Corona Theorem

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(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 representing 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). Note, that after the mentioned identiﬁcation , for f ∈ C(X) we also have

(2.2) hk(µ), ιC (f)i = hµ, fi, i.e. µ = k(µ) ◦ ιC , µ ∈ M(X) Then we obtain an embedding k : M(X) → M(Y ). As a weak-star continuous functional on C(Y ), our k(µ) is a normal measure on Y . The normal measures form the (strongly unique) predual of the space C(Y ) [5, Theorem 6.4.2] and are w* -continuous. By the w* - density of the unit ball of C(X) in that of C(Y ) we have Lemma 2.3. The mapping k is an isometry.

Let us denote by Mφ(X) the set of all representing measures for φ on X. Lemma 2.4. If M ⊂ M(X) is a reducing band for A which contains a representing measure µ ∈ Mφ(X) then it contains all representing measures ν ∈ Mφ(X). Proof. The real measure µ − ν annihilates A, so by the reducing property, it decom- poses into two parts µ − νM ∈M and νs ∈Ms, both of them annihilating A. But νs must be equal 0 as a nonnegative annihilating measure. Now given a bounded linear functional φ on A and its representing measure ν ∈ ∗∗ Mφ(X), we apply the mapping k to obtain an “extension” of φ to A denoted by k(φ). More precisely, for F ∈ A∗∗ ⊂ C(Y ) we have

k(φ)(F )= Fdk(ν), where ν ∈ M (X). Z φ

It is easy to note the independence of the choice of ν ∈ Mφ, applying equality (2.2) to the diﬀerence of any two such measures. From the w*-continuity of k(ν) we deduce that also k(φ) is w* continuous on A∗∗, hence it is a normal measure. CORONA THEOREM 5

If f ∈ A is identiﬁed with its canonical image ιC (f), then from equality (2.2) we deduce that k(φ)(ιC (f)) = φ(f), justifying the name ”extension”. The w* density of ι(A) implies the uniqueness among w* - continuous extensions. These facts including the following Proposition are consequences of Sections 5.4 and 6.5 in [5]: Proposition 2.5. The measure k(µ) is a normal measure on Y = Sp(C(X)∗∗) for any µ ∈ M(X). Also any multiplicative linear functional φ on A has a unique w-* continuous extension acting on A∗∗. This extension k(φ) is also multiplicative. Note that for any h ∈ C(X),µ ∈ M(X) we have

(2.3) ιC (h) · k(µ)= k(h · µ). Indeed, for any F ∈ C(Y ) we have

hιC (h) · k(µ), F i = hk(µ), ιC (h)F i = hιC (h)F,µi = hF, h · µi = hk(h · µ), F i. for any f ∈ C(X). The last but one equality comes from the deﬁnition of the Arens product. This allows us to deﬁne the product g ⊙ µ of a measure µ ∈ M(X) by g ∈ C(Y ) as a measure on X given by the formula

(2.4) hg ⊙ µ, fi := hg · k(µ), ιC (f)i, f ∈ C(X) i.e. g ⊙ µ =(g · k(µ)) ◦ ιC .

In particular, for the constant function f = 1 , since ιC (1) = 1, we have (2.5) hg ⊙ µ, 1i = hg · k(µ), 1i = hk(µ),gi = hg,µi. Here we treat g ∈ C(Y ) as an element of M(X)∗. Lemma 2.6. The mapping M(Y ) ∋ ν 7→ g · ν ∈ M(Y ) is weak-star continuous.

Proof. Let να → ν in w*-topology, which means that for any f ∈ C(Y ) we have f dνα → f dν. Consequently, R R fd(g · ν )= fgdν → fgdν = fd(g · ν). Z α Z α Z Z Lemma 2.7. The mapping M(X) ∋ µ 7→ g ⊙ µ ∈ M(X) is norm continuous.

Proof. Let kµn−µk→ 0. Hence kg·k(µn)−g·k(µ)k≤kgkkk(µn)−k(µ)k→ 0 by the isometric property of k and consequently g⊙µn = g·k(µn)◦ιC → g·k(µ)◦ιC = g⊙µ since kg · k(µn) ◦ ιC − g · k(µ) ◦ ιC k = k(g · k(µn) − g · k(µ)) ◦ ιC k≤kg · k(µn) − g · k(µ)k. If we put g ⊙ M(X)= {g ⊙ µ : µ ∈ M(X)}, we have the following Remark 2.8. For µ ∈ M(X) and g ∈ C(Y ) we have k(g ⊙ µ) = g · k(µ). Conse- quently k(g ⊙ M(X)) ⊂ g · M(Y ). 6 MAREKKOSIEKANDKRZYSZTOFRUDOL

Proof. Our conclusion follows from (2.2), (2.4) and from the weak-star density of ιC (C(X)) in C(Y ). Theorem 2.9. For any idempotent g ∈ A∗∗ the subspace g ⊙ M(X) of M(X) is a reducing band (for A) and g · M(Y )= M({y ∈ Y : g(y)=1}) is a weak-star closed band in M(Y ) reducing for A∗∗. Proof. To show that g ⊙ M(X) is a band, take any h ∈ L1(µ). We need to check whether h · g ⊙ µ is in g ⊙ M(X). By a standard approximation argument, we may even assume that h ∈ C(X). Then for f ∈ C(X) we have

(2.6) hh·(g⊙µ), fi = hg⊙µ, h·fi = hg·k(µ), ιC (f)·ιC (h)i = hg·ιC (h)·k(µ), ιC (f)i

The measure g · ιC (h) · k(µ) is of the needed form g · k(h · µ) with h · µ ∈ M(X) by (2.3). Next we consider the band decomposition ν = ν1 + ν2 with respect to g ⊙ M(X) ⊥ s of a measure ν ∈ A (annihilating A) with ν1 ∈ g ⊙ M(X), ν2 ∈ (g ⊙ M(X)) . We claim that ν1 = g ⊙ ν, ν2 = (1 − g) ⊙ ν. To see this it suffices to show that µ1 := g ⊙ µ ⊥ ν2 := (1 − g) ⊙ ν for any µ ∈ M(X). Here we use the following metric characterisation (in the total variation norms) of singularity (see [5, Prop. 4.2.5]) referring to the duality between C(X) and M(X). Namely µ1 ⊥ ν2 iff kµ1 ± ν2k = kµ1k + kν2k. These two equalities can be verified separately. Let us proceed eg. with the plus sign in “±”. For the nontrivial inequality, given any ǫ > 0 it suffices to show that |hµ1 + ν2, fi| + ǫ ≥ kµ1k + kν2k for some f from the unit ball of C(X). Since the measuresµ ˜1 := g · k(µ) andν ˜2 := (1 − g) · k(ν) are singular (as g(1 − g) = 0), the sum of their values at some F in the unit ball of C(Y ) should be arbitrarily close to kµ˜1 +˜ν2k. Indeed, their total variation norms are approximated (notation ≃) by the values at some F1 and F2 respectively, but then it suffices to take F = gF1+(1−g)F2. (Note that g and 1−g are characteristic functions of disjoint sets, implying kF k ≤ 1 and hµ˜1, F i = hµ˜1, F1i, hν˜2, F i = hν˜2, F2i.) Hence we have

hg · k(µ)+(1 − g) · k(ν), F i = hµ˜1 +˜ν2, F i≃kµ˜1 +˜ν2k = kµ˜1k + kν˜2k The left hand side of this formula (by Remark 2.8) equals hk(g ⊙ µ)+ k((1 − g) ⊙ ν), F i. Since k is an isometry, by the same Remark, the right hand side approximates kg ⊙ µk + k(1 − g) ⊙ νk. By the weak-star density of the image under ιC of the unit ball of C(X) in C(Y ), we may approximate hk(g ⊙ µ)+ k((1 − g) ⊙ ν), F i by hk(g ⊙ µ)+ k((1 − g) ⊙ ν), ιC (f)i for some f from the unit ball of C(X). The latter quantity is equal to hg ⊙ µ + (1 − g) ⊙ ν, fi and is arbitrarily close to kµ˜1k + kν˜2k. Now by Remark 2.8 and by the isometric property of k the latter is equal to kk(g ⊙ µ)k + kk((1 − g) ⊙ ν)k = kg ⊙ µk + k(1 − g) ⊙ νk. Consequently, kg ⊙ µ + (1 − g) ⊙ νk = kg ⊙ µk + k(1 − g) ⊙ νk. Analogously such equality holds for the minus sign in the left-hand side. Hence the singularity of these measures follows. CORONA THEOREM 7

If f ∈ A, then using the fact that k(ν) annihilates A∗∗ and (2.4), we obtain hg ⊙ µ, fi = 0, since g · ι(f) ∈ A∗∗. Hence our band is reducing. σ Throughout the present paper we denote by E the closure in the corresponding weak-star topology of a dual space. If E ⊂ F , where F is a w*-closed subset of a σ dual space, then this E is the closure of E in the induced w* topology on F . For σ example, if E ⊂ Sp(A), then E is the Gelfand topology closure of E. σ σ Theorem 2.10. If M ⊂ M(X) is a band then M(Y ) = k(M) ⊕ k(Ms) . More- σ σ over, there are clopen disjoint subsets E, F of Y such that k(M) = M(E), k(Ms) = M(F ) and E ∪ F = Y . σ σ If M is a reducing band for A then k(M) and k(Ms) are reducing bands for A∗∗. Proof. For any pair of measures µ ∈M, ν ∈Ms we have µ ⊥ ν. Hence kk(µ) ± k(ν)k = kµ ± νk = kµk + kνk = kk(µ)k + kk(ν)k. Consequently, k(µ) ⊥ k(ν) and, as normal mutually singular measures on a hyperstonean set Y , they have disjoint clopen supports. Denote V := {supp k(µ): µ ∈ M}, W := {supp k(ν): ν ∈Ms}.

