Pure states on Cuntz algebras arising from geometric progressions

TL;DR

This paper studies how to extend pure states from smaller Cuntz algebras to larger ones via geometric progressions, classifies these extensions, and describes their GNS representations.

Contribution

It introduces a new embedding of Cuntz algebras based on geometric progressions and classifies all pure state extensions and their GNS representations.

Findings

Necessary and sufficient condition for uniqueness of extensions

Complete classification of extensions up to unitary equivalence

Decomposition of mixing states into convex hull of pure states

Abstract

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.