The isomorphism problem for analytic discs with self-crossings on the boundary
Streszczenie
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 $\mathcal M_V = {\rm Mult}({\mathcal H}_V)$. 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.