[ [σ σ The sets V, W are open and disjoint which gives V ∩ W = ∅. But V is also open σ σ since Y is hyperstonean. Hence V ∩W = ∅. Every weak-star closed band in M(Y ) σ is of the form M(Y ′) for some closed subset Y ′ ⊂ Y and k(M) is a weak-star closed band in M(Y ) for any band M ⊂ M(X) (see [16, Proposition 3]). This implies σ σ σ σ σ σ k(M) = M(V ) and k(Ms) = M(W ) since M(V ) (respectively M(W )) is the σ σ smallest weak-star closed band containing k(M) (respectively k(M)s ). Also by σ σ σ σ [16, Proposition 3] we have V ∪ W = Y . Putting E := V and F := W we obtain the desired conditions. ∗∗ ⊥ Assume now µ ∈ (A ) . This measure is a limit of some net µ = lim k(µα), where ⊥ µα ∈ A and kµαk≤kµk. This inequality can be obtained by the identiﬁcation ∗∗ ⊥ ⊥⊥⊥ ⊥ ∗∗ a s (A ) ∼ A ∼ (A ) . For every α we have a decomposition µα = µα + µα, a s s a s where µα ∈M, µα ∈M and kµαk = kµαk + kµαk. By the latter equality the nets a s ∗∗ k(µα) and k(µα) are bounded and, by the weak-star density of ι(A) in A , their σ elements annihilate A∗∗. Hence they have weak-star adherent points µa ∈ k(M) , σ µs ∈ k(Ms) annihilating A∗∗. Moreover, µ = µa + µs.

∗ Identifying isometrically A with the quotient space M(X)/A⊥ we have the following Theorem 2.11. A band M ⊂ M(X) is reducing for A iﬀ the decomposition A∗ = s M/A⊥ + M /A⊥ is a direct sum with m-norm, i.e s (2.7) k[µ1] + [µ2]k = k[µ1]k + k[µ2]k for any µ1 ∈M,µ2 ∈M .

Proof. If M is reducing, the quotient spaces M/A⊥ and M/A⊥∩M are isometric. The s same holds for M . The equivalence classes [µj] in the latter quotient spaces are 8 MAREKKOSIEKANDKRZYSZTOFRUDOL

⊥ ⊥ s consisting of elements of the form µj +νj, j =1, 2 with ν1 ∈ A ∩M, ν2 ∈ A ∩M , which are mutually singular, hence kµ1 + ν1 + µ2 + ν2k = kµ1 + ν1k + kµ2 + ν2k. Passing to inﬁmum over such νj we deduce (2.7). On the other hand, if M is not a reducing band, for some µ ∈ A⊥ its both components µM,µs in the Lebesgue decomposition w.r. to M are not annihilating: ⊥ ⊥ ⊥ µM ∈/ A ,µs ∈/ A . These measures have positive distance to A , hence if (2.7) holds, the quotient norm of µ = µM + µs should be positive too. But this is the distance of µ from A⊥, contradicting our assumption that µ ∈ A⊥. 3. Gleason parts and idempotents There is a natural decomposition of Sp(A) into Gleason parts -the equivalence classes under the relation kφ − ψk < 2 given by the norm of the corresponding functionals φ, ψ ∈ Sp(A) (see [8]). Denote by MG the smallest band containing all representing measures for a (nontrivial) Gleason part G. From Theorem 2.10 we deduce that the following decomposition holds. Corollary 3.1. σ σ s k(MG) ⊕ k(MG) = M(Y ). Let G be an arbitrary Gleason part of the spectrum, Sp(A) = X of A (referred to in what follows as ”a Gleason part of A” for the sake of brevity). The idea of ∗∗ associating to G an idempotent gG ∈ A has appeared in the Ph. D. thesis of Brian Cole and in a strengthened form most suitable for our purposes it is contained in Theorem 4.17 of [6]. Namely: ∗∗ Proposition 3.2. There exists an idempotent gG ∈ A vanishing on other Gleason ∗∗ parts G1 =6 G of A and equal 1 on G. Moreover gGgG1 =0 and if F ∈ A satisﬁes F (x0)=1 for some x0 ∈ G, while kF k ≤ 1, then FgG = gG. Let us consider a probabilistic measure η on X such that with respect to the norm of A∗ we have (3.1) dist(G, η)=2.

Choose x0 ∈ G. Hence (and by Triangle Inequality) kx0 − ηk = 2. Then there exists a sequence {fn,η} ⊂ A such that kfn,ηk ≤ 1, fn,η(x0) → 1 and fn,η(η) → −1. By ∗∗ Banach-Alaoglu Theorem, the sequence fn,η has a weak-star adherent point fη ∈ A . Moreover, kfηk ≤ 1, fη(x0) = 1 and fη(η)= −1. The points of k(G) belong to the same Gleason part of A∗∗, since k is an isometry. Hence, by Proposition 3.2 applied for F = fη, we obtain fη ≡ 1 on k(G). Since fη(η) = −1 and k(η) is probabilistic, we have fη ≡ −1 [k(η)] a.e. Deﬁne f˜η := (fη + 1)/2. So we have f˜η ≡ 0 [k(η)] a.e. Consequently, by Proposition 3.2,

(3.2) gG ≡ 0 [k(η)] a.e. Then we have

(3.3) gG · k(η)=0 CORONA THEOREM 9 for η singular to all measures representing the points in G. Applying (3.3) to measures representing any x∈ / G we have

(3.4) gG(k(x))=0 for x∈ / G.

If µ0 is any representing measure for a point in G then gG(µ0) = 1 (in the last equality gG is treated as a functional on M(X)). So, by (2.5), gG dk(µ0) = 1. Therefore it must be R

(3.5) gG · k(µ0)= k(µ0), since gG · k(µ0) and k(µ0) are probabilistic. Hence (3.5) holds also for any measure ν absolutely continuous with respect to µ0. Consequently, by Lemma 2.6 and (3.3) we obtain σ σ s Lemma 3.3. gG · ν = ν for ν ∈ k(MG) , gG · ν = 0 for ν ∈ k(MG) and hence σ σ σ s gG · k(MG) = k(MG) , gG · k(MG) = {0}.

Again, by (2.3),(2.4), (3.5) and (3.3), gG ⊙ µ equals µ (respectively 0) for all measures absolutely continuous with respect to (respectively singular to all) measures representing points in G and by Lemma 2.7 we obtain

s Lemma 3.4. gG ⊙µ = µ for µ ∈MG, gG ⊙µ =0 for µ ∈MG and gG ⊙MG = MG, s gG ⊙MG = {0}. By Theorem 2.9 and Lemmas 3.3, 3.4 we deduce the following

Theorem 3.5. The band MG = gG ⊙ M(X) is a reducing band with respect to A, that is for each µ ∈ M(X) we have a unique decomposition µ = µG + µs, where ⊥ ⊥ µG ∈ gG ⊙ M(X) and µs ∈ (1 − gG) ⊙ M(X). If µ ∈ A then also µG,µs ∈ A . σ σ s The bands gG ·M(Y )= k(MG) and (1−gG)·M(Y )= k(MG) are also reducing in M(Y ) with respect to A∗∗.

4. Embedding measures, spectra and parts In Y we introduce the equivalence relation deﬁned for x, y ∈ Y by x ∼ y ⇔ f(x)= f(y) for f ∈ A∗∗. We have the canonical surjection ∗∗ Π1 : Y → Y/∼ ⊂ Sp(A ).

This mapping Π1 transports (by push-forward) measures, deﬁning the map

Π:˜ M(Y ) ∋ µ 7→ Π(˜ µ) ∈ M(Y/∼), where

(4.1) fd Π(˜ µ)= (f ◦ Π ) dµ, f ∈ C(Y/ ). Z Z 1 ∼ 10 MAREKKOSIEKANDKRZYSZTOFRUDOL

∗∗ Deﬁnition 4.1. We have an embedding j : M(Sp(A)) → M(Π1(Y )) ⊂ M(Sp(A )) given by the formula

(4.2) j(µ)= ν if f dν = (f ◦ Π ) dk(µ) for f ∈ C(Y/ ), Z Z 1 ∼ where the measure j(µ) is concentrated on Π1(Y ). The considered dependencies are illustrated below by diagrams: XYk

Π1 j ∗∗ Y/∼ ⊂ Sp(A ) M(X) k M(Y )

e Π j ∗∗ M(Y/∼) ⊂ M(Sp(A )) with the dotted arrows denoting canonical surjections. We have also the restriction mapping

∗∗ C(Sp(A )) ∋ f 7→ f|Y/∼ ∈ C(Y/∼). Lemma 4.2. The projection Π˜ preserves the absolute continuity: µ ≪ ν implies Π(˜ µ) ≪ Π(˜ ν). ˜ −1 −1 Proof. If 0=(Π(ν))(E)= ν(Π1 (E)) for some Borel set E, then 0 = µ(Π1 (E)) = (Π(˜ µ))(E). We also need the following lemma ∗∗ Lemma 4.3. If E ⊂ Y is a reducing set for A then χE = χΠ1(E) ◦ Π1. −1 Proof. On the contrary assume that there is x ∈ Π1 (Π1(E)) \ E. Then we can ﬁnd y ∈ E such that Π1(x) = Π1(y). Hence for point-mass measures δx, δy we have ∗∗ ⊥ ∗∗ δx − δy ∈ (A ) . Since E is a reducing set δx and δy must also annihilate A which is impossible for probabilistic measures. We need the following counterpart of Theorem 2.10 σ σ s Theorem 4.4. If a band M in M(Y/∼) is reducing for A then j(M) and j(M ) σ σ ∗∗ s are reducing bands for A and M(Y/∼) = j(M) ⊕ j(M ) . Moreover, there are clopen disjoint subsets E,˜ F˜ of Y/∼ such that E˜ ∪ F˜ = Y/∼ and σ σ σ σ (4.3) j(M) = M(E˜)= Π(˜ k(M) ), j(Ms) = M(F˜)= Π(˜ k(Ms) ). Proof. By Theorem 2.10 there exist reducing clopen sets E, F ⊂ Y such that s k(M) = χE · M(Y ) and k(M ) = χF · M(Y ). Moreover E ∪ F = Y and we σ σ have the equalities k(M) = M(E), k(Ms) = M(F ). By Lemma 4.3 we have ∗∗ χE = χΠ1(E) ◦ π1, because E is a reducing set for A . Hence χΠ1(E), as a factori- sation of χE is also continuous (in the quotient topology of Y/∼). Let us deﬁne CORONA THEOREM 11

E˜ = Π1(E) and F˜ = Π1(F ). Since the sets E, F are “saturated w.r.to ∼”, we have σ σ σ E˜ ∩ F˜ = ∅. We have that j(M) = Π(˜ k(M) ), since Π(˜ k(M) ) = Π(˜ M(E)) and the latter set is w*-closed by the equality Π(˜ M(E)) = M(E˜). To show the last equality note that for µ ∈ M(E) the support of Π(˜ µ) is contained in E˜, showing the ⊂ relation, (since in the opposite case for y0 ∈ Y˜ \ E˜ using (4.1) for a continuous function equal 0 on E˜ and nonzero at a neighbourhood of y0 we would have a contradiction.) The opposite inclusion can be obtained using Hahn - Banach theorem as follows. Any measure ν on E˜ is a continuous linear functional on C(E˜) contained isometrically (via f 7→ f ◦Π1) in C(E). Norm-preserving extension yields a measure ν1 on E satisfying Π(˜ ν1)= ν.

