Interpolating Sequences For Dual Uniform Algebras

Given a dual uniform algebra $A=X^*$ with maximal ideal space $M_A$, we provide the first sufficient condition in terms of the Gleason distance of $A$ for a sequence in $M_A\cap X$ to be interpolating for $A$. We prove that a sequence in $\mathbb{D}^N$ is uniformly separated if and only if it is interpolating for $H^\infty(\mathbb{D}^N)$ and its sequence of norms satisfies the Blaschke condition, and then use this characterization to classify the interpolating sequences for $H^\infty(\mathbb{D}^N)$ in terms of its Gleason distance. We also study interpolating sequences for $\mathscr{H}^\infty$, the algebra of bounded Dirichlet series, obtaining necessary and sufficient conditions for a sequence in $\mathbb{C}+$ to be interpolating for this space, and relating the geometry of such sequences to that of the interpolating sequences for $H^\infty(\mathbb{C}+)$. Finally, we show that a sequence in the Shilov boundary of the second dual of a uniform algebra $A$ is interpolating for $A^{**}$ if and only if it is discrete for the $w^*$-topology.