Function theory off the complexified unit circle: Möbius-invariant differential operators, strict deformation quantization, and spectral synthesis

We study the Fr\'echet space structure of $\mathcal{H}(\Omega)$, the space of holomorphic functions on $\Omega=\hat{\mathbb{C}}^2 \setminus \{(z,w) \in \hat{\mathbb{C}}^2 \, : \, zw \not=1\}$, the complement of the ``complexified unit circle'' $\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, zw =1\}$. This offers a unified framework for investigating conformally invariant differential operators on the unit disk $\mathbb{D}$ and the Riemann sphere~$\hat{\mathbb{C}}$, which have been studied by Peschl, Aharonov, Minda and many others, within their conjecturally natural habitat. We apply this machinery to a problem in deformation quantization by deriving an explicit formula for the canonical Wick--type star product $\star_{\hbar}$ for all smooth functions defined on the unit disk $\mathbb{D}$ belonging to the observable algebra
$\mathcal{A}(\mathbb{D}):=\{f(z,\overline{z}) \, : \, f\in \mathcal{H}(\Omega)\}$.
This formula is given in form of a factorials series which depends holomorphically on a complex deformation parameter $\hbar$ and easily leads to an asymptotic expansion of the star product $\star_{\hbar}$ in powers of $\hbar$. As another application we give a function--theoretic Runge--type characterization of the ``exceptional'' eigenspaces of the invariant Laplacian on the unit disk, which have been introduced by Helgason and Rudin in the 70s and 80s. Finally, we discuss a recent result of A.~Moucha, who showed that $\mathcal{H}(\Omega)$ and hence the observable algebra $\mathcal{A}(\mathbb D)$ admit spectral synthesis by exhibiting~a Schauder basis of eigenfunctions of the invariant Laplacian on $\Omega$ and the unit disk $\mathbb D$, respectively.

This talk is based on collobarations with M. Heins, D. Kraus, A. Moucha, S. Schleißinger, M. Schötz, T. Sugawa and S. Waldmann.