We can also deﬁne the product g ⊙ µ of a measure µ ∈ M(X) by g ∈ C(Y/∼) by the formula (4.4) hg ⊙ µ, fi := hg · j(µ), ι(f)i, f ∈ C(X) i.e. g ⊙ µ =(g · j(µ)) ◦ ι.

∗∗ The function gG ∈ A (deﬁned in Proposition 3.2) is an idempotent on Y . Hence, being constant on the ﬁbers of Π1, the function gG can be treated as an idempotent ∗∗ on Y/ ∼. Hence its Gelfand transformg ˆG ∈ C(Sp(A )) is also an idempotent in ∗∗ C(Y/∼), because we have a natural inclusion Y/∼ ⊂ Sp(A ). In what follows we omit the symbolˆtreating gG as an element of C(Y/∼). So, by (3.4), we have

(4.5) gG(j(x))=0 for x∈ / G. Also we have the following consequence of Theorem 4.4.

Theorem 4.5. The band MG = gG ⊙ M(X) is a reducing band with respect to A, that is for each µ ∈ M(X) we have a unique decomposition µ = µG + µs, where ⊥ ⊥ µG ∈ gG ⊙ M(X) and µs ∈ (1 − gG) ⊙ M(X). If µ ∈ A then also µG,µs ∈ A . σ σ s The bands gG · M(Y/∼) = j(MG) and (1 − gG) · M(Y/∼) = j(MG) are also ∗∗ reducing in M(Y/∼) with respect to A .

∗∗ Corollary 4.6. If z ∈ Sp(A ) then all its representing measures on Y/∼ belong σ σ s either to j(MG) or to j(MG) .

Proof. Let ν be a representing measure for z. Denote by νG its part belonging σ σ s to j(MG) and by νs its part belonging to j(MG) . From their construction in Theorem 4.5, both νG and νs are nonnegative and if both were nonzero, then we would have kνGk < 1. Hence for gG we have 0 < gG(z) < 1 since gG(z) = gG dν and gG| s σ ≡ 0. But this leads to a contradiction with the the fact that gRG is an MG idempotent. σ σ s If z would have two representing measures ν1 ∈ j(MG) , ν2 ∈ j(MG) then we would have 1 = gG dν1 = gG(z)= gG dν2 = 0 which is a contradiction. R R We also have (4.6) j = Π˜ ◦ k. 12 MAREKKOSIEKANDKRZYSZTOFRUDOL

Using the canonical mapping ι : A → A∗∗ and its adjoint ι∗ : A∗∗∗ → A∗ we have for f ∈ A and ψ ∈ A∗∗∗ hψ, ι(f)i = hι∗(ψ), fi. Applied to ψ = z ∈ Sp(A∗∗) ⊂ A∗∗∗ this yields ∗ ∗ ∗ (4.7) (ι(f))(z)= f(ι (z))=(f ◦ ι )(z), i.e. ι(f)= f ◦ (ι |Sp(A∗∗)). Let us now deﬁne the following extension Λ of ι: ∗ ∗∗ (4.8) Λ : C(Sp(A)) ∋ f 7→ f ◦ (ι |Sp(A∗∗)) ∈ C(Sp(A )) Then its adjoint acts on µ ∈ M(Sp(A∗∗)) by the formula ∗ ∗ (4.9) hΛ µ, fi = hµ, Λfi = hµ, f ◦ (ι |Sp(A∗∗))i, f ∈ C(Sp(A)). ∗∗ ∼ This map Λ can be used to express the adjoint of ιC : C(X) → C(X) = C(Y ) as follows: ∗ ∗ ˜ (4.10) ιC =Λ ◦ Π.

∗ ∗ To simplify the notation we shall write simply ι in place of ι |Sp(A∗∗) from now on. By applying Π˜ to both sides of the formula (2.3) from (4.6) we obtain for h ∈ C(X) j(h·µ)= Π(˜ k(h·µ)) = Π(˜ ιC (h)·k(µ)) = Π((Λ(˜ h)◦Π1)·k(µ)) = Λ(h)·Π(˜ k(µ)) = Λ(h)·j(µ). The last but one equality is veriﬁed using (4.2) by integrating f ∈ C(Sp(A∗∗)) as follows: fd(Π((Λ(˜ h)◦Π )·k(µ))) = f ◦Π d((Λ(h)◦Π )·k(µ)) = (f ·Λ(h))◦Π dk(µ). Z 1 Z 1 1 Z 1 Therefore for any h ∈ C(X) and µ ∈ M(X) we obtain (4.11) Λ(h) · j(µ)= j(h · µ). Lemma 4.7. The mapping j deﬁned in (4.2) is an isometry. It has its left inverse equal to Λ∗. Namely, Λ∗(j(µ)) = µ for µ ∈ M(Sp(A)). Similarly, the right inverse to ι∗ is j, namely (ι∗ ◦ j)(z)= z for any z ∈ Sp(A). Proof. The total variation norm kj(µ)k is the least upper bound of |hh, j(µ)i| with h varying through {h ∈ C(Y/∼): khk ≤ 1}. This norm is not less than kµk = sup{|hf,µi| : f ∈ C(X), kfk ≤ 1}, since the relation x ∼ y implies the relation f(x)= f(y) for f ∈ A. Because Π˜ does not increase the norms and k is (by Lemma 2.3) an isometry, the opposite inequality (kj(µ)k≤kµk) is also true. For f as in (4.9) and for µ ∈ M(Sp(A)) , we have hΛ∗(j(µ)), fi = hj(µ), Λfi = hj(µ), f ◦ ι∗i = hf ◦ ι∗,µi = hΛ(f),µi = hµ, fi. The last but one equality results from (4.9). The second relation follows from the following identities valid for f ∈ A, z ∈ Sp(A): hf, ι∗(j(z))i = hι(f), j(z)i = hf, zi. The last equality corresponds to (2.2) applied for f ∈ A. CORONA THEOREM 13

5. A Hoffman – Rossi type theorem At a key point of further considerations we need an extension of a Hoffman-Rossi’s result, Theorem 6.1 in [4], but the one reaching beyond the setting of that paper. In this purpose we apply duality for real Banach lattices, that will replace some limit passages under the integral sign. Let A be a uniform algebra, such that A = B∗ for some Banach space B and let ϕ ∈ Sp(A). (In our setting A = A∗∗). Since ϕ has real representing measures, the formula ϕ(Re(u)) := Re(ϕ(u)), u ∈A defines ϕ also on the real space ReA := {Re(u): u ∈ A} as well as on its real subspace ReAϕ , where Aϕ := {v ∈A : ϕ(v)=0} is the kernel of ϕ. Now the Riesz- ∗ Kantorovitch theorem (Theorem 1.16 [1]) implies that both MR(X)= CR(X) and its dual, W := CR(Y ) are Dedekind complete (every nonempty bounded above subset has a supremum). In the latter space the lattice suprema of pairs coincide with point-wise least upper bounds (i.e. with suprema in the lattice RY with product order). The important property is that the order-dual coincides for Banach lattices W with its norm-dual W ∗, the set of norm - bounded linear functionals (Theorem 1.33 [1]). This applies also for CR(X), whose dual is MR(X), the space of real-valued ∗ bounded measures on X. In turn, MR(X) = CR(Y ) and MR(X) is weak-star dense ∗ (after canonical embedding) in MR(Y )= CR(Y ) . Let us begin with the easier part of Hoffman-Rossi Theorem, which in our situation is the following implication, relating some positivity condition with approximation by elements of the sum P +Re Aϕ. Here P denotes the cone of nonnegative functions in C(Sp(A)) and

Re Aϕ = {Re u : u ∈Aϕ}. Lemma 5.1. If u ∈ Re C(Sp(A)) has non-negative integrals udλ ≥ 0 against all measures λ representing ϕ ∈ Sp(A), then u lies in the norm-closureR of the cone P + Re Aϕ. Proof. We use similar arguments as in the proof of the version of Hoffman-Rossi Theorem given in [20]. Assume u is not in this closure, u 6∈ P + Re Aϕ. Then there is a measure µ ∈ Re M(Sp(A)) such that µ(u) < inf{µ(v): v ∈ P + Re Aϕ}. Since P + Re Aϕ is a cone – the infimum must be 0. In particular, for v ∈ P we have µ(v) ≥ 0, hence µ ≥ 0, µ =6 0 and we may assume that µ(1) = 1. Since Re Aϕ is a linear subspace of Re C(Sp(A)), and since µ(v) ≥ 0 for v ∈ Re Aϕ, µ must vanish on Re Aϕ. Since µ is real, µ vanishes on Aϕ, and since µ(1) = 1, µ is a representing measure for ϕ, which leads to a contradiction. We need also the following lemma comparing upper limits taken point-wise and in the lattice sense (where modulus is also a lattice modulus): Lemma 5.2. Let E be a Banach lattice, E∗ its dual and let h ∈ E∗ be a weak-star ∗ adherent point of an order – bounded sequence {fn} of functionals in E . Then ∗ lim sup |fn| defined as infk(supn≥k |fn|) exists in E and for any positive ν ∈ E we 14 MAREKKOSIEKANDKRZYSZTOFRUDOL have

(5.1) lim sup |fn|(ν) ≤ (lim sup |fn|)(ν). Moreover, in the lattice sense we have

(5.2) |h| ≤ (lim sup |fn|).

