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arXiv · 2610.26556

Commutative Counterexamples to Arveson's Hyperrigidity Conjecture

Boris Bilich·Marcel Scherer
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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 $\ell^p$ 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.

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Boris Bilich, Marcel Scherer. 2026-10-19. Commutative Counterexamples to Arveson's Hyperrigidity Conjecture. https://arxiv.org/abs/2610.26556

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