Countably generated Douglas algebras

Abstract

Under a certain assumption of f f and g g in L ∞ {L^\infty } which is considered by Sarason, a strong separation theorem is proved. This is available to study a Douglas algebra [ H ∞ , f ] [{H^\infty },\,f] generated by H ∞ {H^\infty } and f f . It is proved that (1) ball ( B / H ∞ + C ) (B/{H^\infty } + C) does not have exposed points for every Douglas algebra B B , (2) Sarason’s three functions problem is solved affirmatively, (3) some characterization of f f for which [ H ∞ , f ] [{H^\infty },\,f] is singly generated, and (4) the M M -ideal conjecture for Douglas algebras is not true.