∗ Proof. Since E as a dual Banach lattice is Dedekind complete, Φk := supn≥k |fn| ∗ ∗ exists in E for every k ≥ 0 and similarly Φ = lim sup |fn| := infk Φk exists in E . We have |fn|(ν) ≤ Φn0 (ν) for n ≥ n0, ν ∈ E, ν ≥ 0. Hence supn≥n0 (|fn|(ν)) ≤ Φn0 (ν). By the symmetric version of [19, Lemma 1.7.4], lim sup |fn| is a weak-star limit of ∞ the sequence {Φk}k=1. Hence Φk(ν) → Φ(ν) for k →∞, ν ∈ E. On the other hand supn≥k(|fn|(ν)) → lim sup(|fn|(ν)) for k →∞. This implies (5.1) Let µ ∈ E, µ ≥ 0. Let ν ∈ E, |ν| ≤ µ. Applying (5.1) to the right-side term of

|h(ν)| ≤ lim sup |fn(ν)| ≤ lim sup(|fn|(|ν|)) we conclude that

|h(ν)| ≤ (lim sup |fn|)(|ν|) ≤ (lim sup |fn|)(µ). Hence

sup |h(ν)| ≤ (lim sup |fn|)(µ). |ν|≤µ But the left hand side of the last inequality is equal to |h|(µ) (see [1, Corollary 3.26]), which ﬁnishes the proof.

We are going to apply the surjection Π1 deﬁned at the beginning of Section 4, ∗∗ where Π1 : Y → Y/∼ ⊂ Sp(A ). Here we need to distinguish the set ∆ of those linear-multiplicative functionals on A∗∗ that fail to extend to elements of Sp(C(X)∗∗). Denote ∗∗ ∆ := Sp(A ) \ Y/∼ and Z := Y ∪ ∆.

∗∗ Let us deﬁne a mapping Π˜ 1 : Z → Sp(A ) as follows:

Π˜ 1(z) = Π1(z) for z ∈ Y, Π˜ 1(z)= z for z ∈ ∆. We now introduce a specific topology on Z. For any closed subset E of Sp(A∗∗) ˜ −1 −1 define E := Π1 (E) = Π1 (E) ∪ (∆ ∩ E). Denote by T the weakest topology whose family of closed sets contains the following familyτ ˜ of subsets of Z: e τ˜ := {(W ∩ E) ∪ (∆ ∩ E): W is closed in Y, E is closed in Sp(A∗∗)}. If we take W = eY in the definition ofτ ˜, then we see that for a closed set E in ∗∗ ˜ ˜ −1 Sp(A ) the set E = Π1 (E) is closed in Z. Lemma 5.3. The restriction of T to Y (respectively to ∆) coincides with the initial topologies in these subsets. The set Y is compact in T . The mapping Π˜ 1 : Z → Sp(A∗∗) is continuous with respect to the T topology on Z. CORONA THEOREM 15

Proof. It follows directly form the definition ofτ ˜ that Π˜ 1 is continuous and that the restriction of T to ∆ coincides with the initial topology of ∆. To compare the two topologies on Y take an arbitrary set W closed in Y and note ^ ^ that W =(W ∩ Sp(A∗∗)) ∩ Y and the latter set is equal ((W ∩ Sp(A∗∗)) ∪ ∆) ∩ Y , which is T -closed. Consequently, if W is closed in Y then it is also closed in T restricted to Y . Conversely, the intersections with Y of sets inτ ˜ come from intersections of closed sets in Y , hence are closed and these both topologies of Y must be equal. The significance of this topology is explained by the following fact Lemma 5.4. Let u : Sp(A∗∗) → R be a continuous function.Then its extension u˜ : Z → R defined by setting u˜ := u ◦ Π˜ 1 is T continuous. σ Since the set k(G) is compact and Π1 is w*-continuous, we obtain the following σ σ Lemma 5.5. Π1(k(G) )= j(G) . ∗∗ In what follows along with Π1, we use another natural projection π : Sp(A ) → X defined so that π(y) = x if δy|A = δx. This notation is justified if one considers A ∗∗ as a subset of A and Sp(A) as the set {δx : x ∈ X}. The main result of the present section concerns a nontrivial Gleason part G ⊂ X = Sp(A) of the spectrum of our uniform algebra A, satisfying the following conditions Assumption 5.6. (1) j(G)= π−1(G) ∗∗ (2) any measure µ on Sp(A ) representing a point y0 ∈ j(G) has supp(µ) ⊂ σ j(G) (3) there is a sequence of compact subsets Km ⊂ G such that Km ⊂ int(Km+1), ∗∗ Km = G and such that for any norm-bounded by 1 sequence fj ∈ A Swith |fj(x)| → 1 for all x ∈ j(G) the functions |fj| converge uniformly on any fixed j(Km)

Remark 5.7. Observe that by (1) we have j(Km)= j(G). Instead of an equivalence, we shall only useS one implication contained in the first part of the following extension of the Hoffman - Rossi result (see [4], [13]). Theorem 5.8. Let A be a uniform algebra, whose spectrum has a nontrivial Gleason part G ⊂ X = Sp(A) satisfying the Assumption 5.6 and consider a point ϕ ∈ G, so ∗∗ that j(ϕ)= y0 ∈ Sp(A ) . σ (i) If u ∈ Re C(Sp(A∗∗)) satisfies u ≤ 0, u ≡ 0 on j(G) , then there is h ∈ A∗∗ such that khk ≤ 1, ϕ(h)=1 and |h| ≤ eu˜ in the lattice sense in M(X)∗. ∗∗ σ (ii) If µ is a measure on Y/∼ representing a point z0 ∈ Sp(A ) \ j(G) , then µ σ σ is singular to j(MG) = M(j(G) ). Proof. Let u be like in (i). By Assumption 5.6 (2) the integrals considered in Lemma 5.1 are zero, hence nonnegative and consequently u ∈ P + Re Aϕ. One has a se- ∗∗ ∗∗ quence {gn} ⊂ (A )ϕ and functions vn ∈ C(Sp(A )), vn ≥ 0 such that vn + Re gn 16 MAREKKOSIEKANDKRZYSZTOFRUDOL tends uniformly to u on Sp(A∗∗). Hence for any x ∈ Sp(A∗∗) the upper limit of |egn (x)| = eRe gn (x) ≤ e(vn+Re gn)(x) satisfies (5.3) lim sup |egn (x)| ≤ lim sup e(vn+Re gn)(x) = lim e(vn+Re gn)(x)= eu(x) . As in Lemma 5.4 we can extend a continuous function on Sp(A∗∗) to a continuous −1 function on Z, constant on every fiber Π1 ({x}) for x ∈ Y/∼ (omitting the tilde over such a function for the sake of convenience). Every function gn, as an element ∗∗ −1 of A , is also constant on every fiber Π1 ({x}). So the estimate (5.3) holds on Z. ∗∗ Let x0 ∈ Sp(A ) and let µ be its representing measure concentrated on Y/∼. ∗∗ Such a measure exists since Y/∼ contains the Shilov boundary of A . Then from the equalities eRehgn,µi = |ehgn,µi| = |egn(x0)| we obtain by (5.3) (5.4) lim sup eRehgn,µi = lim sup eRe gn(x0) ≤ eu(x0).

The sequence Re gn is bounded from above. The space C(Y )R identified with ∗∗ Re g (C(X)R) is Dedekind complete, as a dual Banach lattice. Hence lim sup(e n ) understood as a lattice upper limit is a continuous function on Y . gn The sequence e is bounded by the boundedness from above of Re gn. Let h be its weak-star adherent point. Notice that since gn(ϕ) = 0, we have h(ϕ) = 1. Moreover, by Lemma 5.2 (5.5) |h| ≤ lim sup |egn | = lim sup(eRe gn ), where the formula holds in the lattice sense. The lattice considered here is the real dual of M(X). Consequently the above formula holds also on M(Y )=(M(X)∗∗). We have Re gn ≤ Re gn + vn, where the right hand side tends uniformly to u. For every ε > 0 we can find n0 ∈ N such that k Re gn + vn − uk < ε for n > n0. Consequently Re gn ≤ u + ε for n > n0 on Z and also in the lattice sense if we consider both sides of the equality as elements of C(Y ). By letting ǫ → 0 we see that the inequality (5.6) lim sup |egn | = lim sup(eRe gn ) ≤ eu holds in the lattice sense in C(Y ) -and by w*-density – on M(Y ). By the uniform convergence and by uniform continuity of the exponential function on bounded from above sets, also evn+Re gn tend uniformly to eu. Recall that eu ≤ 1. Hence (by the positivity of vn) one find constants Mn ≥ 1 with lim Mn = 1 such Re gn that e ≤ Mn. gn gn To apply [8, VI2.1(iv)] we can replace e by e /Mn, having norms ≤ 1, while gn(x) Re gn(x) gn(x) |e /Mn| → 1 for x ∈ j(G). Hence for x ∈ j(G) we have e = |e | → 1, gn since e (ϕ) = 1. Moreover, by Assumption 5.6 (3) there are compact subsets Km of Re gn G such that G = m Km and the convergence e → 1 is uniform on each j(Km). By Remark 5.7 weS have j(G)= m j(Km) and consequently −1 S −1 k(G) = Π1 (j(G)) = Π1 (j(Km)) = k(Km). [m [m Take an arbitrary ε > 0. By the same Assumption, for every m we can find a −1 continuous function fm : Z → [0, 1 − ε] equal 1 − ε on k(Km) = Π1 (j(Km)) and CORONA THEOREM 17 zero on Z \ int(k(Km+1)). The uniform convergence to 1 on k(Km+1) implies that Re gn e ≥ 1 − ε on k(Km+1) for n large enough. Therefore our conditions imposed Re gn on fm imply that for any n large enough e ≥ fm everywhere on Z. The lattice Re gn lim inf e is therefore greater than or equal to fm. By letting m →∞ and ε → 0 we finally deduce that this lattice lower limit is equal 1 on j(G) -and as a continuous σ σ function - also on k(G) . By (5.6) we have lim sup(eRe gn ) ≤ 1 on k(G) and since the point-wise limit lies between the lattice lower- and upper limits, we deduce that σ the point-wise limit satisfies lim eRe gn =1 on k(G) . Hence also σ (5.7) lim Re gn =0 on j(G) . To prove (ii) note at the beginning that each Gleason part of A∗∗ containing j(G) must be contained in π−1(G). So, by the assumptions, j(G) is the whole Gleason ∗∗ part of A . By the Assumption 5.6 (2), any representing measure for y0 ∈ j(G) is σ σ concentrated on j(G) . Observe that j(G) as a weak-star compact set is also closed ∗∗ σ ∗∗ in Gelfand topology of Sp(A ). Since z0 ∈/ j(G) , we can find u ∈ Re C(Sp(A )) σ taking values in the interval [−1, 0] such that u ≡ 0 on j(G) and u(z0)= −1. The function u satisfies the assumptions of Lemma 5.1 applied for the algebra A = A∗∗. σ Hence, if z0 would have a representing measure µ on j(G) , then for the sequence gn chosen for this u as at the beginning of the proof we would have by (5.7)

