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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.11709 |
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| _version_ | 1866915024668196864 |
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| author | Scherer, Marcel |
| author_facet | Scherer, Marcel |
| contents | 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 $$
\{ T\in\mathcal{B}(H);\ T\ \mathrm{normal}\ \mathrm{and}\ σ(T)\subset \mathrm{ex}(K) \}.
$$ Our approach relies on geometric properties of $K$ and generalizes previous results by Brown. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Hyperrigidity Conjecture for compact convex sets in $\mathbb{R}^2$ Scherer, Marcel Functional Analysis Primary 46L05, 46L07, 47B15 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 $$ \{ T\in\mathcal{B}(H);\ T\ \mathrm{normal}\ \mathrm{and}\ σ(T)\subset \mathrm{ex}(K) \}. $$ Our approach relies on geometric properties of $K$ and generalizes previous results by Brown. |
| title | The Hyperrigidity Conjecture for compact convex sets in $\mathbb{R}^2$ |
| topic | Functional Analysis Primary 46L05, 46L07, 47B15 |
| url | https://arxiv.org/abs/2411.11709 |