Linear functionals on the Cuntz algebra

For a pure state $p'$ on $\mathcal{O}_n$, which is an extension of a pure state $p$ on $\mathrm{UHF}_n$ with the property that if $(\mathcal{H}_{p'},\pi_{p'},\omega_{p'})$is a corresponding representation, then $\pi_{p'}(\mathrm{UHF}_n)=B(\mathcal{H}_{p'})$, $p'$ induces a unital shift of $B(\mathcal{H})$ of the Powers index $n$. We describe states $p$ on $\mathrm{UHF}_n$ by using sequences of unit vectors in $\mathbb{C} ^n$. We study the linear functionals on the Cuntz algebra $\mathcal{O}_n$ whose restrictions are the product pure state on $\mathrm{UHF}_n$. We find conditions on the sequence of unit vectors for which the corresponding linear functionals on $\mathcal{O}_n$become states under these conditions.