lim Re gn(z0) = lim Re gn dµ = lim Re gn dµ =0. n→∞ n→∞ Z Z

u(z0) −1 But e = e which contradicts (5.4) for x0 = z0. Note that by Corollary 4.6 if σ at least one measure on (Y/∼) representing z0 does not belong to j(MG) then all σ measures on (Y/∼) representing z0 are singular to j(MG) . Remark 5.9. The part of Theorem 5.8 (i) concerning h ∈ A∗∗ is not used in the remaining part of our paper, but it seems proper to include it here. Indeed, it has the nearest formulation to one of the equivalent conditions of the band version of the Hoﬀman-Rossi theorem in [4].

6. Quotient algebras (Algebras of H∞- type) One of the approaches to study properties of H∞(G), the algebra of bounded analytic functions on a given domain G ⊂ CN , is to consider its abstract counterpart, ∞ the algebra H (MG) corresponding to a nontrivial Gleason part G for a function algebra A. The band of measures MG corresponding to G is now considered as a Banach ∗ space, so that its dual space MG carries its weak-star topology. By

∞ H (MG) ∗ we have denoted the weak-star closure of A in MG. Note that there is a unique ∞ meaning for the value f(x) of f ∈ H (MG) at any point x ∈ G. ∞ Proposition 6.1. The algebra H (MG) is isometrically and algebraically isomor- ∗∗ phic to A / ⊥ ∗∗ . MG∩A 18 MAREKKOSIEKANDKRZYSZTOFRUDOL

∗∗ Proof. Let g ∈ A . Then there is a net {gα} ⊂ A such that hgα,µi→hg,µi for µ ∈ M(X). Denote by f the restriction of g to MG (note that MG is a subspace of M(X) and A∗∗ ⊂ C(X)∗∗ = M(X)∗, cf. Remark 2.2). Then f depends only on ∗∗ the equivalence class [g] ∈ A / ⊥ ∗∗ of g and also hg ,µi→hf,µi for µ ∈ M , MG∩A α G ∞ hence f ∈ H (MG). This means that the mapping ∗∗ ∞ (6.1) A / ⊥ ∗∗ ∋ [g] 7→ g|M ∈ H (MG) MG∩A G is well deﬁned. ′ ′ ′ From the obvious relations kg k≥kg |MG k for any g ∈ [g], we have k[g]k ≥ kg|MG k. ∞ On the other hand, let f ∈ H (MG). By Lemma 2.2 of [15], we can ﬁnd a net {fα} ⊂ A such that fα → f and kfαk≤kfk for each α. Hence, by Banach-Alaoglu ∗∗ Theorem, the net {fα} has an adherent point g ∈ A . So hf,µi = hg,µi for µ ∈MG ∗∗ and kgk≤kfk. Consequently the norm of [g] in A / ⊥ ∗∗ is less than or equal MG∩A to kfk. This means that the mapping (6.1) is isometric and surjective. By the Nagasawa’s Theorem (see [22] V.31), it is also isomorphic (preserves both linear and multiplicative structure).

∞ Corollary 6.2. G can be identiﬁed as a subset of the spectrum of H (MG).

∞ By Proposition 6.1, we identify the algebra H (MG) with the quotient algebra ∗∗ ⊥ ∗∗ ∗∗ A / ⊥ ∗∗ . Note that M ∩ A is a weak-star closed ideal in the algebra A . MG∩A G (see [16, Proposition 3, (4)]). ∗∗ As in the remarks preceding Theorem 4.5, the function gG ∈ A (deﬁned before ∗∗ Lemma 3.3) will be treated as a function on Y/∼ ⊂ Sp(A ).The fact that gG is also an idempotent on Y/∼ is useful in proving the following result:

∞ Proposition 6.3. The spectrum of H (MG) can be identified with the set of those ∗∗ σ elements of Sp(A ) which have all representing measures in j(MG) . σ Proof. If ν ∈ j(MG) for all ν representing for z then ∗∗ ⊥ ∗∗ ⊥ (6.2) (A ∩MG)(z) ≡ 0, in other words z(A ∩MG) ≡ 0 where z is considered as a functional on A∗∗. This means that z is also a well defined ∗∗ element of the spectrum of A / ⊥ ∗∗ . MG∩A ∗∗ On the other hand assume now that z is a well defined functional on A / ⊥ ∗∗ . MG∩A It corresponds to a functionalz ˜ on A∗∗ obtained by composing z with the canonical ∗∗ ⊥ surjection. Then we must havez ˜ ≡ 0 on A ∩MG. Ifz ˜ would have a representing σ measure ν∈ / j(MG) then by Corollary 4.6, it would have a representing measure σ s ν ∈ j(MG) . Now by Theorem 4.5 we have gG = 1 a.e. with respect every measure σ ∗∗ ⊥ in j(MG) , and consequently 1 − gG ∈ A ∩ MG which means that 1 − gG is ∗∗ equivalent to zero in the quotient algebra A / ⊥ ∗∗ . Since we have assumed that z MG∩A ∗∗ is a well defined functional on A / ⊥ ∗∗ we conclude that 1−gˆ (˜z) = 0. This leads MG∩A G σ s to a contradiction since gG ≡ 0 on j(MG) and consequentlyg ˆG(˜z)= gG dν = 0. R CORONA THEOREM 19

From the last Proposition and from Theorem 5.8 (ii) by duality arguments we deduce the following result. Theorem 6.4 (First Abstract Corona Theorem). Assume that G satisﬁes the As- ∞ sumption 5.6. Then the canonical image of G is dense in the spectrum of H (MG) in its Gelfand topology. ∞ Proof. Let x ∈ Sp(H (MG)). By Proposition 6.3 we can treat x as an element of ∗∗ σ Sp(A ) having a representing measure νx ∈ j(MG) . By Theorem 5.8 (ii), we have σ ∞ x ∈ j(G) . Hence as an element of Sp(H (MG)), x is in the closure of G. 7. Inductive limits We will also consider higher - level biduals with their Arens products, deﬁned recursively for n ∈ Z+ by ∗∗ ∗∗ A0 = A, A1 = A ,..., An+1 =(An) and linked by the corresponding canonical injections ιn : An → An+1 and by

ιm,n = ιm−1 ◦ ιm−2 ◦···◦ ιn : An → Am, n < m, ιn,nf = f.

Note that ιn+1,n = ιn. Since these maps are isometries and unital homomorphisms, many properties of the inductive limits of C∗ - algebras (cf.[21]) remain valid in this setting. Let us recall that if we consider An as a subset of An+1, then

A := LimAn −→ ∞ is the completion of the union n=1 An in the natural norm defined using the isometry of the mappings ιn . Alternatively,S there is a more general approach. One defines on the disjoint sum ({n} × An) the equivalence relation S (n, f) ∼ (m, h) ⇔ ( ιk,n(f)= ιk,m(h) for k = max{n, m}) and then one passes to the quotient space. The algebraic operations +, · are defined on the equivalence classes of these elements so that

[(n, f)] + [(m, h)] = [(k, ιk,n(f)+ ιk,m(h))], [(n, f)] · [(m, h)] = [(k, ιk,n(f) · ιk,m(h))], similarly for the multiplications by scalars. The norm is deﬁned by k[(n, f)]k := kfk, making the canonical mappings

αn : An ∋ f 7→ [(n, f)] ∈ LimAn −→ isometric, unital homomorphisms. The inductive limit, LimAn is the completion of −→ this quotient space together with the sequence of canonical isometric embeddings ι∞,n : An →A. Proposition 7.1. The limits of spectra exhibit the ,,inductive - projective” duality:

(7.1) Sp(LimAn) = Lim Sp(An) . −→ ←− ∗ The projective system is connected by the mapings ιn : Sp(An+1) ∋ ψ 7→ ψ ◦ ιn ∈ Sp(An) and the equality means a canonical isometric isomorphism, carrying the product topology of Gelfand topologies of Sp(An) onto the Gelfand topology of Sp(LimAn). −→ 20 MAREKKOSIEKANDKRZYSZTOFRUDOL

