CORONA THEOREM MAREK KOSIEK AND KRZYSZTOF RUDOL Abstract. For a wide class of domains G ⊂ Cd including balls and polydisks we prove the density of their canonical image in the spectrum of H∞(G). This is the Corona Theorem and we establish first its abstract version for certain uniform algebras by duality methods. These methods include the study of idempotents and of bands of measures corresponding to Gleason parts, embeddings in bidual algebras with their Arens product and inductive limits of higher order biduals. The essential tools are properties of supports of normal measures on hyperstonean spaces, analysis on ordered Banach lattices and uniform bounds for operators used to solve the Gleason problem concerning generators of the ideal of analytic functions vanishing at a given point. Contents 1. Introduction 1 2. Reducing bands 3 3. Gleason parts and idempotents 8 4. Embedding measures, spectra and parts 9 5. A Hoffman – Rossi type theorem 13 6. Quotient algebras (Algebras of H∞- type) 17 7. Inductive limits 19 8. Embedded idempotent 22 9. Abstract Corona Theorems 27 10. Stability of G 28 11. Classical setting 29 Additional Remarks 30 arXiv:2106.15683v1 [math.FA] 29 Jun 2021 References 30 1. Introduction The original statement of Corona Problem was whether Ω is dense in the spectrum Sp(H∞(Ω)) of the Banach algebra H∞(Ω) of bounded analytic functions on a domain Ω in the space Cd, where d ∈ N. (By a domain we understand an open, connected and bounded set. C denotes the complex plane.) The natural embedding is through the evaluation functionals δz : f 7→ f(z) at the points z ∈ Ω and we consider Sp(A) as the set of nonzero linear, multiplicative functionals on A, with Gelfand (=weak- star) topology. The history of the research of the problem is outlined in [7]. The 2020 Mathematics Subject Classification. Primary 30H80, 32A65, 46J15; Secondary 32A38, 46B42, 46J20. 1 2 MAREKKOSIEKANDKRZYSZTOFRUDOL equivalent formulation (see [12, Chapter X], [9, Thm. V.1.8]) asserts the existence ∞ n of solutions (g1,...,gn) ∈ H (Ω) of the B´ezout equation (1.1) f1(z)g1(z)+ ··· + fn(z)gn(z)=1, z ∈ Ω given for any n a finite corona data, which by definition is an n-tuple (f1,...,fn) ∈ H∞(Ω)n satisfying a uniform estimate |f1(z)| + ··· + |fn(z)| ≥ c> 0, z ∈ Ω. Lennart Carleson’s solution of the problem in the case where Ω is the open unit disc D, on the complex plane [2] and most of the subsequent attempts to extend it for domains Ω ⊂ Cd were based on this equivalent formulation. Comparatively little research has followed the direct topological formulation of the problem. A concept that sheds some light on the geometry of the maximal ideal space of a uniform algebra A (denoted here as Sp(A)) is the notion of Gleason parts defined as equivalence classes under the relation kφ − ψk < 2 given by the norm of the corresponding functionals φ, ψ ∈ Sp(A). These parts form a decomposition of Sp(A). In the most important case A will be the algebra A(G) of analytic functions on a sufficiently regular domain G ⊂ Cd, which extend to continuous functions on G -the Euclidean closure of G. Then by connectedness and Harnack’s inequalities, the entire set G corresponds to a single non-trivial Gleason part. Trivial (i.e. one-point) parts correspond to the boundary points x ∈ ∂G -at least in the unit ball case. On the way to our main result we were dealing with a number of different Banach algebras. Our aim was to formulate some abstract counterparts of corona theorem. As a result we have obtained the classical result for a large class of domains, including balls and polydisks in Cd. Given a non-empty, compact Hausdorff space X, we consider a closed, unital subalgebra A of C(X) separating the points of X (a uniform algebra on X). Without loss of generality, we further suppose that A is a natural algebra in the sense that Sp(A)= X, so that any nonzero multiplicative and linear functional on A is of the form δx : A ∋ f 7→ f(x) ∈ C for some x ∈ X. The second dual of A, endowed with Arens multiplication ,,·” is then a uniform algebra, denoted A∗∗ and its spectrum will be denoted by Sp(A∗∗), while Y := Sp(C(X)∗∗) (denoted in [5] by X) is a hyperstonean space corresponding to X. Here one should bear in mind the Arens-regularity of closed subalgebras of C∗ algebras, so that e the right and left Arens products coincide and are commutative, w-* continuous extensions