The positivity of linear functionals on Cuntz algebras associated to unit vectors

We study the linear functional $\rho$ on the Cuntz algebra $\mathcal{O}_{n}$ associated to a sequence $\langle \eta _{m} \rangle$ of unit vectors $\eta _{m}$ in $\mathbb{C}^{n}$ that is a generalization of the Cuntz state. We prove that $\rho$ is positive if and only if $\langle \eta _{m} \rangle$ is a constant sequence.