On the completion problem for algebra H∞

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On the completion problem for algebra H ∞ ✩ Alexander Brudnyi Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4, Canada

a r t i c l e

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Article history: Received 21 December 2012 Accepted 25 January 2014 Available online xxxx Communicated by K. Ball Keywords: Operator corona problem Completion problem Bounded holomorphic function Maximal ideal space Banach holomorphic vector bundle

a b s t r a c t We study the completion problem to an invertible operatorvalued function for the class of bounded holomorphic functions on the unit disk D ⊂ S with relatively compact images in the space of bounded linear operators between complex Banach spaces. In particular, we prove that in this class of functions the operator-valued corona problem and the completion problem are not equivalent, and establish an Oka-type principle asserting that the completion problem is solvable if and only if it is solvable in the class of continuous operator-valued functions on D with relatively compact images. © 2014 Elsevier Inc. All rights reserved.

1. Introduction 1.1. Corona problem Let H ∞ be the Banach space of bounded holomorphic functions on the unit disk D ⊂ C equipped with the supremum norm and H ∞ (L(X, Y )) be the Banach space of holomorphic functions F on D with values in the space of bounded linear operators X → Y between complex Banach spaces X, Y with norm F  := supz∈D F (z)L(X,Y ) . By IX we denote the identity operator X → X. ✩

Research supported in part by NSERC. E-mail address: [email protected].

0022-1236/$ – see front matter © 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jfa.2014.01.023

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The following problem was posed by Sz.-Nagy [20] in 1978: Operator Corona Problem. Suppose F ∈ H ∞ (L(H1 , H2 )), where H1 , H2 are separable Hilbert spaces, satisfies F (z)x  δx for all x ∈ H1 , z ∈ D, where δ > 0 is a constant. Does there exist G ∈ H ∞ (L(H2 , H1 )) such that G(z)F (z) = IH1 for all z ∈ D? This problem is of importance in operator theory (angles between invariant subspaces, unconditionally convergent spectral decompositions) and in control theory (the stabilization problem). It is also related to the study of submodules of H ∞ and to many other subjects of analysis, see [16,15,21,22,27] and references therein. Obviously, the condition imposed on F is necessary since it implies existence of a uniformly bounded family of left inverses of F (z), z ∈ D. The question is whether this condition is sufficient for the existence of a bounded holomorphic left inverse of F . In general, the answer is known to be negative (see [21,23,25] and references therein). But in some specific cases it is positive. In particular, the famous Carleson corona theorem [5] stating that n for {fi }ni=1 ⊂ H ∞ the Bezout equation i=1 gi fi = 1 is solvable with {gi }ni=1 ⊂ H ∞ as soon as max1in |fi (z)| > δ > 0 for every z ∈ D means that the answer is positive when dim H1 = 1, dim H2 = n < ∞. More generally, the answer is positive as soon as dim H1 < ∞ or F is a “small” perturbation of a left invertible function F0 ∈ H ∞ (L(H1 , H2 )) (for example, if F − F0 belongs to H ∞ (L(H1 , H2 )) with values in the class of Hilbert Schmidt operators), see [24] and references therein. For a long time there were no positive results in the case dim H1 = ∞. The first positive results were obtained in [28] where the following more general problem was studied. Problem 1. Let X1 , X2 be complex Banach spaces and F ∈ H ∞ (L(X1 , X2 )) be such that for each z ∈ D there exists a left inverse Gz of F (z) satisfying supz∈D Gz  < ∞. Does there exist G ∈ H ∞ (L(X2 , X1 )) such that G(z)F (z) = IX1 for all z ∈ D? Since in this general setting the answer is negative, it was suggested in [28] to in∞ vestigate the problem for the case of F ∈ Hcomp (L(X1 , X2 )), the space of holomorphic functions on D with relatively compact images in L(X1 , X2 ).1 In particular, it was shown, see [28, Th. 2.1], that the answer is positive for F that can be uniformly approximated  by finite sums fk (z)Lk , where fk ∈ H ∞ and Lk ∈ L(X1 , X2 ). The question of whether ∞ each F ∈ Hcomp (L(X1 , X2 )) can be obtained in that form is closely related to the still open problem about the Grothendieck approximation property for H ∞ . (The strongest result in this direction [2, Th. 9] states that H ∞ has the approximation property “up to logarithm”.) Another, not involving the approximation property for H ∞ , approach that ∞ led to the solution of Problem 1 for Hcomp spaces was proposed in [4]. 1

∞ The notation Hcomp had been introduced in [28].

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Theorem A. (See [4, Theorem 1.5].) Let X1 , X2 be complex Banach spaces and ∞ F ∈ Hcomp (L(X1 , X2 )) be such that for every z ∈ D there exists a left inverse Gz of ∞ F (z) satisfying supz∈D Gz  < ∞. Then there exists G ∈ Hcomp (L(X2 , X1 )) such that G(z)F (z) = IX1 for all z ∈ D. 1.2. Completion problem In fact, in [4] a particular case of the following problem was investigated. Given complex Banach spaces X1 , X2 and Y by LI loc (X1 , X2 , Y ) we denote the class of functions F ∈ H ∞ (L(X1 , X2 )) such that for every z ∈ D there exists a left inverse Gz of F (z) satisfying supz∈D Gz  < ∞ and Ker G0 is isomorphic to Y . Operator Completion Problem. Let F ∈ LI loc (X1 , X2 , Y ). Do there exist functions H ∈ H ∞ (L(X1 ⊕ Y, X2 )) and G ∈ H ∞ (L(X2 , X1 ⊕ Y )) such that H(z)G(z) = IX2 ,

G(z)H(z) = IX1 ⊕Y

 and H(z)X = F (z) 1

for all z ∈ D?

Seemingly much stronger, this problem is equivalent to the operator corona problem for X1 , X2 being separable Hilbert cases. This result, known as the Tolokonnikov lemma, is proved in full generality in [24]. (Thus in this case the operator completion problem has a positive solution as soon as the operator corona problem has it.) In general, the operator completion problem is much more involved. Some sufficient conditions for its ∞ solvability for Hcomp spaces were given in [4]. To formulate the result, by GL(X) we denote the group of invertible elements of the Banach algebra L(X) of bounded linear operators on a complex Banach space X equipped with the operator norm. Also, by LI comp (X1 , X2 , Y ) we denote the subset of LI loc (X1 , X2 , Y ) containing operators with relatively compact images. Theorem B. (See [4, Theorem 1.3].) Suppose that GL(Y ) is connected. Then for every ∞ F ∈ LI comp (X1 , X2 , Y ) there exist functions H ∈ Hcomp (L(X1 ⊕ Y, X2 )) and G ∈ ∞ Hcomp (L(X2 , X1 ⊕ Y )) such that H(z)G(z) = IX2 ,

G(z)H(z) = IX1 ⊕Y

and

 H(z)X = F (z) 1

for all z ∈ D.

