Boundaries for algebras of analytic functions on function module Banach spaces

We consider the uniform algebra A∞(BX) of continuous and bounded functions that are analytic on the interior of the closed unit ball BX of a complex Banach function module X. We focus on norming subsets of BX, i.e., boundaries, for such algebra. In particular, if X is a dual complex Banach space whose centralizer is infinite‐dimensional, then the intersection of all closed boundaries is empty. This also holds in case that X is an ℓ∞‐sum of infinitely many Banach spaces and further, the torus is a boundary.