  Motivated by recent work on strict deformation quantization of the unit disk
and the Riemann sphere, we study the Fr\'echet space structure of the set of
holomorphic functions on the complement $\Omega:=\{(z,w)\in
\hat{\mathbb{C}}^2\, :\, z\cdot w\not=1\}$ of the complexified unit circle
${\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w=1\}}$. We also characterize
the subgroup of all biholomorphic automorphisms of $\Omega$ which leave the
canonical Laplacian on $\Omega$ invariant.
