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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.14340 |
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| _version_ | 1866917088773275648 |
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| author | Bhattacharya, Saptak |
| author_facet | Bhattacharya, Saptak |
| contents | Given $n\in\mathbb{N}$ any point on the closed unit disk $\overline{\mathbb{D}}$ can be written as the average of $n$ points on the unit circle $\mathbb{S}^1$. Here we discuss a non-commutative version of this result. We prove that for any Hilbert space $\mathcal{H}$ and a state $ϕ:B(\mathcal{H})\to\mathbb{C}$, $\{ϕ(U): U\,\mathrm{ unitary}\}=\overline{\mathbb{D}}$. We also show that if $\dim$ $\mathcal{H}$ is finite, for any $w\in\overline{\mathbb{D}}$ we can choose a unitary $U$ with atmost $3$ distinct eigenvalues such that $ϕ(U)=w$. Lastly, we prove the divisibility property for any state $ϕ$ on $B(\mathcal{H})$ where $\mathcal{H}$ is infinite-dimensional, showing that $\{ϕ(P) : P^*=P^2=P\}=[0,1]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14340 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some non-commutative averaging theorems Bhattacharya, Saptak Functional Analysis 15A60, 47L05 Given $n\in\mathbb{N}$ any point on the closed unit disk $\overline{\mathbb{D}}$ can be written as the average of $n$ points on the unit circle $\mathbb{S}^1$. Here we discuss a non-commutative version of this result. We prove that for any Hilbert space $\mathcal{H}$ and a state $ϕ:B(\mathcal{H})\to\mathbb{C}$, $\{ϕ(U): U\,\mathrm{ unitary}\}=\overline{\mathbb{D}}$. We also show that if $\dim$ $\mathcal{H}$ is finite, for any $w\in\overline{\mathbb{D}}$ we can choose a unitary $U$ with atmost $3$ distinct eigenvalues such that $ϕ(U)=w$. Lastly, we prove the divisibility property for any state $ϕ$ on $B(\mathcal{H})$ where $\mathcal{H}$ is infinite-dimensional, showing that $\{ϕ(P) : P^*=P^2=P\}=[0,1]$. |
| title | Some non-commutative averaging theorems |
| topic | Functional Analysis 15A60, 47L05 |
| url | https://arxiv.org/abs/2511.14340 |