As a corollary one obtains Corollary. Theorem B is valid in one of the following cases (a) dimC Y < ∞; (b) X2 is isomorphic to a Hilbert space or c0 or one of the spaces p , 1  p  ∞; (c) X2 is isomorphic to one of the spaces Lp [0, 1], 1 < p < ∞, or C[0, 1] and X1 is not isomorphic to X2 .

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Proof. (a) In this case GL(Y ) ∼ = GLn (C) for some n ∈ N, i.e., GL(Y ) is connected. (b) The hypotheses of the theorem imply that Y is a complemented subspace of X2 . Assuming without loss of generality that dimC Y = ∞, we obtain that Y is isomorphic to a Hilbert space or c0 or p , 1  p  ∞, respectively, see [18,12]. Then GL(Y ) is contractible, see, e.g., [14] and references therein. (c) In this case we have X2 ∼ = X1 ⊕ Y and X1  X2 . This implies that Y is isomorphic to Lp [0, 1], 1 < p < ∞, or C[0, 1], respectively, see [1,13]. Then GL(Y ) is contractible, see, e.g., [14]. 2 The set of functions F satisfying assertions of Theorem B will be denoted by LI 0comp (X1 , X2 , Y ). (In this notation the theorem states that if GL(Y ) is connected, then LI comp (X1 , X2 , Y ) coincides with LI 0comp (X1 , X2 , Y ).) In the present paper we describe the structure of the set LI comp (X1 , X2 , Y ) (Theorem 2.1), establish an Oka-type principle asserting that the operator completion ∞ if and only if it is solvable in the class Ccomp problem is solvable in the class Hcomp of continuous operator-valued functions on D with relatively compact images (Theo∞ rem 2.3), and prove that in the class Hcomp the operator-valued corona problem is not equivalent to the completion problem, i.e., that there exist X1 , X2 , Y such that LI comp (X1 , X2 , Y ) \ LI 0comp (X1 , X2 , Y ) = ∅ (Theorem 2.5). Our proofs rely heavily upon some important results on Banach holomorphic vector bundles on the maximal ideal space of the algebra H ∞ presented in Section 3. 2. Main results ∞ By GL(Hcomp (L(X2 ))) we denote the complex Lie group of invertible elements of ∞ ∞ (L(X2 )) and by GL0 (Hcomp (L(X2 ))) its connected compothe Banach algebra Hcomp ∞ nent containing the unit. Group GL(Hcomp (L(X2 ))) acts on LI comp (X1 , X2 , Y ) by the formula  ∞   G(F )(z) := G(z)F (z), G ∈ GL Hcomp L(X2 ) , F ∈ LI comp (X1 , X2 , Y ), z ∈ D.

The set of all orbits under this action is denoted by SOcomp (X1 , X2 , Y ). By {F } ∈ SOcomp (X1 , X2 , Y ) we denote the orbit of F ∈ LI comp (X1 , X2 , Y ). In the next result we are working in the category of complex Banach manifolds and their morphisms, see, e.g., [10] for basic definitions. Theorem 2.1. LI comp (X1 , X2 , Y ) is an open subset of the complex Banach space ∞ (L(X1 , X2 )). Hcomp Let CF be the connected component of LI comp (X1 , X2 , Y ) containing F . Then  ∞   π F : GL0 Hcomp L(X2 ) → CF , π F (G) := G(F ), is a holomorphic principal bundle on CF with fibre TF := (π F )−1 (F ) a complex Lie ∞ (L(X2 ))). subgroup of GL0 (Hcomp

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In the terminology of [19], CF is a complex Banach homogeneous space under the ∞ action of GL0 (Hcomp (L(X2 ))). Remark 2.2. Results similar to Theorem 2.1 are valid also for other uniform algebras and are equivalent to the solvability of the corresponding operator corona problems for them, see Theorems 5.1, 5.2. Since X1 ⊕ Y ∼ = X2 , in what follows without loss of generality we will identify these spaces. Then each F ∈ LI comp (X1 , X2 , Y ) has a form  F =

F1 F2



    ∞ ∞ where F1 ∈ Hcomp L(X1 ) and F2 ∈ Hcomp L(X1 , Y ) .

  Since LI 0comp (X1 , X2 , Y ) = {IX1 } ∈ SOcomp (X1 , X2 , Y ), where IX1 := IX0 1 , the theorem implies that LI 0comp (X1 , X2 , Y ) is the union of some connected components of LI comp (X1 , X2 , Y ). In this case the complex Lie group TIX1 consists of elements of a form  G :=

IX1 0

E D



 ∞     ∞ where D ∈ GL0 Hcomp L(Y ) , E ∈ Hcomp L(Y, X1 ) .

Next, let Ccomp (Z) denote the Banach space of continuous functions on D with relatively compact images in a complex Banach space Z. Theorem 2.3 (Oka-type principle). F ∈ LI 0comp (X1 , X2 , Y ) if and only if there exist H ∈ Ccomp (L(X1 ⊕ Y, X2 )) and G ∈ Ccomp (L(X2 , X1 ⊕ Y )) such that H(z)G(z) = IX2 ,

G(z)H(z) = IX1 ⊕Y

and

H(z)|X1 = F (z)

for all z ∈ D.

Remark 2.4. The classical Oka principle for a commutative unital complex Banach algebra relates some of its properties via the Gelfand transform to similar ones for the algebra of continuous functions on its maximal ideal space, see, e.g., [19] and references therein. By contrast, the Oka-type principle of Theorem 2.3 is stronger because while ∞ each function in Hcomp (Z) admits a continuous extension to the maximal ideal space ∞ of H there are functions in Ccomp (Z) that cannot be continuously extended to it (see Remark 6.1). For a complex Banach space X by GL0 (X) we denote the connected component of GL(X) containing the identity map IX . It is a clopen normal subgroup of GL(X). By C(GL(X)) := GL(X)/GL0 (X) we denote the discrete group of connected components of GL(X). A long-standing problem is to determine the groups that arise as groups C(GL(X)) for complex Banach spaces X. The simplest examples of complex Banach spaces whose

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linear groups are not connected are c0 ⊕ p and q ⊕ p , 1  q < p < ∞; their groups of connected components are isomorphic to Z, see [6,14]. For a complex Banach space Z by X2 EX : LI comp (X1 , X2 , Y ) → LI comp (X1 , X2 ⊕ Z, Y ⊕ Z) 2 ⊕Z

we denote a continuous injective map defined by the formula X2 X2 EX (F )(z) := EX ◦ F (z), 2 ⊕Z 2 ⊕Z

F ∈ LI comp (X1 , X2 , Y ), z ∈ D,

X2 where EX : X2 → X2 ⊕ Z is the natural embedding. 2 ⊕Z Our next result supplements [4, Th. 1.3] (see Theorem B above).

