Fr\'echet algebraic deformation quantization of the Poincar\'e disk

TL;DR

This paper develops a Fréchet algebraic deformation quantization of the Poincaré disk, introducing a continuous star product topology and analyzing its properties, with applications to group algebras and infinite matrices.

Contribution

It provides an explicit construction of a Fréchet topology for deformation quantization on the Poincaré disk, ensuring continuity of the star product and exploring its algebraic and topological features.

Findings

The star product on D_n forms a strongly nuclear K"othe space.

The symmetry group SU(1, n) acts smoothly by automorphisms.

Evaluation functionals are continuous positive functionals.

Abstract

Starting from formal deformation quantization we use an explicit formula for a star product on the Poincar\'e disk D_n to introduce a Fr\'echet topology making the star product continuous. To this end a general construction of locally convex topologies on algebras with countable vector space basis is introduced and applied. Several examples of independent interest are investigated as e.g. group algebras over finitely generated groups and infinite matrices. In the case of the star product on D_n the resulting Fr\'echet algebra is shown to have many nice features: it is a strongly nuclear K\"othe space, the symmetry group SU(1, n) acts smoothly by continuous automorphisms with an inner infinitesimal action, and evaluation functionals at all points of D_n are continuous positive functionals.