Proof. The projective limit consists of sequences ϕ =(φn) ∈ Sp(An) satisfying ∗ Z Q (7.2) ιn(φn+1)= φn for any n ∈ +, ∗ where (ιn(φn+1))(f)= φn+1(ιn(f)) for any f ∈ An. ∗ Consequently also ιk,n(φk) = φn for any n ≤ k. The corresponding functional ϕ acts on LimAn assigning the value φn(fn) to [(n, fn)]. Due to (7.2), the latter is −→ independent of the representative of the equivalence class. It extends by continuity (having norm 1) onto the whole of LimAn and is multiplicative and linear. −→ Conversely, given ϕ ∈ Sp(LimAn) we obtain the sequence of homomorphisms −→ φn = ϕ ◦ αn satisfying (7.2) and belonging to Sp(An). Finally, the product topology describes the convergence at each coordinate separately, which implies the point-wise convergence on αn(An) -a dense subset of LimAn. Since the norms of linear multiplicative functionalsS are bounded by 1, this −→ convergence takes place also at the whole inductive limit. We also have a sequence of C∗-algebras: ∗∗ C(X)= C(Sp(A)),C(Sp(A )),...,C(Sp(An)),... Like in the case of A, we can extend the deﬁnition (4.8) of Λ from Section 4 to ∗ obtain Λn : C(Sp(An)) → C(Sp(An+1)) : f 7→ f ◦ ιn and for n < m we deﬁne

Λm,n =Λm−1 ◦···◦ Λn : C(Sp(An)) → C(Sp(Am)). These objects form an inductive system of isometric embeddings. Then by using ∗ the surjections Λ : M(Sp(A1)) → M(Sp(A0)) deﬁned in (4.9) and their analogues ∗ ∗ Λn : M(Sp(An+1)) → M(Sp(An)) and Λm,n : M(Sp(Am)) → M(Sp(An)) we obtain a projective system of spaces M(Sp(An)). To avoid some technicalities we restrict our attention to the closed unit balls M(Sp(An))[1] of these measure spaces. Proposition 7.2. We have (up to natural isomorphisms) the equalities

(i) C(Sp(A)) = LimC(Sp(An)), −→ (ii) M(Sp(A))[1] = LimM(Sp(An))[1]. ←−

Proof. The inductive limit algebra LimC(Sp(An)) is clearly symmetric (contains −→ complex conjugates of its elements) and separates the points of Sp(A). Indeed, the latter set (as a projective limit) is a compact subset of the cartesian product Sp(An). Now a pair (ζ,ξ) of its distinct points has at least one diﬀerent (say, kQ-th) coordinate (ζk,ξk), separated by some fk ∈ Ak. The image of this fk in LimC(Sp(An)) separates this pair of points. By the Stone - Weierstrass theorem, −→ we deduce (i). The measures from the unit balls M(Sp(An))[1] are linear functionals on C(Sp(An)) of norms not exceeding 1. Hence to show (ii) we may transfer the reasoning from the proof of Proposition 7.1, where linear and multiplicative functionals were considered. (To obtain the counterpart of (ii) describing the whole space M(Sp(An)) we would have to impose some ﬁnite bound on the norms of the members of a sequence CORONA THEOREM 21

∗ in the ”algebraic” projective limit. This is so because the linking mappings Λn+1,n are contractions, not isometries.) Also by formula (4.2) we obtain a chain of embeddings

jn : M(Sp(An)) → M(Sp(An+1)), where ∼ is now replaced by ∼n, the relation of equality on An. We also deﬁne jn,m : M(Sp(An)) → M(Sp(Am)) for m ≥ n by

jn,m = jm−1 ◦ jm−2 ◦···◦ jn : M(Sp(An)) → M(Sp(Am)), n < m, jn,n = id. Here and in the formulae that follow we use id to denote the identity map on the space apparent from the context. Let us now consider for some n a functional z ∈ Sp(An) and a measure µ ∈ M(Sp(An)). Denote by jn,∞(z), jn,∞(µ), their embeddings into the projective limits Lim Sp(An), ←− LimM(Sp(An)) -respectively, deﬁned by ←− ∗ ∗ (7.3) jn,∞(z)=(ιn,0(z),...,ιn,n−1(z),z,jn,n+1(z), jn,n+2(z),... ) ∗ ∗ (7.4) jn,∞(µ)=(Λn,0(µ),..., Λn,n−1(µ),µ,jn,n+1(µ), jn,n+2(µ),... ) It is important to note that the so deﬁned sequences do belong to the corresponding projective limits. This follows from Lemma 4.7 applied at the levels of the algebras Am for any m > n. By Lemma 4.7 we have also ∗ ∗ (7.5) Λ∞,n ◦ jn,∞ = id, ι∞,n ◦ jn,∞ = id for n =1, 2 ....

In further considerations we are applying these mappings to obtain some separa- tion results for the images of nontrivial Gleason parts G of the spectrum X = Sp(A) of a uniform algebra A ⊂ C(X). The essential role is played by idempotents. Such functions assume 0 and 1 as the only values - and are characteristic functions of σ some sets. Recall that E denotes the Gelfand topology closure of E. ∗∗ As in Section 4 we regard the quotient space Y/∼ of Y = Sp(C(X) ) as a subset of Sp(A∗∗). Since A∗∗ is a subalgebra of C(Y ), any f ∈ A∗∗ is a function on Y , constant on the equivalence classes [y]∼. Such functions can be factored by some ∗∗ f∼ : Y/∼ → C so that f(y)= f∼([y]∼). Hence A is isometrically isomorphic to the ∗∗ algebra {f∼ : f ∈ A }.

Let Gα be any fixed Gleason part of Anα different from j0,nα (G). ∞ Lemma 7.3. The union n=1 jn,∞(Sp(An)) is dense in Sp(A). Moreover, the union of jnα,∞(Gα) taken overS all parts Gα different from j0,nα (G) is dense in the set σ Sp(A) \ j0,∞(G) . Proof. The first equality follows from Proposition 7.1. The second one is a consequence of the first equality and of the formula for closures of unions of two sets.

We extend the deﬁnitions of Λm,n to the case m = ∞ by setting ∗ (7.6) Λ∞,n(f)= f ◦ ι∞,n|Sp(A), f ∈ C(Sp(An)) 22 MAREKKOSIEKANDKRZYSZTOFRUDOL

By applying (4.11) to each Λn and jn, then to their compositions Λm,n and jn,m -we deduce the following Lemma for n ∈ N,n

Lemma 7.4. For h ∈ C(Sp(An)),µ ∈ M(Sp(An)) we have

Λm,n(h) · jn,m(µ)= jn,m(h · µ).

When m = ∞, the left - hand side has to be defined as the sequence (ν0, ν1,... ), where for k > n we let νk =Λk,n(h) · jn,k(µ). Proof. It suffices to prove the result for m =+∞, since for finite m it results directly from (4.11). By this “finite m -case” the sequence {νk} of the above form is equal to the sequence (jn,k(h · µ)). The latter has k-th terms with k > n the same as the sequence defined by jn,∞(µ). The previous terms must coincide since both sequences belong to the projective limit.

8. Embedded idempotent The following generalisation of the Assumption 5.6 (1), needed in this section, will be later shown to hold in the most important classical setting ∗ −1 (8.1) j0,n(G)=(ιn,0) (G) for n =1, 2,..., ∞. Similarly, the following assumption taken now - will be satisﬁed for A = A(G) if G is a regular domain in Cd. Namely, assume that (8.2) G is open in Sp(A). ∗ This assumption together with (8.1) and the w* continuity of ιn,0 implies the openness of j0,n(G) for any n. By gn (n =1, 2,... ) we shall denote the unique idempotent in An associated with Gleason part j0,n−1(G). Such idempotents were described in Proposition 3.2 and discussed in paragraphs preceding Theorem 4.5. The sequence ι∞,n(gn) is decreasing on Sp(A), since ιn(gn)gn+1 = gn+1 by Proposition 3.2. Hence its point-wise limit

g = lim ι∞,n(gn) is a Borel -measurable function and an idempotent. For any ﬁnite Borel measure µ on Sp(A) the convergence of integrals takes place: gdµ = lim ι∞,n(gn) dµ. Denote by F the “ﬁnite level points” of Sp(A), namelyR let R

F = jn,∞(Sp(An)). [n Denote by E the set of points, where g is discontinuous. Now g is a limit of a −1 −1 decreasing sequence of idempotents and we have g ({1}) = n(ι∞,n(gn)) ({1}) -which is a closed set containing E. Since g assumes only twoT values: 0 or 1, its discontinuity points form the boundary of g−1({1}). This boundary has no interior points. Hence E is nowhere dense. On the other hand, we can show that g is continuous on F and F (8.3) ∩ E = ∅, g|F\j0,∞(G) ≡ 0. CORONA THEOREM 23

To see this, consider x ∈ F, say x = jm,∞(y) ∈ jm,∞(Sp(Am)), where y ∈ Sp(Am). If y∈ / j0,m(G), then for k = m + 1 by (4.5) we have gk(jm,k(y)) = 0. Hence (ι∞,k(gk))(x) = 0. The sequence being decreasing implies that its limit, g is equal 0 −1 on the open set (ι∞,m+1(gm+1)) ({0}) containing x, hence x∈ / E in this case. This also proves the second part of (8.3). In the remaining case we have x ∈ j0,∞(G), which is open (by (8.1) and (8.2)) and g = 1 on this open set. Denote by Ω the spectrum of A. Let B(Ω) be the set of bounded Borel-measurable functions on Ω. We can extend by continuity the restriction g|Ω\E to the Stone- Cechˇ compactiﬁ- cation K := β(Ω \ E). For f ∈ C(Ω \ E) ∩ B(Ω) let κ(f) ∈ C(K) be the (unique) continuous extension on K of f|Ω\E. Note that due to this uniqueness κ is multiplicative. For x ∈ Ω let us deﬁne the set

Φx = {y ∈ K : (κ(f))(y)= f(x) for any f ∈ C(Ω)}.