Theorem 2.5. Suppose that C(GL(Y )) contains a non-torsion element. Then there exists a complex Banach space X such that (a) LI comp (X, X ⊕ Y, Y ) \ LI 0comp (X, X ⊕ Y, Y ) = ∅ and the cardinality of the orbit space SOcomp (X, X ⊕ Y, Y ) is at least of that of the continuum; (b)   X2 EX LI comp (X1 , X2 , Y ) ⊂ LI 0comp (X1 , X2 ⊕ X, Y ⊕ X) 2 ⊕X for all possible X1 and X2 . Remark 2.6. In all known examples nontrivial groups C(GL(Y )) satisfy the condition of the theorem. Therefore it is natural to ask about the existence of complex Banach spaces Y with nontrivial torsion groups C(GL(Y )). As the space X one can take, e.g., p (Y ), 1  p < ∞, see Section 4. Since Theorem A holds for each F ∈ LI comp (X, X ⊕ Y, Y ) \ LI 0comp (X, X ⊕ Y, Y ), part (a) of the theorem ∞ . Although shows that the analog of Tolokonnikov’s lemma is not valid in the class Hcomp ∞ such F are not completed to functions in GL(Hcomp (L(X ⊕ Y ))), it is not clear whether they can be completed to invertible operator-valued functions in H ∞ (L(X ⊕ Y )). In what follows, Z n denotes the n-fold direct sum of a complex Banach space Z. Theorem 2.7. Suppose LI comp (X, X ⊕ Y, Y ) \ LI 0comp (X, X ⊕ Y, Y ) = ∅. For each S ∈ SOcomp (X, X ⊕ Y, Y ) \ {IX } there exist a number NS ∈ N and an element FS ∈ LI comp (X, X ⊕ Y NS , Y NS ) such that   F ⊕ FS ∈ LI 0comp X 2 , X 2 ⊕ Y NS +1 , Y NS +1

for all F ∈ S.

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Remark 2.8. If the quotient C(GL(Y ))/[C(GL(Y )), C(GL(Y ))] is a finitely generated abelian group, then each NS is bounded from above by maximum of its rank and 1, see Remark 8.5. For instance, this is applied to complex Banach spaces Y with C(GL(Y )) = Zn , n ∈ N (see [7, Prop. 3] for the corresponding examples). 3. Banach vector bundles on the maximal ideal space of H ∞ This section contains some important results on Banach holomorphic vector bundles on the maximal ideal space of the algebra H ∞ used in the proofs of our main theorems. 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 Carleson corona theorem 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 be X-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))}. 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∈I Ui × X by the equivalence relation: Uj × X u × x ∼ u × gij (u)x ∈ Ui × X. The projection p : E → M (H ∞ ) is induced by natural projections Ui × X → Ui , i ∈ I. A morphism ϕ : (E1 , X1 , p1 ) → (E2 , X2 , p2 ) of holomorphic Banach vector bundles ∼ on M (H ∞ ) is a continuous map which sends each vector space p−1 1 (w) = X1 linearly −1 ∞ ∼ to vector space p2 (w) = X2 , w ∈ M (H ), and such that ϕ|D : E1 |D → E2 |D is a holomorphic map of complex Banach manifolds. If, in addition, ϕ is bijective, then ϕ is called an isomorphism. We say that a holomorphic Banach vector bundle (E, X, p) on M (H ∞ ) is holomorphically trivial if it is isomorphic to the trivial bundle EX := M (H ∞ ) × X. (For the basic facts of the theory of bundles, see, e.g., [9].) Let E → M (H ∞ ) be a holomorphic Banach vector bundle with fibre X defined on an open cover U = (Ui )i∈I of M (H ∞ ) by a cocycle g = {gij ∈ O(Ui ∩ Uj ; GL(X))}. By EC(GL(X)) we denote the principal bundle on M (H ∞ ) with fibre C(GL(X)) defined on U by the cocycle qX (g) = {qX (gij ) ∈ C(Ui ∩ Uj ; C(GL(X)))}; here qX : GL(X) → GL(X)/GL0 (X) =: C(GL(X)) is the (continuous) quotient homomorphism onto the discrete group of connected components of GL(X).

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Theorem C. (See [4, Theorem 1.1].) E is holomorphically trivial if and only if the associated bundle EC(GL(X)) is trivial in the category of principal bundles with discrete fibres. Corollary. (See [4, Corollary 1.2].) E is holomorphically trivial in one of the following cases: (1) The image of each function gij in the definition of E belongs to GL0 (X) (e.g., this is true if GL(X) is connected); (2) E is trivial in the category of continuous Banach vector bundles. In particular, the result is valid for spaces X with contractible group GL(X). The class of such spaces include infinite-dimensional Hilbert spaces, spaces p and Lp [0, 1], 1  p  ∞, c0 and C[0, 1], spaces Lp (Ω, μ), 1 < p < ∞, of p-integrable measurable functions on an arbitrary measure space Ω, and some classes of reflexive symmetric function spaces; the class of spaces X with connected but not simply connected group GL(X) include finite dimensional Banach spaces, finite direct products of James spaces etc., see, e.g., [14] and references therein. There are also Banach spaces X whose linear groups GL(X) are not connected (e.g., the groups of connected components of spaces c0 ⊕ p and p ⊕ q , 1  p < q < ∞, are isomorphic to Z), see [6,14,7]. In this case we have Theorem 3.1. Suppose the complex Banach space X is such that C(GL(X)) contains a non-torsion element. Then there exist not holomorphically trivial holomorphic Banach vector bundles on M (H ∞ ) with fibre X. Meanwhile, every Banach holomorphic vector bundle on M (H ∞ ) is stably holomorphically trivial: Theorem 3.2. For a complex Banach space X there exists a complex Banach space Z such that for each holomorphic Banach vector bundle E on M (H ∞ ) with fibre X the Whitney sum E ⊕ EZ is holomorphically trivial. In particular, the Čech cohomology groups H k (M (H ∞ ), OE ) = 0 for all k  1, where OE is the sheaf of germs of holomorphic sections of E. We will show that the statement is valid for Z = p (X) with 1  p < ∞. The second part of the corollary generalizes a similar statement established in [3, Th. 1.4] for E being holomorphically trivial. Further, by SHT (Y, X1 ) we denote the isomorphism classes of holomorphic Banach vector bundles E on M (H ∞ ) with fibres Y such that the Whitney sums E ⊕ EX1 are holomorphically trivial. Recall that SOcomp (X1 , X2 , Y ) stands for the orbit space (i.e., the set of all orbits) ∞ under the action of the group GL(Hcomp (L(X2 ))) on LI(X1 , X2 , Y ).

