Fréchet algebraic deformation quantization of the Poincaré disk

Abstract

Abstract. Starting from formal deformation quantization we use an explicit formula for a star product on the Poincaré disk 𝔻 n $\mathbb {D}_n$ to introduce a Fréchet 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 $\mathbb {D}_n$ the resulting Fréchet algebra is shown to have many nice features: it is a strongly nuclear Köthe space, the symmetry group SU ( 1 , n ) $\mathrm {SU}(1, n)$ acts smoothly by continuous automorphisms with an inner infinitesimal action, and evaluation functionals at all points of 𝔻 n $\mathbb {D}_n$ are continuous positive functionals.