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Rational Cuntz states peak on the free disk algebra

Robert T. W. Martin, Eli Shamovich

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 Clouâtre and Thompson.

Rational Cuntz states peak on the free disk algebra

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 Clouâtre and Thompson.
Paper Structure (6 sections, 16 theorems, 56 equations)

This paper contains 6 sections, 16 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Non-commutative Hardy space and its multipliers
  4. Non-commutative rational functions
  5. Peaking states and representations
  6. Main result

Key Result

Lemma 2.1

Let $\mathfrak{r} \in \mathbb{C} _0 \ \mathclap{\, <}{\left( \right.} \, \mathfrak{z} \, \mathclap{ \, \, \, \, >}{\left. \right)}$. If $(A , B , C , D )$ is a minimal FM realization of $\mathfrak{r}$, then

Theorems & Definitions (30)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition
  • proof
  • Lemma 2.4
  • proof
  • ...and 20 more