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 $α$ satis