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The isomorphism problem for analytic discs with self-crossings on the boundary

Mikhail Mironov

Abstract

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.

The isomorphism problem for analytic discs with self-crossings on the boundary

Abstract

Suppose is the unit disc embedded in the -dimensional unit ball and attached to the unit sphere. Consider the space , the restriction of the Drury-Arveson space to the variety , and its multiplier algebra . The isomorphism problem is the following: Is equivalent to ? A theorem of Alpay, Putinar and Vinnikov states that for without self-crossings on the boundary is the space of bounded analytic functions on . We consider what happens when there are self-crossings on the boundary and prove that if algebraically, then and must have the same self-crossings up to a unit disc automorphism. We prove that an isomorphism between and can only be given by a composition with a map from to . 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 's with the same self-crossing pattern such that their multiplier algebras are all mutually non-isomorphic.

Paper Structure

This paper contains 16 sections, 28 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. The isomorphism problem
  3. Complete Pick spaces
  4. Analytic Discs
  5. First order equivalence and examples
  6. Organization of the paper
  7. Analytic Discs
  8. Properties of functions in Hf and Mf
  9. Proof of Theorem \ref{['thrm: algebras to crossings']}
  10. Proof of Theorem \ref{['thrm: isomorphic algebras to automorphism']}
  11. First order equivalence and examples
  12. Proof of Theorem \ref{['thrm: multipliers to bilip']}
  13. Proof of Theorem \ref{['thrm: multiplier isomorphism two points to two auto']}
  14. Proof of Theorem \ref{['thrm: f_r and f_s']}
  15. Proof of Theorem \ref{['thrm: f_r-r and f_s-s']}
  16. ...and 1 more sections

Key Result

Theorem A

Let $\mathcal{H}$ be a separable irreducible reproducing kernel Hilbert space on $X$. Then $\mathcal{H}$ is complete Pick if and only if for some $d \in \mathbb{N} \cup \{ \infty \}$ there is a map $b: X \to \mathbb{B}_d$ and a nowhere vanishing function $\delta: X \to \mathbb{C}$ such that Meaning that up to rescaling and composition $\mathcal{H}$ is essentially $\mathcal{H}_V$ for some variety

Theorems & Definitions (46)

  • Definition 1
  • Theorem A
  • Definition 2
  • Theorem B
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Theorem 3
  • Corollary 3
  • ...and 36 more