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Representations of the infinite Cuntz algebra
Abstract
We define an intrinsic unital completely positive (ucp) map from the Cuntz algebra to the Pythagorean with finitely many generators. This permits to lift states and cyclic representations. This process restricts to the Cuntz-Toeplitz algebras and the contraction algebras. Taking inductive limits we obtain an effective machinery for constructing states on the infinite Cuntz algebra 𝒪∞. Combining this approach with previous techniques of the first author we obtain large families of pairwise inequivalent irreducible representations of 𝒪∞: one family being in an obvious bijection with the unit sphere of Hilbert's space.
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BibTeX
@misc{brothier2026representations,
title = {{Representations of the infinite Cuntz algebra}},
author = {Arnaud Brothier and David. E. Evans},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08094},
eprint = {2610.08094},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08094},
}
RIS
TY - UNPB
TI - Representations of the infinite Cuntz algebra
AU - Arnaud Brothier
AU - David. E. Evans
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.08094
UR - https://arxiv.org/abs/2610.08094
AB - We define an intrinsic unital completely positive (ucp) map from the Cuntz algebra to the Pythagorean with finitely many generators. This permits to lift states and cyclic representations. This process restricts to the Cuntz-Toeplitz algebras and the contraction algebras. Taking inductive limits we obtain an effective machinery for constructing states on the infinite Cuntz algebra 𝒪_∞. Combining this approach with previous techniques of the first author we obtain large families of pairwise inequivalent irreducible representations of 𝒪_∞: one family being in an obvious bijection with the unit sphere of Hilbert's space.
ER -