to A∗∗ of the product from A. By M(X) we denote the space (dual to C(X)) of complex, regular Borel measures on X, with total variation norm. We use the standard notation E⊥ for the annihi- lator of a set E ⊂ C(X) consisting of all µ ∈ M(X) satisfying f dµ = 0 for any f ∈ E. A closed subspace M of M(X) is called a band of measuresR if µ ∈M holds for any measure absolutely continuous with respect to |ν| for some measure ν ∈M. CORONA THEOREM 3 Then the set Ms of all measures singular to all µ ∈ M is also a band, forming a 1 s direct-ℓ -sum decomposition: M(X) = M ⊕1 M . We denote the corresponding Lebesgue–type decomposition summands by µM and µs, so that µ = µM + µs with µM ∈M, µs ∈Ms. A reducing band for the algebra A is the one satisfying µM ∈ A⊥ for any µ ∈ A⊥. By a representing (respectively complex representing) measure for φ ∈ Sp(A)= X we mean a nonnegative (respectively -complex) measure µ ∈ M(X) such that φ(f)= f dµ for any f ∈ A. ZX Denote by MG the smallest band containing all representing measures for a point φ from a non-trivial Gleason part G. It actually does not depend on the choice of φ ∈ G. The weak-star closure (in σ(C(X)∗∗∗,C(X)∗∗) -topology in Y ) of (the σ canonical image of) a Gleason part G ⊂ X of A will be denoted by G . The first sections contain some preliminary results stated in the form needed in the sequel, mostly known in different formulations. The first essential results are in Section 5, where using Banach lattices techniques we develop some modifications of the results of Hoffman - Rossi [13] and Seever [20]. In the next section we deduce an abstract corona theorem under some technical assumptions from the previous section. To a Gleason part of A there corresponds an idempotent g in the second dual algebra A∗∗. Its construction can be found in papers published nearly 50 years ago (like [20]), but Theorem 4.17 of the recent monograph [6] contains a version more convenient to apply in the present work. To overcome the resulting difficulties, we pass to a higher level duals. To this end we develop in Section 7 a technique of inductive limits of higher level bidual algebras and such ”tower algebras” appear helpful in dealing with these idempotents. The main purpose of this construction was to show that the support of any rep- resenting measure for a point of the canonical image of G in the spectrum of A∗∗ is contained in the Gelfand closure of that set. The difficulties follow from the fact that such inclusion can fail to hold without additional assumptions on G. Using this result we formulate another two abstract versions of corona theorem in Section 9. In the next section we verify the abstract assumptions used previously -for the concrete algebras over strictly pseudoconvex domains in Cd. This allows us to obtain the final results in Section 11. 2. Reducing bands Let A be a uniform algebra and denote by X its spectrum, i.e. let X = Sp(A). By M(X) we denote the space (dual to C(X)) of complex, regular Borel measures on X, with total variation norm. Definition 2.1. A set E ⊂ X is reducing (for A) if M(E) := {χE · µ : µ ∈ M(X)} is a reducing band in M(X) w.r. to A. Here χE · µ(∆) = µ(∆ ∩ E) and generally, we use the χ · µ to indicate the measure multiplied by some ”density” function χ. 4 MAREKKOSIEKANDKRZYSZTOFRUDOL For example, E is a reducing set for A (in a particular case) if χE ∈ A, since ⊥ ν ∈ A implies E dν = 0. R Remark 2.2. The Arens product in A∗∗ is separately continuous in w* topologies and the w* density of the canonical image in the bidual space can be used to deter- mine such product by iteration. The bidual A∗∗ of A with the Arens multiplication can be considered as a closed subalgebra of C(X)∗∗ (closed both in norm and w-* topologies). The canonical inclusion ι : A → A∗∗ is then the restriction of the cor- ∗∗ responding ιC : C(X) → C(X) ≃ C(Y ) and is both a linear and multiplicative homomorphism. Let us denote by J the isomorphism J : C(Y ) → M(X)∗. Here Y is the hyperstonean envelope of X. The property of strongly unique preduals and the corresponding results collected in [5, Theorem 6.5.1] allow us to treat the −1 action of the isomorphism J on ιC (C(X)) as identity, hence we shall omit J in all formulae (except the next one). To any measure µ ∈ M(X) we can assign the measure k(µ) ∈ M(Y ) given by the duality formula (2.1) hk(µ), fi = hJ (f),µi f ∈ C(Y ), µ ∈ M(X).