The isomorphism problem for analytic discs with self-crossings on the boundary
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914984062091264 |
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| author | Mironov, Mikhail |
| author_facet | Mironov, Mikhail |
| contents | Suppose $V$ is the unit disc $\mathbb{D}$ embedded in the $d$-dimensional unit ball $\mathbb{B}_d$ and attached to the unit sphere. Consider the space $\mathcal{H}_V$, the restriction of the Drury-Arveson space to the variety $V$, and its multiplier algebra $\mathcal{M}_V = \operatorname{Mult}(\mathcal{H}_V)$. The isomorphism problem is the following: Is $V_1 \cong V_2$ equivalent to $\mathcal{M}_{V_1} \cong \mathcal{M}_{V_2}$?
A theorem of Alpay, Putinar and Vinnikov states that for $V$ without self-crossings on the boundary $\mathcal{M}_V$ is the space of bounded analytic functions on $V$. We consider what happens when there are self-crossings on the boundary and prove that if $\mathcal{M}_{V_1} \cong \mathcal{M}_{V_2}$ algebraically, then $V_1$ and $V_2$ must have the same self-crossings up to a unit disc automorphism. We prove that an isomorphism between $\mathcal{M}_{V_1}$ and $\mathcal{M}_{V_2}$ can only be given by a composition with a map from $V_1$ to $V_2$. In the case of a single simple self-crossing we show that there are only two possible candidates for this map and find these candidates. Finally, we provide a continuum of $V$'s with the same self-crossing pattern such that their multiplier algebras are all mutually non-isomorphic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_16966 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The isomorphism problem for analytic discs with self-crossings on the boundary Mironov, Mikhail Functional Analysis Complex Variables Suppose $V$ is the unit disc $\mathbb{D}$ embedded in the $d$-dimensional unit ball $\mathbb{B}_d$ and attached to the unit sphere. Consider the space $\mathcal{H}_V$, the restriction of the Drury-Arveson space to the variety $V$, and its multiplier algebra $\mathcal{M}_V = \operatorname{Mult}(\mathcal{H}_V)$. The isomorphism problem is the following: Is $V_1 \cong V_2$ equivalent to $\mathcal{M}_{V_1} \cong \mathcal{M}_{V_2}$? A theorem of Alpay, Putinar and Vinnikov states that for $V$ without self-crossings on the boundary $\mathcal{M}_V$ is the space of bounded analytic functions on $V$. We consider what happens when there are self-crossings on the boundary and prove that if $\mathcal{M}_{V_1} \cong \mathcal{M}_{V_2}$ algebraically, then $V_1$ and $V_2$ must have the same self-crossings up to a unit disc automorphism. We prove that an isomorphism between $\mathcal{M}_{V_1}$ and $\mathcal{M}_{V_2}$ can only be given by a composition with a map from $V_1$ to $V_2$. In the case of a single simple self-crossing we show that there are only two possible candidates for this map and find these candidates. Finally, we provide a continuum of $V$'s with the same self-crossing pattern such that their multiplier algebras are all mutually non-isomorphic. |
| title | The isomorphism problem for analytic discs with self-crossings on the boundary |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2410.16966 |