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Commutative Counterexamples to Arveson's Hyperrigidity Conjecture
Abstract
We construct counterexamples to Arveson's hyperrigidity conjecture for operator systems in commutative C*-algebras. In each example, every irreducible representation of the C*-envelope is a boundary representation, but hyperrigidity fails. The first example is the operator system of continuous affine functions on an infinite-dimensional weighted ℓp ball. In the second infinite-dimensional example, we start with commuting operators and construct a nontrivial dilation. The final example comes from a compact truncated epigraph of a strictly convex function and yields a finite-dimensional operator system.
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BibTeX
@misc{bilich2026commutative,
title = {{Commutative Counterexamples to Arveson's Hyperrigidity Conjecture}},
author = {Boris Bilich and Marcel Scherer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26556},
eprint = {2610.26556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26556},
}
RIS
TY - UNPB
TI - Commutative Counterexamples to Arveson's Hyperrigidity Conjecture
AU - Boris Bilich
AU - Marcel Scherer
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26556
UR - https://arxiv.org/abs/2610.26556
AB - We construct counterexamples to Arveson's hyperrigidity conjecture for operator systems in commutative C^*-algebras. In each example, every irreducible representation of the C^*-envelope is a boundary representation, but hyperrigidity fails. The first example is the operator system of continuous affine functions on an infinite-dimensional weighted ℓ^p ball. In the second infinite-dimensional example, we start with commuting operators and construct a nontrivial dilation. The final example comes from a compact truncated epigraph of a strictly convex function and yields a finite-dimensional operator system.
ER -