Function Theory off the complexified unit circle: Fr\'echet space structure and automorphisms

TL;DR

This paper investigates the structure of holomorphic functions on a complexified domain related to the unit circle, and characterizes automorphisms preserving the Laplacian, contributing to complex analysis and deformation quantization.

Contribution

It analyzes the Fréchet space structure of holomorphic functions on a specific complex domain and characterizes its automorphisms that preserve the Laplacian, linking to deformation quantization.

Findings

Describes the Fréchet space structure of holomorphic functions on the domain.

Characterizes automorphisms leaving the Laplacian invariant.

Connects complex analysis with deformation quantization.

Abstract

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.