paradoxical isometric actions of $\mathbb{F}_2$ on the sphere. In non-ergodic case, we prove that the relative norm averaging property implies that $D\rtimes_r G$ satisfies the $D^G$-valued selfless, if $G$ satisfies PHP.