Permutative representations of the $2$-adic ring $C^*$-algebra

TL;DR

This paper extends the concept of permutative representations to the 2-adic ring C*-algebra, classifies irreducible permutative representations, and explores their extensions from the Cuntz algebra, revealing a rich structure of representations and pure states.

Contribution

It generalizes permutative representations to the 2-adic ring C*-algebra and classifies irreducible permutative representations, establishing their extension properties and connections to pure states.

Findings

Every permutative representation of  is extendable to  with uniqueness.

Irreducible permutative representations of  are classified via those of the Cuntz algebra.

Most pure states of  have the unique pure extension property.

Abstract

The notion of permutative representation is generalized to the $2$-adic ring $C^*$-algebra $\mathcal{Q}_{2}$. Permutative representations of $\mathcal{Q}_2$ are then investigated with a particular focus on the inclusion of the Cuntz algebra $\mathcal{O}_2\subset\mathcal{Q}_2$. Notably, every permutative representation of $\mathcal{O}_2$ is shown to extend automatically to a permutative representation of $\mathcal{Q}_2$ provided that an extension whatever exists. Moreover, all permutative extensions of a given representation of $\mathcal{O}_2$ are proved to be unitarily equivalent to one another. Irreducible permutative representations of $\mathcal{Q}_2$ are classified in terms of irreducible permutative representations of the Cuntz algebra. Apart from the canonical representation of $\mathcal{Q}_2$, every irreducible representation of $\mathcal{Q}_2$ is the unique extension of an irreducible permutative representation of $\mathcal{O}_2$. Furthermore, a permutative representation of $\mathcal{Q}_2$ will decompose into a direct sum of irreducible permutative subrepresentations if and only if it restricts to $\mathcal{O}_2$ as a regular representation in the sense of Bratteli-Jorgensen. As a result, a vast class of pure states of $\mathcal{O}_2$ is shown to enjoy the unique pure extension property with respect to the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$.