# Journal of Operator Theory

Volume 82, Issue 1, Summer 2019 pp. 197-236.

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

**Authors**:
Valeriano Aiello (1), Roberto Conti (2), Stefano Rossi
(3)

**Author institution:**(1) Department of Mathematics, Vanderbilt University,
1362 Stevenson Center, Nashville, TN 37240,
U.S.A.

(2) Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza
Universit\`{a} di Roma, Via A. Scarpa 16, I-00161 Roma, Italy

(3) Dipartimento di Matematica, Universit\`{a} di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy

**Summary: **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 \textit{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 $\mathcal{O}_2\subset \mathcal{Q}_2$.

**DOI: **http://dx.doi.org/10.7900/jot.20181apr19.2188

**Keywords: **$2$-adic ring $C^*$-algebra, $C^*$-algebras, dyadic integers, permutative representations, Cuntz algebras

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