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Theorem 3.3. There exists a one-to-one correspondence Φ : SOcomp (X1 , X2 , Y ) → SHT (Y, X1 ) that sends the orbit LI 0comp (X1 , X2 , Y ) to the isomorphism class of the trivial bundle EY . 4. Proofs of Theorems 3.1, 3.2 and 3.3 Proof of Theorem 3.1. According to the Arens–Royden theorem the first Čech cohomology group H 1 (M (H ∞ ), Z) is isomorphic to (H ∞ )−1 / exp(H ∞ ), where (H ∞ )−1 is the group of invertible elements of H ∞ . In particular, H 1 (M (H ∞ ), Z) = 0. Consider an integer-valued continuous cocycle {cij } defined on an open cover (Ui )i∈I of M (H ∞ ) representing a nonzero element of H 1 (M (H ∞ ), Z). Let a := {an : n ∈ Z} ⊂ C(GL(X)) be the subgroup generated by a non-torsion element a. Then a ∼ = Z. Let p : B → M (H ∞ ) be the principal bundle with (discrete) fibre C(GL(X)) defined on the cover (Ui ) by C(GL(X))-valued cocycle {acij } (see constructions of bundles in the previous section). Lemma 4.1. B is not trivial in the category of principal bundles on M (H ∞ ) with fibre C(GL(X)). Proof. Suppose, on the contrary, that B is trivial in this category of bundles. Then there exist continuous functions gi : Ui → C(GL(X)), i ∈ I, such that gj (x) = gi (x) · acij (x)

for all x ∈ Ui ∩ Uj , i, j ∈ I.

(4.1)

Next, the restriction of {cij } to the open cover (D ∩ Ui )i∈I of D represents an element of the cohomology group H 1 (D, Z). Since D is contractible, this group is trivial. In particular, there exist continuous functions ci : Ui ∩ D → Z such that cj (x) = ci (x) + cij (x) for all x ∈ Ui ∩ Uj , i, j ∈ I. This and (4.1) imply that equations g(x) := gi (x) · a−ci (x) ,

x ∈ D ∩ Ui , i ∈ I,

determine a continuous function on D with values in C(GL(X)). Since C(GL(X)) is a discrete space and D is connected, g(x) = g(0) for all x ∈ D. This implies that the range of each function g −1 (0) · gi : D ∩ Ui → C(GL(X)) belongs to subgroup a. Further, since D is an open dense subset of M (H ∞ ) and Ui ⊂ M (H ∞ ) is open, the closure cl(D ∩ Ui ) of D ∩ Ui in M (H ∞ ) contains Ui . Since gi is continuous on Ui and a is a closed subset of C(GL(X)), extending (g −1 (0) · gi )|D∩Ui by continuity we get that the range of each g −1 (0) · gi : Ui → C(GL(X)) belongs to a. Then equations (cf. (4.1)) g −1 (0) · gj (x) = g −1 (0) · gi (x) · acij (x)

for all x ∈ Ui ∩ Uj , i, j ∈ I,

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show that cocycle {acij } determines the trivial bundle in the category of principal bundles on M (H ∞ ) with fibre a ∼ = Z. Equivalently, cocycle {cij } represents 0 in H 1 (M (H ∞ ), Z), a contradiction. 2 Now, consider an element b ∈ GL(X) such that qX (b) = a. We set gij := bcij . Then {gij } is a holomorphic 1-cocycle on cover (Ui ) with values in GL(X). It determines a Banach holomorphic vector bundle E → M (H ∞ ) with fibre X. By the definition the associated bundle EC(GL(X)) coincides with B and therefore by Lemma 4.1 is not trivial. Hence, E is not holomorphically trivial. 2 Proof of Theorem 3.2. Let E be a holomorphic Banach vector bundle on M (H ∞ ) with fibre X. For a fixed 1  p < ∞ consider the trivial holomorphic Banach vector bundle Ep (X) on M (H ∞ ) with fibre p

 (X) :=

x = (xn )n∈N ; xn ∈ X,



∞ 

1/p xn pX

<∞ .

n=1

Then the Whitney sum E ⊕ Ep (X) is a topologically trivial holomorphic Banach vector bundle on M (H ∞ ) (see, e.g., [17, Sect. 7] for a similar argument). Therefore E ⊕ Ep (X) is holomorphically trivial by [4, Cor. 1.2]. In particular, the cohomology group H k (M (H ∞ ), OE ) is a complemented subspace of the complex vector space H k (M (H ∞ ), OE⊕Ep (X) ). Since E ⊕ Ep (X) is holomorphically trivial, due to [3, Th. 1.4] the latter group is trivial for k  1. This implies that H k (M (H ∞ ), OE ) = 0 for k  1 as well. 2 Proof of Theorem 3.3. The construction of Φ goes along the lines of the arguments in [4, Sect. 8] and is similar to that in the matrix case, see, e.g., [11]; so we will just briefly describe it referring to the cited papers for details. According to [3, Prop. 1.3] for any complex Banach space X the restriction map to ∞ D induces an isometric isomorphism between spaces O(M (H ∞ ); X) and Hcomp (X). In what follows we identify these spaces. Then each F ∈ LI comp (X1 , X2 , Y ) is presented in a form (recall that X2 = X1 ⊕ Y ) 

F1 F2



        where F1 ∈ O M H ∞ ; L(X1 ) and F2 ∈ O M H ∞ ; L(X1 , Y ) .

Due to the arguments of [4, Sect. 8], there exist a finite open cover (Ui )1ik of M (H ∞ ) and operators Fi ∈ O(Ui ; L(Y, X1 ⊕ Y )), Fi :=

  F1i F2i

    where F1i ∈ O Ui ; L(Y, X1 ) and F2i ∈ O Ui ; L(Y ) ,

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such that  Bi :=

F1 F2

   F1i ∈ O Ui ; GL(X1 ⊕ Y ) .  F2i

Then Bi−1 Bj ∈ O(Ui ∩ Uj ; GL(X1 ⊕ Y )) has a form  Cij :=

IX1 0

Eij Dij



    where Dij ∈ O Ui ∩ Uj ; GL(Y ) , Eij ∈ O Ui ∩ Uj ; L(Y, X1 ) .

The holomorphic 1-cocycle {Dij } determines a holomorphic Banach vector bundle E on M (H ∞ ) with fibre Y , and the holomorphic 1-cocycle {Eij } determines an element δ of the cohomology group H 1 (M (H ∞ ), Hom(E, EX1 )), see, e.g., the argument in [4, Sect. 9]. Due to Theorem 3.2, δ = 0 and hence the holomorphically trivial Banach vector bundle on M (H ∞ ) with fibre X1 ⊕ Y defined by the cocycle {Cij } is holomorphically isomorphic to EX1 ⊕ E. Similarly to the matrix case, one easily shows that going, in the above construction, over the class of operators Fi defined on open covers (Ui ) of M (H ∞ ) for which the corresponding operators Bi ∈ O(Ui ; GL(X1 ⊕ Y )) complete F we obtain Banach holomorphic vector bundles on M (H ∞ ) with fibres Y isomorphic to E. Also, we obtain the same isomorphism class of bundles if instead of F we consider any element of the orbit G(F ), ∞ (L(X1 ⊕ Y ))). Thus for the orbit {F } ∈ SOcomp (X1 , X2 , Y ) containing G ∈ GL(Hcomp F we can define   Φ {F } := {E}

(:= the isomorphism class of E).

