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A well known result of Ando says that H∞(D) has a unique predual. There have been two natural extensions of this result to non-commutative algebras: Ueda showed that finite maximal subdiagonal algebras have unique preduals. In a second direction, Davidson and Wright showed that free semigroup algebras have unique preduals. In these notes, we explore a different natural generalization of this result: Let A be a finitely connected domain in the plane. We show that H∞(A) has a unique isometric predual. We also prove a couple of theorems about the structure of the unique predual.

Preduals of $H^{\infty }$ of finitely connected domains