These are one-point sets for x ∈ Ω \ E. For any y ∈ K treated as a linear-multiplicative functional δy on K its composition with κ, namely δy ◦ κ -as an element of Sp(C(Ω)) corresponds to some x ∈ Ω. Then y ∈ Φx and this x is uniquely determined. We can now deﬁne a ”projection” K ΠΩ : K → Ω by K ΠΩ (y)= x iﬀ y ∈ Φx. The continuity of this projection follows from the fact that the topology of compact space Ω is the weakest topology making all of the functions f ∈ C(Ω) continuous. The same applies to K and to C(K). The convergence yα → y for some net in K means that h(yα) → h(y) for any h ∈ C(K). In particular for any f ∈ C(Ω) K K κ κ and for xα = ΠΩ (yα), x = ΠΩ (y) we have f(xα)=( (f))(yα) → ( (f))(y)= f(x), since κ(f) ∈ C(K). K Since (as a continuous image of compactum) - closed set ΠΩ (K) contains Ω \ E K which is dense in Ω, it also contains Ω, which proves the surjectivity of ΠΩ . β Denote by Ag the Banach subalgebra of C(K) generated by the continuous ex- β tensions to K of the restrictions {f|Ω\E : f ∈A∪{g}}. We treat Ag as an algebra on K. Also denote by Ag (respectively by Cg) the Banach subalgebra of the algebra of bounded Borel functions on Ω with sup-norm, generated by A∪{g} (respectively C(Ω) ∪{g}). This algebra Ag is in fact a uniform closure of the set of functions of the form (fg + h) with f, h running through A (since g2 = g). A similar statement applies to Cg with f, h running through C(Ω). Now extending by continuity the restrictions to Ω \ E of the functions F ∈ Cg from Ω \ E to K = β(Ω \ E) we obtain β the Banach algebra Cg . We have the following

Lemma 8.1. Any functional φ ∈ Sp(C(Ω)) = Ω extends to γ ∈ Sp(Cg). Similarly, β β φ extends to δ ∈ Sp(Cg ). Moreover, the algebras Cg and Cg are isometrically isomorphic. 24 MAREKKOSIEKANDKRZYSZTOFRUDOL

Proof. Let us verify, for example, that for w = fg + h with f, h ∈ C(Ω) the formula (8.4) γ(w) := φ(f)g(φ)+ φ(h) defines a linear multiplicative functional on Cg. The definition does not depend on the representation: if fg + h = f1g + h1 -the equality holds point-wise on Ω, since g = 1 on E. Hence (fg + h)(φ)=(f1g + h1)(φ), which evaluated gives 2 φ(f1)g(φ)+ φ(h1)= φ(f)g(φ)+ φ(h). Since g (φ)= g(φ), the multiplicativity of γ comes from a direct calculation. β The fact that φ extends also to a multiplicative functional on Cg can be shown β as follows: Every φ ∈ Ω \ E has a unique trivial extension to Sp(Cg ). If φ ∈ E −1 then φ belongs to the closure of (Ω \ E) ∩ g ({1}) and g(φ) = 1. Let {φα} be −1 net in (Ω \ E) ∩ g ({1}) such that φα → φ. Since K is compact, {φα} has an ′ ′ adherent point γ in K. If w ∈ Cg and w is a continuous extesion of w|Ω\E to K ′ ′ then γ (w ) = γ(w), where γ is a functional on Cg defined as in (8.4). Hence the ′ norm of w in Cg which is equal to supΩ |w| is less or equal to supK |w |. The isometry follows from the above fact and from that the norm (kw′k = ′ ′ ′ ′ supK |w |) is greater than or equal to supΩ\E |w | and supK |w | equals supΩ\E |w | (by continuity of w on K and by the density of Ω \ E in K). Lemma 8.2. The set F is dense in K.

Proof. Take a point x ∈ K and its open neighborhood Vx. Then Vx ∩ (Ω \ E) is open in Ω \ E, hence equal to W ∩ (Ω \ E) for some open set W in Ω. By Lemma 7.3, the set F is dense in Sp(A), so that W ∩ F =6 ∅. Now by (8.3),

F ∩ Vx = F ∩ Vx ∩ (Ω \ E)= F ∩ (W \ E)= F ∩ W =6 ∅.

Let Kg denote the quotient space of K obtained by identifying any two points x, y ∈ K satisfying for any F ∈ Cg the relation F (x)= F (y). Therefore Cg can be treated as a C* -algebra of continuous functions on Kg separating the points and containing the constants, hence equal C(Kg). Any measure µ ∈ M(Ω), treated as a functional on C(Ω) extends uniquely to a functional on bounded Borel functions on Ω, in particular on functions from Cg. By the latest identification, µ defines a bounded linear functional on C(Kg), represented by a uniquely determined measure k(µ). When µ is a delta-measure we deduce from Lemma 8.1 that k(µ) as a multiplicative linear functional corresponds to a unique point of Sp(Cg). We have therefore also a map k : Ω → Sp(Cg) (which is an identity on the points from Ω\E). For µ ∈ M(Ω) and f ∈ Cg (the latter space is identified with C(Kg)) we have

(8.5) h(κ∗ ◦ k)(µ), fi = hk(µ), κ(f)i = hµ, fi, hence κ∗ ◦ k is an identity map on M(Ω): (8.6) κ∗ ◦ k = id. CORONA THEOREM 25

If moreover h ∈ Cg, then

κ(f) d(κ(h)k(µ)) = κ(h)κ(f) d(k(µ)) = κ(h · f) d(k(µ)) Z Z Z = hf dµ = fd(hµ)= κ(f) d(k(hµ)). Z Z Z Hence we have (8.7) κ(h)k(µ)= k(hµ). Lemma 8.3. If µ ∈ M(Ω) is an annihilating measure for A, then k(µ) annihilates β the algebra Ag .

Proof. Since g = lim ι∞,n(gn) is a limit of a sequence ι∞,n(gn) of elements of A, our claim results from the Lebesgue dominated convergence theorem.

Let us treat now the points of j0,∞(G) as the delta-measures and denote by τ the τ Gel’fand topology on Kg (and by -the corresponding closure operation). Then we obtain the following k τ κ −1 β Corollary 8.4. The set (j0,∞(G)) =( (g)) ({1}) is clopen and reducing for Ag in Kg. κ β Proof. Since (g) is a continuous idempotent belonging to Ag , the right-hand term is a clopen, reducing set. It remains to show the equality. Note that for any n the mapping k acts as identity on the range of jn,∞. Since κ(g) equals 1 on k(j0,∞(G)), by continuity it equals 1 also on its closure, proving “⊂”. To show the opposite −1 note, that by (8.3) the set k(F) \ k(j0,∞(G)) is disjoint with (κ(g)) ({1})-a clopen subset of Kg. This pre-image is also disjoint with the closure of k(F) \ k(j0,∞(G)). By Lemma 8.2 the set k(F) is dense in Kg, hence k(j0,∞(G)) is dense in (κ(g))−1({1}). We have ∗ κ∗ k τ σ (8.8) (ι∞,n ◦ )( (j0,∞(G)) )= j0,n(G) . ∗ κ∗ k τ Indeed, since ι∞,n and are continuous and (j0,∞(G)) is τ- compact, the contain- ment of j0,n(G) in the left-hand side implies the same for its closure. The opposite σ ∗ κ∗ inclusion is proved as follows. Take x∈ / j0,n(G) . Then its pre-image under ι∞,n ◦ κ∗−1 ∗−1 σ is disjoint from (ι∞,n(j0,n(G) )). The latest is a τ- closed set, which by (7.5) and (8.6) contains k(j0,∞(G)). Hence the pre-image of {x} is also disjoint from τ k(j0,∞(G)) . This means that x is not a member of the left-hand side of (8.8). In the last conclusion we were using the equivalence: S ∩ h−1(T )= ∅⇔ h(S) ∩ T = ∅ ∗ κ∗ k τ with T = {x}, h = ι∞,n ◦ ,S = (j0,∞(G)) . By the weak-star density of ιn+1,n(An) in An+1, by induction and from the w* -continuity of jn,m(µ) we obtain the ﬁrst part of the following 26 MAREKKOSIEKANDKRZYSZTOFRUDOL

⊥ Lemma 8.5. If a measure µ ∈ M(Sp(An)) annihilates An (µ ∈ An ), then for m > n ⊥ ⊥ k β ⊥ we have jn,m(µ) ∈ Am. Moreover, jn,∞(µ) ∈A and (jn,∞(µ)) ∈ (Ag ) .

Proof. By the norm-density of the union n ι∞,n(An) in A, it suﬃces to show for ι∞,m(f) with f ∈ Am,m>n that ι∞,m(fS)d jn,∞(µ) = 0 . The integral is the same as f djn,m(µ) = 0. The last claimR follows from Lemma 8.3. R For n =0, 1,... let

(8.9) κn = κ ◦ ι∞,n and kn = k ◦ jn,∞.

For µ ∈ M(Sp(An)) and h ∈ Cg, we extend the deﬁnition (4.4) of the product ⊙ :

(8.10) fd(h ⊙ µ) := κ (f) d(h · k (µ)) = hκ (f) d(k (µ)) Z Z n n Z n n for f ∈ C(Sp(An)), i.e. k κ κ∗ k h ⊙ µ := (h · n(µ)) ◦ n = n(h · n(µ)).

Inserting in (8.10) f ∈ An, h = g and applying Lemma 8.5 we obtain that g ⊙ M(Sp(An)) is a reducing band for An. From Theorem 4.5 we deduce that gn+1 ⊙ M(Sp(An)) = Mj0,n(G). We have for any n the inequality g ≤ ι∞,n(gn). In κ κ other words, (g) ≤ n(g). Hence g ⊙ M(Sp(An)) ⊂ Mj0,n(G). Since gn = 1 on j0,n(G) for n =1, 2 ... and consequently g =1 on kn(j0,n(G)), using Lemma 2.4 we conclude that the band g⊙M(Sp(An)) contains all representing measures for j0,n(G).

Since Mj0,n(G) is a minimal band containing all such representing measures, we have g ⊙ M(Sp(An)) = Mj0,n(G). Consequently, by (8.8) and the equality in Corollary 8.4, we have Theorem 8.6. If G is an open Gleason part in Sp(A) satisfying (8.1) then support of every representing measure µ0 for An at any point x0 in j0,n(G) lies in the Gelfand closure of j0,n(G).