Starting with a bundle E representing the class {E} ∈ SHT (Y, X1 ) and reversing the above arguments we construct an element F ∈ LI comp (X1 , X2 , Y ) such that Φ({F }) = {E}. It is the matter of definitions to check that the map Φ : SOcomp (X1 , X2 , Y ) → SHT (Y, X1 ) is injective and that F ∈ LI 0comp (X1 , X2 , Y ) if and only if the bundles of the class Φ({F }) are holomorphically trivial. 2 5. Proof of Theorem 2.1 5.1. A general result We will prove a more general statement. Let A be a unital closed subalgebra of the Banach algebra Cb (X) of bounded complex-valued continuous functions on a paracompact Hausdorff space X equipped with supremum norm. For a complex Banach space B by A(X; B) we denote the subspace of bounded B-valued continuous functions f on X such that ϕ(f ) ∈ A for all ϕ ∈ B∗.

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Let X1 be a complemented subspace of a complex Banach space X2 . Assume that (*) A(X; L(X2 )) is a Banach algebra with respect to the multiplication given by the pointwise composition of operators. (This is valid, e.g., if A = Cb (X) or A = H ∞ (X), where X is a complex manifold.) Then the group GL(A(X; L(X2 ))) of invertible elements of A(X; L(X2 )) acts on A(X; L(X1 , X2 )) by the formula: G(F )(x) := G(x)F (x),

     G ∈ GL A X; L(X2 ) , F ∈ A X; L(X1 , X2 ) , x ∈ X.

By LI A (X1 , X2 ) we denote the set of F ∈ A(X; L(X1 , X2 )) for which there exists G ∈ A(X; L(X2 , X1 )) such that G(x)F (x) = IX1 for all x ∈ X. Theorem 5.1. LI A (X1 , X2 ) is an open subset of the Banach space A(X; L(X1 , X2 )). Let CF be a connected component of LI A (X1 , X2 ) containing F . Then    π F : GL0 A X; L(X2 ) → CF ,

π F (G) := G(F ),

is a holomorphic principal bundle on CF with fibre TF := (π F )−1 (F ) a complex Lie subgroup of GL0 (A(X; L(X2 ))). Proof. Let F ∈ LI A (X1 , X2 ). Then there exists G ∈ A(X; L(X2 , X1 )) such that 1 G(x)F (x) = IX1 for all x ∈ X. Let F1 ∈ A(X; L(X1 , X2 )) satisfy F1 − F  < ε := 2G . We have   sup G(x)F1 (x) − IX1 L(X

1 ,X2 )

x∈X

   = sup G(x)F (x) − IX1 + G(x) F1 (x) − F (x) L(X x∈X

1 ,X2 )

 G · F1 − F  <

1 . 2

Hence, GF1 ∈ GL(A(X; L(X1 ))) and (GF1 )−1 G is left inverse to F1 (see condition (*)). Thus, the open ball of radius ε with center at F belongs to LI A (X1 , X2 ), that is, LI A (X1 , X2 ) is an open subset of A(X; L(X1 , X2 )). To establish the second statement, first we show that M := π F (GL0 (A(X; L(X2 )))) satisfies the conclusions of the theorem and then prove that M = CF . To this end, we denote the tangent space to a complex Banach manifold Y at y ∈ Y by Yy , and the differential at y of a holomorphic map f : Y → Z of complex Banach manifolds Y and Z by dfy . By e we denote the unit of GL0 (A(X; L(X2 ))). Due to [19, Prop. 1.5], it suffices to prove that Ker dπeF is a complemented subspace of GL0 (A(X; L(X2 )))e and dπeF : GL0 (A(X; L(X2 )))e → LI A (X1 , X2 )F is surjective.

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Since LI A (X1 , X2 ) is an open subset of the Banach space A(X; L(X1 , X2 )), the Banach space LI A (X1 , X2 )F is isometrically isomorphic to A(X; L(X1 , X2 )). Also, by definition GL0 (A(X; L(X2 )))e is isometrically isomorphic to A(X; L(X2 )). Then dπeF maps A(X; L(X2 )) to A(X; L(X1 , X2 )) by the formula: dπeF (H)(x) := H(x)F (x) for all x ∈ X. Next, since F ∈ LI A (X1 , X2 ), there is G ∈ A(X; L(X2 , X1 )) such that G(x)F (x) = IX1 for all x ∈ X. Consider subspace V := {HG; H ∈ A(X; L(X1 , X2 ))} ⊂ A(X; L(X2 )). By the definition, dπeF maps V isomorphically onto A(X; L(X1 , X2 )). This implies that A(X; L(X2 )) = V ⊕ Ker dπeF and dπeF is surjective. Now, from the results of [19] we obtain that M is an open connected subset of A(X; L(X1 , X2 )) and π F : GL0 (A(X; L(X2 ))) → M is a holomorphic principal bundle on M with fibre TF := {G ∈ GL0 (A(X; L(X2 ))); π F (G) = F } a complex Lie subgroup of GL0 (A(X; L(X2 ))). To finish the proof of the theorem let us show that M = CF . In fact, let H ∈ CF . Then (since CF is open connected) there exists a path γ : [0, 1] → CF such that γ(0) = F and γ(1) = H. Since γ([0, 1]) ⊂ LI A (X1 , X2 ), using the fact (following from the definition of a principal bundle) that each map π p , p ∈ CF , is open and compactness of γ([0, 1]) we obtain that there is a finite partition 0 = t0 < t1 < · · · < tn = 1 and functions Gi ∈ GL0 (A(X; L(X2 ))) such that γ(ti ) = Gi (γ(ti−1 )), 1  i  n. Thus, H = (Gn · · · G1 )(F ), i.e. H ∈ CF . The proof of the theorem is complete. 2 A converse statement is also true (we retain the notation of Theorem 5.1): Theorem 5.2. If F ∈ A(X; L(X1 , X2 )) is such that M := π F (GL0 (A(X; L(X2 )))) is an open subset of A(X; L(X1 , X2 )) and π F : GL0 (A(X; L(X2 ))) → M is a holomorphic principal bundle with fibre TF , then F ∈ LI A (X1 , X2 ). Proof. By the definition of a holomorphic fiber bundle, there exists a holomorphic section s : U → GL0 (A(X; L(X2 ))) of the bundle π F : GL0 (A(X; L(X2 ))) → M defined in an open neighbourhood U ⊂ M of F (i.e., π F ◦ s = Id). Since M is an open subset of A(X; L(X1 , X2 )), the differential dsF maps A(X; L(X1 , X2 )) into A(X; L(X2 )) and is such that dπeF ◦ dsF = IA(X;L(X1 ,X2 )) .

(5.1)

According to our assumption X2 = X1 ⊕ Y for a subspace Y of X2 . Then each H ∈ A(X; L(X1 , X2 )) has a form  H=

H1 H2



    where H1 ∈ A X; L(X1 ) and H2 ∈ A X; L(X1 , Y ) .

