Function Theory off the complexified unit circle: Fréchet space structure and automorphisms

Motivated by recent work on strict deformation quantization of the unit disk and the Riemann sphere, we study the Fréchet space structure of the set of holomorphic functions on the complement $\mathrm{\Omega }:=\{(z,w)\in {\widehat{\mathbb{C}}}^2:z\cdot w≠1\}$ of the complexified unit circle $\{(z,w)\in {\widehat{\mathbb{C}}}^2:z\cdot w=1\}$. We also characterize the subgroup of all biholomorphic automorphisms of $\mathrm{\Omega }$ which leave the canonical Laplacian on $\mathrm{\Omega }$ invariant.