Boundary representations from constrained interpolation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913994476879872 |
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| author | Ayun, Gal Ben Shamovich, Eli |
| author_facet | Ayun, Gal Ben Shamovich, Eli |
| contents | In this paper, we study $C^*$-envelopes of finite-dimensional operator algebras arising from constrained interpolation problems on the unit disc. In particular, we consider interpolation problems for the algebra $H^\infty_{\text{node}}$ that consists of bounded analytic functions on the unit disk that satisfy $ f(0) = f(λ)$ for some $0 \neq λ\in \mathbb{D}$. We show that there exist choices of four interpolation nodes that exclude both $0$ and $λ$, such that if $I$ is the ideal of functions that vanish at the interpolation nodes, then $C^*_e(H^\infty_{\text{node}}/I)$ is infinite-dimensional. This differs markedly from the behavior of the algebra corresponding to interpolation nodes that contain the constrained points studied in the literature. Additionally, we use the distance formula to provide a completely isometric embedding of $C^*_e(H^\infty_{\text{node}}/I)$ for any choice of $n$ interpolation nodes that do not contain the constrained points into $M_n(G^2_{nc})$, where $G^2_{nc}$ is Brown's noncommutative Grassmannian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11027 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary representations from constrained interpolation Ayun, Gal Ben Shamovich, Eli Operator Algebras Complex Variables In this paper, we study $C^*$-envelopes of finite-dimensional operator algebras arising from constrained interpolation problems on the unit disc. In particular, we consider interpolation problems for the algebra $H^\infty_{\text{node}}$ that consists of bounded analytic functions on the unit disk that satisfy $ f(0) = f(λ)$ for some $0 \neq λ\in \mathbb{D}$. We show that there exist choices of four interpolation nodes that exclude both $0$ and $λ$, such that if $I$ is the ideal of functions that vanish at the interpolation nodes, then $C^*_e(H^\infty_{\text{node}}/I)$ is infinite-dimensional. This differs markedly from the behavior of the algebra corresponding to interpolation nodes that contain the constrained points studied in the literature. Additionally, we use the distance formula to provide a completely isometric embedding of $C^*_e(H^\infty_{\text{node}}/I)$ for any choice of $n$ interpolation nodes that do not contain the constrained points into $M_n(G^2_{nc})$, where $G^2_{nc}$ is Brown's noncommutative Grassmannian. |
| title | Boundary representations from constrained interpolation |
| topic | Operator Algebras Complex Variables |
| url | https://arxiv.org/abs/2501.11027 |