Inner functions and the maximal ideal space of $H^{\infty }(U^{n})$

Abstract:

For the case of the polydisc, Range has shown that the Shilov boundary ${\partial _n}$ of ${H^\infty }({U^n})$ is a proper subset of $\tau {X_n}$, the set of all restrictions of complex homomorphisms of ${L^\infty }({T^n})$ to ${H^\infty }({U^n})$. In this paper, we show that $\tau {X_n}$ is a proper subset of those complex homomorphisms of ${H^\infty }({U^n})$ which are unimodular on the class of all inner functions.