Rational Cuntz states peak on the free disk algebra

TL;DR

This paper explores the convex structure of states on the Cuntz algebra using non-commutative realization theory, revealing that many Cuntz states are peak states for the free disk algebra.

Contribution

It introduces a novel application of realization theory to analyze Cuntz states and identifies a broad class of these states as peak states in the free disk algebra.

Findings

Many Cuntz states are peak states for the free disk algebra.

Non-commutative Clark measures relate to isometric NC rational multipliers.

Application of realization theory provides new insights into the convex structure of states.

Abstract

We apply realization theory of non-commutative rational multipliers of the Fock space, or free Hardy space of square--summable power series in several non-commuting variables to the convex analysis of states on the Cuntz algebra. We show, in particular, that a large class of Cuntz states which arise as the `non-commutative Clark measures' of isometric NC rational multipliers are peak states for Popescu's free disk algebra in the sense of Clou\^atre and Thompson.