Holomorphic Banach Vector Bundles on the Maximal Ideal Space of H and the Operator Corona Problem of Sz.-nagy

We establish triviality of some holomorphic Banach vector bundles on the maximal ideal space M(H∞) of the Banach algebra H∞ of bounded holomorphic functions on the unit disk D ⊂ C with pointwise multiplication and supremum norm. We apply the result to the study of the Sz.-Nagy operator corona problem. 1. Formulation of Main Results 1.1. We continue our study started in [Br2] of analytic objects on the maximal ideal space M(H) of the Banach algebra H of bounded holomorphic functions on the unit disk D ⊂ C with pointwise multiplication and supremum norm. The present paper deals with holomorphic Banach vector bundles defined on M(H) and the operator corona problem posed by Sz.-Nagy. Recall that for a commutative unital complex Banach algebra A with dual space A the maximal ideal space M(A) of A is the set of nonzero homomorphisms A → C equipped with the Gelfand topology, the weak topology induced by A. It is a compact Hausdorff space contained in the unit ball of A. In the case of H evaluation at a point of D is an element of M(H), so D is naturally embedded into M(H) as an open subset. The famous Carleson corona theorem [C] asserts that D is dense in M(H). Let U ⊂ M(H) be an open subset and X be a complex Banach space. A continuous function f ∈ C(U ;X) is said to beX-valued holomorphic if its restriction to U ∩ D is X-valued holomorphic in the usual sense. By O(U ;X) we denote the vector space of X-valued holomorphic functions on U . Let E be a continuous Banach vector bundle on M(H) with fibre X defined on an open cover U = (Ui)i∈I of M(H ) by a cocycle {gij ∈ C(Ui ∩ Uj ;GL(X))}; here GL(X) is the group of invertible elements of the Banach algebra L(X) of bounded linear operators on X equipped with the operator norm. We say that E is holomorphic if all gij ∈ O(Ui ∩ Uj ;GL(X)). In this case E|D is a holomorphic Banach vector bundle on D in the usual sense. Recall that E is defined as the quotient space of the disjoint union ⊔i∈IUi ×X by the equivalence relation: Uj ×X ∋ u× x ∼ u× gij(u)x ∈ Ui ×X. The projection p : E → X is induced by natural projections Ui ×X → Ui, i ∈ I. 2000 Mathematics Subject Classification. Primary 30D55. Secondary 30H05.