Pure states on Cuntz algebras arising from geometric progressions

Let ${\cal O}_n$ denote the Cuntz algebra for $n\geq 2$. We introduce an embedding $f$ of ${\cal O}_m$ into ${\cal O}_n$ arising from a geometric progression of Cuntz generaters of ${\cal O}_n$. By identifying ${\cal O}_m$ with $f({\cal O}_m)$, we extend Cuntz states on ${\cal O}_m$ to ${\cal O}_n$. We show (i) a necessary and sufficient condition of the uniqueness of the extension, (ii) the complete classification of all such extensions up to unitary equivalence of their GNS representations, and (iii) the decomposition formula of a mixing state into a convex hull of pure states. The complete set of invariants of all GNS representations by such pure states is given as a certain set of complex unit vectors.