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arXiv · 2411.11709

The Hyperrigidity Conjecture for compact convex sets in $\mathbb{R}^2$

Marcel Scherer
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Abstract

We prove that for every compact, convex subset $K\subset\mathbb{R}^2$ the operator system $A(K)$, consisting of all continuous affine functions on $K$, is hyperrigid in the C*-algebra $C(\mathrm{ex}(K))$. In particular, this result implies that the weak and strong operator topologies coincide on the set $$ \{ T\in\mathcal{B}(H);\ T\ \mathrm{normal}\ \mathrm{and}\ \sigma(T)\subset \mathrm{ex}(K) \}. $$ Our approach relies on geometric properties of $K$ and generalizes previous results by Brown.

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Marcel Scherer. 2024-11-18. The Hyperrigidity Conjecture for compact convex sets in $\mathbb{R}^2$. https://arxiv.org/abs/2411.11709

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