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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 $\mathcal{O}_2\subset \mathcal{Q}_2$.

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

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