Pseudo-hyperbolic distance and Gleason parts of the algebra of bounded hyper-analytic functions on the big disk

Abstract

Let $G$ be the compact group of all characters of the additive group of rational numbers, and let $H_G^\infty$ be the Banach algebra of so-called bounded hyper-analytic functions on the big-disk $\Delta_G$. We characterize the pseudo-hyperbolic distance of the algebra $H_G^\infty$ in terms of the pseudo-hyperbolic distance of the algebra $H^\infty$ and establish relationships between Gleason parts in $M(H_G^\infty)$ and $M(H^\infty)$.