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Applying (5.1) to the constant function H :=

IX  0

1

∈ A(X; L(X1 , X2 )) we obtain

    G11 (x) G12 (x) F1 (x) IX1 = 0 G21 (x) G22 (x) F2 (x)         



dsF (H)

F

for all x ∈ X.

H

(Here G11 ∈ A(X; L(X1 )), G12 ∈ A(X; L(Y, X1 )), G21 ∈ A(X; L(X1 , Y )), G22 ∈ A(X; L(Y )).) Hence, G := (G11 , G12 ) ∈ A(X; L(X2 , X1 )) is the left inverse of F . This means that F ∈ LI A (X1 , X2 ). 2 5.2. Proof of Theorem 2.1 We apply Theorem 5.1 to the following the extension of H ∞ to M (H ∞ ) by the ∞ (B) for isometrically isomorphic to Hcomp in [3, Prop. 1.3]). Since Theorem A (see ∞ (L(X1 , X2 )), Hcomp

setting: X = M (H ∞ ) and A := O(M (H ∞ )), Gelfand transform. In this case A(X; B) is any complex Banach space B (cf. the proof Subsection 1.1) holds for the Banach space

LI O(M (H ∞ )) (X1 , X2 ) =



LI comp (X1 , X2 , Y ),

Y

where the union is taken over all subspaces Y ⊂ X2 such that X1 ⊕ Y ∼ = X2 . As in the proof of Theorem 5.1 one easily shows that each LI comp (X1 , X2 , Y ) is an open subset of LI O(M (H ∞ )) (X1 , X2 ). Thus each connected component of LI comp (X1 , X2 , Y ) is an open subset of a connected component of LI O(M (H ∞ )) (X1 , X2 ). On the other hand, since M (H ∞ ) is connected, each connected component of LI O(M (H ∞ )) (X1 , X2 ) belongs to one of LI comp (X1 , X2 , Y ) (cf. the proof of Theorem 5.1). This implies that connected components of sets LI comp (X1 , X2 , Y ) and LI O(M (H ∞ )) (X1 , X2 ) coincide. Now, the required result follows from Theorem 5.1. 6. Proof of Theorem 2.3 Let Cb be the Banach algebra of complex-valued bounded continuous functions on D equipped with supremum norm and M (Cb ) be its maximal ideal space. Since for Cb the corona theorem is valid, D is an open dense subset of M (Cb ). Repeating word-forword the arguments from the proof of [3, Prop. 1.3] one obtains that for any complex Banach space Z the restriction map to D induces an isometrical isomorphism between C(M (Cb ); Z) and Ccomp (Z). Next, the adjoint map to the continuous embedding H ∞ → Cb induces a continuous surjective map π : M (Cb ) → M (H ∞ ) which is identity on D. Let E be a holomorphic Banach vector bundle on M (H ∞ ) representing the class {Φ({F })} ∈ SHT (Y, X1 ), where {F } ∈ SOcomp (X1 , X2 , Y ) is the orbit of F from

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∞ the statement of the theorem with respect to the action of GL(Hcomp (L(X2 ))), see ∗ Theorem 3.3. Consider the pullback bundle π E on M (Cb ). One easily shows, using arguments similar to those of the proof of Theorem 3.3, that if F ∈ LI comp (X1 , X2 , Y ), X2 := X1 ⊕ Y , is completed to an operator-valued function in GL(Ccomp (L(X2 ))) (this is the hypothesis of the theorem!), then the bundle π ∗ E is topologically trivial. Further, for such F consider the associated principal bundle EC(GL(Y )) on M (H ∞ ) with fibre C(GL(Y )) := GL(Y )/GL0 (Y ). The triviality of π ∗ E implies that the principal bundle π ∗ EC(GL(Y )) → M (Cb ) is trivial. In turn, this implies that it has a continuous section s : M (Cb ) → π ∗ EC(GL(Y )) . Suppose that EC(GL(Y )) is defined on a finite open cover (Ui )i∈I of M (H ∞ ) by a continuous locally constant cocycle {cij } with values in C(GL(Y )) (cf. the construction of Section 3). Then on the open cover (π −1 (Ui ))i∈I of M (Cb ) the section s is determined by a family of continuous functions si : π −1 (Ui ) → C(GL(Y )) with relatively compact images (because s is continuous and is defined on a compact Hausdorff space) such that

sj (x) = si (x) · π ∗ cij (x) for all x ∈ π −1 (Ui ) ∩ π −1 (Uj ), i, j ∈ I. Since π|D = Id, the restriction s|D is a continuous section of the principal bundle (EC(GL(Y )) )|D on D, that is, sj (x) = si (x) · cij (x) for all x ∈ (Ui ∩ D) ∩ (Uj ∩ D), i, j ∈ I.

(6.1)

Since each si is continuous on the open subset Ui ∩ D of D with relatively compact image in the discrete space C(GL(Y )), it is constant on each connected component of Ui ∩ D and its image is finite. Thus, each si admits a continuous extension sˆi : Ui → C(GL(Y )) (this follows from [26, Th. 3.2] if we identify image of si with a finite subset of C). Since each Ui ∩ D is dense in Ui , extending (6.1) by continuity we obtain sˆj (x) = sˆi (x) · cij (x) for all x ∈ Ui ∩ Uj , i, j ∈ I. This shows that the family sˆ = {ˆ si }i∈I determines a continuous section of the principal ∞ bundle EC(GL(Y )) → M (H ), i.e., this bundle is trivial. Now, applying Theorem C of Section 3 we conclude from the previous statement that E → M (H ∞ ) is a holomorphically trivial bundle. Thus, according to Theorem 3.3, F ∈ LI 0comp (X1 , X2 , Y ) as required. The converse statement, if F ∈ LI 0comp (X1 , X2 , Y ), then it can be completed to an operator-valued function in GL(Ccomp (L(X2 ))), is obvious. The proof of the theorem is complete. Remark 6.1. Observe that the map π : M (Cb ) → M (H ∞ ) is not a homeomorphism. For otherwise, every f ∈ Cb admits a continuous extension to M (H ∞ ). This implies, by the Stone–Weierstrass theorem, that f is the uniform limit of a sequence of complex polynomials in variables z1 and z2 taking values in H ∞ and H ∞ , respectively.

