LINEAR FUNCTIONALS ON $\mathcal{O}_n$ AND PRODUCT PURE STATES OF UHF

For a sequence $\{{\eta}_m\}_m$ of unit vectors in $\mathbb{C}^n$, we consider the associated linear functional ${\omega}$ on the Cuntz algebra $\mathcal{O}_n$. We show that the restriction ${\omega}{\mid}_{UHF_n}$ is the product pure state of a subalgebra $UHF_n$ of $\mathcal{O}_n$ such that ${\omega}{\mid}_{UHF_n}={\otimes}{\omega}_m$ with ${\omega}_m({\cdot})$ < ${\cdot}{\eta}_m,{\eta}_m$ >. We study product pure states of UHF and obtain a concrete description of them in terms of unit vectors. We also study states of $UHF_n$ which is the restriction of the linear functionals on $O_n$ associated to a fixed unit vector in $\mathbb{C}^n$.