Extensions of Operator Systems, Hyperrigidity and Quantum Double Suspension
Abstract
We introduce a class of extensions of unital operator systems satisfying a certain Toeplitz type property and show that these extensions induce corresponding extensions of the C*-algebras they generate. We prove that an extension in this class is hyperrigid in the C*-algebra it generates if and only if both operator systems from which it is constructed are hyperrigid in the C*-algebras they generate. We then introduce the quantum double suspension of operator systems as a special case of this construction and show that its iterates yield the Hong-Szymański quantum double suspensions at the C*-algebra level, including the Vaksman-Soibelman quantum spheres. In particular, we construct hyperrigid operator systems that generate the C*-algebras of these quantum spheres. As a further application, we show that every counterexample to Arveson's hyperrigidity conjecture gives rise to a sequence of counterexamples. We identify the Šilov ideal and the C*-envelope of these extensions and derive a K-theoretic condition under which each such sequence contains infinitely many pairwise non-isomorphic counterexamples. In particular, Scherer's counterexample gives rise to infinitely many pairwise non-isomorphic counterexamples. Finally, we compute the propagation number, an invariant of operator systems introduced by Connes and van Suijlekom, for this class of extensions.
Related
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BibTeX
@misc{bhatt2026extensions,
title = {{Extensions of Operator Systems, Hyperrigidity and Quantum Double Suspension}},
author = {Vibhor Bhatt and Satyajit Guin and Palash Sharma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18129},
eprint = {2610.18129},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18129},
}
RIS
TY - UNPB
TI - Extensions of Operator Systems, Hyperrigidity and Quantum Double Suspension
AU - Vibhor Bhatt
AU - Satyajit Guin
AU - Palash Sharma
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.18129
UR - https://arxiv.org/abs/2610.18129
AB - We introduce a class of extensions of unital operator systems satisfying a certain Toeplitz type property and show that these extensions induce corresponding extensions of the C^*-algebras they generate. We prove that an extension in this class is hyperrigid in the C^*-algebra it generates if and only if both operator systems from which it is constructed are hyperrigid in the C^*-algebras they generate. We then introduce the quantum double suspension of operator systems as a special case of this construction and show that its iterates yield the Hong-Szymański quantum double suspensions at the C^*-algebra level, including the Vaksman-Soibelman quantum spheres. In particular, we construct hyperrigid operator systems that generate the C^*-algebras of these quantum spheres. As a further application, we show that every counterexample to Arveson's hyperrigidity conjecture gives rise to a sequence of counterexamples. We identify the Šilov ideal and the C^*-envelope of these extensions and derive a K-theoretic condition under which each such sequence contains infinitely many pairwise non-isomorphic counterexamples. In particular, Scherer's counterexample gives rise to infinitely many pairwise non-isomorphic counterexamples. Finally, we compute the propagation number, an invariant of operator systems introduced by Connes and van Suijlekom, for this class of extensions.
ER -