Functions not vanishing on trivial Gleason parts of Douglas algebras

Abstract

Let B B denote a closed subalgebra of L ∞ {L^\infty } containing the space of bounded analytic functions. Let M ( B ) M(B) denote the maximal ideal space of B B . Let f f be a function in B B such that f f does not vanish on any Gleason part consisting of a single point. We show that if g g is a function in B B such that | g | ≤ | f |  on  M ( B ) \left | g \right | \leq \left | f \right |{\text { on }}M(B) , then g / f ∈ B g/f \in B .