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In particular, every f ∈ Cb has radial limits almost everywhere on the unit circle, a contradiction (e.g., f (z) = sin((1 − |z|)−1 ), z ∈ D, does not satisfy this requirement). 7. Proof of Theorem 2.5 As a complex Banach space X we take p (Y ) for some 1  p < ∞. (a) According to Theorem 3.2 every holomorphic Banach vector bundle on M (H ∞ ) with fibre Y determines an element of SHT (Y, X). Also, according to Theorem 3.1 there exists a holomorphic Banach vector bundle E on M (H ∞ ) which is not holomorphically trivial. Finally, according to Theorem 3.3 there exists an element {F } ∈ SOcomp (X, X ⊕ Y, Y ) such that Φ({F }) = {E}. Thus F ∈ / LI comp (X, X ⊕ Y, Y ) \ LI 0comp (X, X ⊕ Y, Y ). Further, the construction of the bundle in Theorem 3.1 and Theorem 3.3 imply that the cardinality of the orbit space SOcomp (X, X ⊕ Y, Y ) is at least of that of the Čech cohomology group H 1 (M (H ∞ ), Z) ∼ = (H ∞ )−1 / exp(H ∞ ). The latter group is isomorphic to the quotient group Γ := {f ∈ BM OA; | Re f | < ∞}/H ∞ (see, e.g., [8] for the basic facts of harmonic analysis). This quotient considered as a real vector space is nontrivial; hence, Γ has the cardinality at least of that of the continuum. On the other hand, the cardinality of Γ does not exceed the cardinality of BM OA. Since, by the C. Fefferman theorem, the latter is dual to the separable Banach space H 1 (D), it has the cardinality of the continuum. These imply that Γ ∼ = H 1 (M (H ∞ ), Z) has the cardinality of the continuum as well which completes the proof of part (a) of the theorem. (b) The implication   X2 LI comp (X1 , X2 , Y ) ⊂ LI 0comp (X1 , X2 ⊕ X, Y ⊕ X) EX 2 ⊕X for all possible X1 and X2 , follows straightforwardly from the identity  X2  Φ EX (F ) = {E ⊕ EX }, 2 ⊕X

  where {E} = Φ {F } ,

and Theorems 3.2 and 3.3. We leave the details to the reader. The proof of the theorem is complete. 8. Proof of Theorem 2.7 8.1. Auxiliary results In the proof we use the following results. Lemma 8.1. Any principal bundle on M (H ∞ ) whose fibre is a finite group is trivial.

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Proof. Let B → M (H ∞ ) be a principal bundle with finite fibre G defined on a finite open cover (Ui )i∈I of M (H ∞ ) by a continuous G-valued cocycle {cij }. Since D is contractible, the restriction B|D is trivial, that is, there exist continuous maps ci : Ui ∩ D → G, i ∈ I, such that cj (x) = ci (x) · cij (x) for all x ∈ (Ui ∩ D) ∩ (Uj ∩ D), i, j ∈ I. As in the proof of Theorem 2.3 we deduce that each ci has a continuous extension cˆi : Ui → G such that cˆj (x) = cˆi (x) · cij (x) for all x ∈ Ui ∩ Uj , i, j ∈ I. This shows that B is a trivial principal bundle. 2 For a complex Banach space Y let us embed GL(Y ) into GL(Y 2 ) by the formula i(G) := IY ⊕ G,

G ∈ GL(Y ).

By D : GL(Y ) → C(GL(Y 2 )) we denote the composite homomorphism qY 2 ◦ i. Lemma 8.2. GY := D(GL(Y )) is an abelian subgroup of C(GL(Y 2 )). Proof. We have, for all G1 , G2 ∈ GL(Y ), G1 ⊕ G2 = L · (G2 ⊕ G1 ) · L−1 ,

 where L :=

0 IY

IY 0

 .

Since L ∈ GL0 (Y 2 ) (note that ln L is well defined!), the previous identity implies D(G1 · G2 ) = D(G1 ) · D(G2 ) = qY 2 (IY ⊕ G1 ) · qY 2 (G2 ⊕ IY ) = qY 2 (G2 ⊕ G1 ) = qY 2 (G1 ⊕ G2 ) = qY 2 (IY ⊕ G2 ) · qY 2 (G1 ⊕ IY ) = D(G2 · G1 ).

2

Let E → M (H ∞ ) be a holomorphic Banach vector bundle with fibre Y defined on a finite open cover (Ui )i∈I of M (H ∞ ) by a holomorphic GL(Y )-valued cocycle {cij }. Passing to a finite refinement of (Ui ), if necessary, without loss of generality we may assume that the image of each cij is relatively compact in GL(Y ). Then the associated with the bundle E ⊕ EY principal bundle (E ⊕ EY )C(GL(Y 2 )) on M (H ∞ ) with fibre C(GL(Y 2 )) is defined on the cover (Ui ) by the continuous cocycle {D(cij )}; here the image of each D(cij ) is a finite subset of the abelian group GY . Consider a subgroup SE ⊂ GY generated by values of all functions D(cij ), i, j ∈ I. Since I is a finite set, SE is a finitely generated abelian group. Then SE ∼ = FE ⊕ TE , where FE ∼ = ZNE , NE ∈ Z+ , and TE is a finite abelian group. Suppose that NE  1 and let g1 , . . . , gNE ⊂ SE be generators of FE . Then we have

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D(cij ) =

NE 

nij,k ek + tij ,

(8.1)

k=1

where {nij,k }, 1  k  NE , are continuous integer-valued cocycles on (Ui ) and {tij } is a continuous cocycle on (Ui ) with values in TE . Let us choose L1 , . . . , LNE ∈ GL(Y ) such that D(Lk ) = ek , 1  k  NE . Consider a holomorphic Banach vector bundle Ek → M (H ∞ ) defined on the cover (Ui ) by the −n cocycle {cij,k }, where cij,k := Lk ij,k , 1  k  NE .  := E ⊕ (NE E  ) is a holomorphically trivial Banach vector bundle on Lemma 8.3. E k=1 k M (H ∞ ) with fibre Y NE +1 . Proof. Consider an embedding iNE : GL(Y ) → GL(Y NE +1 ), iNE (G) := IYNE ⊕ G,

G ∈ GL(Y ).

One easily checks that for the homomorphism DNE := qY NE +1 ◦ iNE : GL(Y ) → C(GL(Y NE +1 )), there exists a homomorphism hNE : GY → C(GL(Y NE +1 )) such that DNE = hNE ◦ D.

(8.2)

 is defined on the open cover (Ui ) of M (H ∞ ) by the cocycle {˜ Next, the bundle E cij }, where c˜ij := cij ⊕ cij,1 ⊕ · · · ⊕ cij,NE ,

i, j ∈ I.

Applying arguments similar to those of the proof of Lemma 8.2 together with (8.1) and (8.2), we have, for each x ∈ Ui ∩ Uj and i, j ∈ I,      qY NE +1 c˜ij (x) = qY NE +1 IYNE ⊕ cij (x) · cij,1 (x) · · · cij,NE (x)    = hNE D cij (x) · cij,1 (x) · · · cij,NE (x) ⊂ hNE (TE ).  principal bundle E C(GL(Y NE +1 )) on M (H ∞ ) with fibre Hence, the associated with E C(GL(Y NE +1 )) is defined on the cover (Ui ) by the cocycle {qY NE +1 (˜ cij )} with values in the finite abelian group hNE (TE ). According to Lemma 8.1, the principal bundle B → M (H ∞ ) with fibre hNE (TE ) defined by the cocycle {qY NE +1 (˜ cij )} is trivial. Since  B is the subbundle of EC(GL(Y NE +1 )) , the latter is a trivial bundle as well. Therefore  is holomorphically according to [4, Th. 1.1] (see Section 3, Theorem C), the bundle E trivial. 2

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8.2. Proof of Theorem 2.7 Fix an orbit S ∈ SOcomp (X, X ⊕ Y, Y ) \ {IX }. Then Φ(S) is the isomorphism class of some holomorphic Banach vector bundle E → M (H ∞ ) with fibre Y such that E ⊕ EX is holomorphically trivial, see Theorem 3.3. Consider the subgroup SE ⊂ GY associated with E as in Lemma 8.3. First, we assume that rank SE := NE  1. Then the bundle E  :=

NE k=1

(8.3)

Ek constructed by E as in Lemma 8.3 is well defined.

Lemma 8.4. E  ⊕ EX → M (H ∞ ) is a holomorphically trivial Banach vector bundle. Proof. We retain the notation of Lemma 8.3. According to (8.1) we have, for all x ∈ Ui ∩ Uj , i, j ∈ I,  −1 cij (x) = cij,1 (x) · · · cij,NE (x) · t˜ij (x),

(8.4)

where t˜ij ∈ C(Ui ∩ Uj , GL(Y )) is such that D(t˜ij (x)) = tij (x). The associated with E  ⊕ EX principal bundle (E  ⊕ EX )C(GL(Y NE ⊕X)) → M (H ∞ ) with fibre C(GL(Y NE ⊕ X)) is defined on the open cover (Ui ) of M (H ∞ ) by the cocycle {qY NE ⊕X (cij,1 ⊕ · · · ⊕ cij,NE ⊕ IX )}. Using arguments similar to those of Lemma 8.3 and (8.4) we obtain, for all x ∈ Ui ∩ Uj , i, j ∈ I,   qY NE ⊕X cij,1 (x) ⊕ · · · ⊕ cij,NE (x) ⊕ IX     = qY NE ⊕X IYNE −1 ⊕ cij,1 (x) · · · cij,NE (x) ⊕ IX     = qY NE ⊕X IYNE −1 ⊕ t˜ij (x) · cij (x)−1 ⊕ IX     −1 = qY NE ⊕X IYNE −1 ⊕ t˜ij (x) ⊕ IX · qY NE ⊕X IYNE −1 ⊕ cij (x) ⊕ IX =: gij (x) · hij (x). (We omit “IYNE −1 ⊕” if NE = 1.) Since tij (x) ∈ TE , {gij } is a continuous cocycle on (Ui ) with values in a finite abelian subgroup G ⊂ C(GL(Y NE ⊕ X)). In turn, {hij } is a continuous cocycle on (Ui ) with values in an abelian subgroup H ⊂ C(GL(Y NE ⊕ X)). Moreover, the subgroup of C(GL(Y NE ⊕ X)) generated by G and H is abelian, and since E ⊕ EX is topologically trivial, the principal bundle with fibre H constructed by the cocycle {hij } is trivial. These imply that (E  ⊕ EX )C(GL(Y NE ⊕X)) is isomorphic to the principal bundle B → M (H ∞ ) with fibre C(GL(Y NE ⊕ X)) defined by the cocycle {gij }. Note that due to Lemma 8.1 the principal bundle on M (H ∞ ) with fibre G constructed by this cocycle is trivial. Since it is a subbundle of B, the latter is a trivial bundle as well. Thus we have proved that (E  ⊕ EX )C(GL(Y NE ⊕X)) → M (H ∞ ) is a trivial principal bundle. Now

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according to [4, Th. 1.1] (see Section 3, Theorem C), the bundle E  ⊕ EX → M (H ∞ ) is holomorphically trivial. 2 From this lemma and Theorem 3.3 follow that there exists an element FE  ∈ LI comp (X, X ⊕ Y NS , Y NS ) such that Φ(FE  ) is the isomorphism class of the bundle E  . Moreover, applying further this theorem and Lemma 8.3 we obtain, in addition, that F ⊕ FE  ∈ LI 0comp (X 2 , X 2 ⊕ Y NE +1 , Y NE +1 ) for all F ∈ S. Since our construction depends on the fixed element E of the isomorphism class Φ(S) only, we may define NS := NE  and FS := FE  . This completes the proof of the theorem under the hypothesis of condition (8.3). Now suppose that in the above notation rank SE := NE = 0.

(8.5)

Then SE is a finite abelian subgroup of C(GL(Y 2 )), and, so, by Lemma 8.1 the bundle E ⊕ EY → M (H ∞ ) is holomorphically trivial. In this case, we set FS := IX , NS := 1. Theorem 3.3 implies easily that F ⊕FS ∈ LI 0comp (X 2 , X 2 ⊕Y NS +1 , Y NS +1 ) for all F ∈ S, as required. The proof of the theorem is complete. Remark 8.5. Suppose that C(GL(Y ))/[C(GL(Y )), C(GL(Y ))] is a finitely generated abelian group of rank r  0. Then the group GY is a finitely generated abelian group of rank  r, see Lemma 8.2. This implies that rank SE  rank GY  r, see Lemma 8.3. In this case NS  max{r, 1} for all S ∈ SOcomp (X, X ⊕ Y, Y ) \ {IX }. References [1] D. Alspach, P. Enflo, E. Odell, On the structure of separable Lp spaces, 1 < p < ∞, Studia Math. 60 (1977) 79–90. [2] J. Bourgain, O. Reinov, On the approximation properties for the space H ∞ , Math. Nachr. 122 (1983) 19–27. [3] A. Brudnyi, Stein-like theory for Banach-valued holomorphic functions on the maximal ideal space of H ∞ , Invent. Math. 193 (2013) 187–227. [4] A. Brudnyi, Holomorphic Banach vector bundles on the maximal ideal space of H ∞ and the operator corona problem of Sz.-Nagy, Adv. Math. 232 (2013) 121–141. [5] L. Carleson, Interpolations by bounded analytic functions and the corona theorem, Ann. of Math. 76 (1962) 547–559. [6] A. Douady, Une espace de Banach dont le groupe linéaire n’est pas connexe, Indag. Math. 68 (1965) 787–789. [7] I.S. Edelstein, B.S. Mityagin, Homotopy type of linear groups of two classes of Banach spaces, Funct. Anal. Appl. 4 (3) (1970) 221–231. [8] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [9] D. Husemoller, Fibre Bundles, third edition, Springer-Verlag, New York, 1994. [10] S. Lang, Differential Manifolds, Addison–Wesley, Reading, MA, 1972. [11] V. Lin, Holomorphic fiberings and multivalued functions of elements of a Banach algebra, Funct. Anal. Appl. 7 (1973) 43–51, English translation. [12] J. Lindenstrauss, On complemented subspaces of m, Israel J. Math. 5 (1967) 153–156. [13] J. Lindenstrauss, A. Pelczynski, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971) 225–249.

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