Proof. Let δ0 be the Dirac measure at x0 (which is a representing measure). Hence ⊥ µ := µ0 − δ0 is an annihilating measure: µ ∈ An . The proof will be concluded by σ showing that the restricted measure µ|j0,n(G) also annihilates An, since the diﬀerence σ σ µ − µ|j0,n(G) is the nonnegative measure µ0 restricted to the complement of j0,n(G) and its integral of the constant function 1 ∈ An would be zero. Using the notation k β ⊥ (8.9), we have n(µ) ∈ (Ag ) , by the previous Lemma. From Corollary 8.4 -the k k τ β measure n(µ) restricted to the set 0(G) also annihilates Ag . Consequently, An is κ∗ k τ annihilated by n( n(µ)|k0(G) ). By (8.8), we have k k τ κ∗ −1 σ n(x0) ∈ 0(G) ⊂ ( n) (j0,n(G) ), and consequently

(8.11) kn(µ)|k τ ≤ kn(µ)| κ∗ −1 σ = kn(µ| σ ) 0(G) ( n) (j0,n(G) ) j0,n(G) CORONA THEOREM 27

The last equality is proved as follows. By (8.7) and Lemma 7.4, we have

κn(h)kn(µ)= kn(hµ), for h ∈ C(Sp(An)).

Since we consider regular Borel measures, we may approximate the restrictions ν|F = χF · ν of measures ν to Borel subsets F by the measures of the form h · ν where h is a continuous function bounded by 1. This allows us to extend the last formula replacing continuous functions by characteristic functions of closed sets. Then we use the equality κ κ∗ −1 n(χF )= χ( n) (F ). The measure µ| σ − κ∗ (k (µ)| τ ) is by (7.5) and (8.6) equal to the measure j0,n(G) n n k0(G) κ∗ k σ κ∗ k τ n( n(µ|j0,n(G) )) − n( n(µ)|k0(G) ) which is by (8.11) positive and annihilates An, κ∗ k τ since n( n(µ)|k0(G) ) annihilates An, hence it must be equal to 0. Consequently, σ µ|j0,n(G) annihilates An. 9. Abstract Corona Theorems For a uniform algebra A, by (8.1) and by Theorem 8.6, the first two parts of the Assumption 5.6 are satisfied. Hence by Theorem 6.4 we have Theorem 9.1 (Second Abstract Corona Theorem). Let G be an open Gleason part in Sp(A) satisfying (8.1) and the Assumption 5.6 (3). Then the canonical image of ∞ G is dense in the spectrum of H (MG) in its Gelfand topology. Note that the Assumption 5.6 (3) is satisfied if on G the Gelfand and norm topologies are equal. In such case as Kn we may take the sets 1 K = {x ∈ G : kx − xk ≤ 2 − }, n 0 n where x0 is an arbitrarily chosen point of G. Therefore we have also the following result Theorem 9.2 (Third Abstract Corona Theorem). Let G be an open Gleason part in Sp(A) satisfying (8.1), on which the Gelfand and norm topologies are equal. Then the ∞ canonical image of G is dense in the spectrum of H (MG) in its Gelfand topology. Proof. We need to show that the Assumption 5.6 (3) holds in this setting. We first compare three topologies on j(G), which is a topological subspace of j(M(Sp(A))). The latter space has its weakest topologies, say T1, T2 making continuous all functionals from ι(A) and all functionals from A∗∗-respectively. The first topology is weaker than the second one and both are weaker than the norm topology. The same applies to their restrictions to j(G), where the first and the norm topologies coincide, by our assumption. More precisely, we compare here the images under j of the Gelfand and norm topologies on G. We also use the fact that j restricted to G is a bijective isometry, by the assumption (8.1). Hence the three topologies are equal on j(G) and the sets j(Km) are compact in the norm topology. This will imply the uniform convergence of sequences |fj| appearing in ssumption 5.6 (3), because a fixed ǫ > 0 we cover j(Km) by a finite number of balls of radii ǫ and use the point-wise (hence uniform) convergence at their centres. 28 MAREKKOSIEKANDKRZYSZTOFRUDOL

10. Stability of G Let A = A(G), where G ⊂ Cd is a strictly pseudoconvex (”spsc.”) domain and let us ﬁx a point λ = (λ1,...λd) ∈ G. The Gleason problem was asking whether the ideal {f ∈ A(G): f(λ)=0} is generated by the shifted coordinate functions zk − λk for k ≤ d. The positive solution in the case of spsc domains with smooth boundaries was established in [11]. It has been even strengthened in [14] as follows: The continuous linear operators A ∋ f 7→ h{k} ∈ A, k =1,...,d solving the Gleason problem exist, which means that

d {k} (10.1) f(z) − f(λ)= (zk − λk)h (z), z ∈ G, f ∈ A. Xk=1

In other words, when f(λ) and λk are treated as constant functions in A and zk ∈ A as the coordinate functions, we have

d {k} (10.2) f − f(λ)= (zk − λk)h Xk=1

{k} and for some constant C depending only on λ the sup-norms bounds satisfy kh kG ≤ CkfkG. Here the sup-norm over G is the same as the norm in A. For polydomains the analogous decompositions are easily available by fixing all but one variables method. In the case of strictly pseudoconvex domains the results can be found in [14]. This implies, in particular, that the spectrum of A(D) is mapped in a bijective way to G by the Gelfand transform (ˆz1,..., zˆd) of the coordinate functions. Hence the Gelfand topology coincides with the Euclidean topology on G. Hence G is open in the spectrum of A(G). ∗∗ Now for f ∈ A = A1 we can find a net of elements fα ∈ A converging w* to f and such that kfαk≤kfk. Then we apply these operators obtaining a tuple {1} {d} (hα ,...,hα ) satisfying (10.2). By passing to a subnet we may assume that also {k} {k} {k} each net hα is converging w-* to some h with kh k ≤ Ckfk with the same C as above. Since these nets consist of analytic functions on G that are uniformly bounded by Ckfk, a normal family argument shows the locally uniform convergence and the analyticity of h{k} and f on G. The point-wise convergence follows easily from the w-* convergence. Moreover, h{k} ∈ A∗∗ for any k =1,...,d. ∗∗∗∗ By induction, we can continue obtaining the decompositions of f ∈ A = A2 (and so on) of the form (10.2). This equation implies that the fibers over λ of the spectrum of A∗∗ (respectively A∗∗∗∗) consist of one point - the evaluation functional at λ. Indeed, applying a linear multiplicative functional ϕ such that ϕ(zk)= λk to both sides of (10.2), we deduce that ϕ(f)= f(λ). Let ϕ be a linear multiplicative functional on A∗∗∗∗ lying in its Gleason part containing the image of G. Then its restriction to A is of the form δλ for some CORONA THEOREM 29 element λ of G, hence it is in the fiber over λ, which is just shown to be a singleton δλ. Summarising, we have the following result. Theorem 10.1. For A = A(G), where G ⊂ Cd is a strictly pseudoconvex do- 2 main with C boundary or a polydomain, the Gleason part of Sp(An) containing the canonical image j0,n(G) of G is equal to j0,n(G). A similar description holds for the Gleason part of Sp(A) containing the canonical image j0,∞(G) of G. This part equals j0,∞(G). Moreover the Gelfand topology of Sp(A) and the norm topology of A∗ coincide on G. Also the assumption (8.1) on ”one-point fibers” is satisfied. Proof. By the previous remarks, only the case n = ∞ requires justification. Suppose φ ∈ Sp(A) belongs to the fiber over a point λ =(λ1,...,λd) ∈ G. This means that for any k = 1,...,d we have φ(ι∞,0(zk)) = λk. Here zk denotes the corresponding coordinate function on G. Then for f ∈ An also φ(ι∞,n(f)) = f(j0,n(λ)). Being equal on a dense subset ι∞,n(An), the continuous functionals φ and j0,∞(λ) coincide. The equality of the twoS topologies follows from the decomposition (10.1) and from the uniform bound mentioned after (10.2). 11. Classical setting

Let z ∈ G and let νz be its representing measure with respect the algebra A. For ∞ f ∈ A we have f(z)= f dνz. By the weak-star density of A in H (MG) we also ∞ have f(z)= hf, νzi for Rf ∈ H (MG). Lemma 11.1. The value f(z) does not depend on the choice of representing mea- ∞ sure. It can be assigned to f ∈ H (MG):

f(z)= f dν . Z z d ∞ Proposition 11.2. If G is a bounded domain in C and f ∈ H (MG) then the deﬁned above mapping f|G : G ∋ z 7→ f(z) is a bounded analytic function of z ∈ G.

Proof. Let z0 ∈ G. The analyticity of f|G on G can be established by routine methods, like [16, Proposition 16], or by normal family arguments. In the next theorem we consider a bounded domain of holomorphy G ⊂ Cn such that its closure, G is the spectrum of A(G). For this it suﬃces to assume either that G is an intersection of a sequence of domains of holomorphy, that is, G has a Stein neighbourhoods basis [18], or that it has a smooth boundary [10]. Moreover G is a Gleason part for A(G) (see remarks in the Introduction ). Here A(G) plays the role of our initial uniform algebra A. For the purpose of the proof we assume additionally that G is a strongly starlike domain. By this we mean that there exists a point x0 ∈ G such that for the translate

Gx0 := G − x0 we have for ρ > 1 that Gx0 ⊂ ρ · Gx0 . It is easy to see that convex domains are strongly starlike with respect to any of its points. Theorem 11.3. If G is a strongly starlike domain in Cd satisfying the above as- ∞ ∞ sumptions, then the algebras H (G) and H (MG) are isometrically isomorphic. The associated homeomorphism of spectra preserves the points of G. 30 MAREKKOSIEKANDKRZYSZTOFRUDOL

∞ ∞ Proof. By Proposition 11.2 we have {f|G : f ∈ H (MG)} ⊂ H (G) and the norm ∞ of every element h ∈ H (MG) is greater or equal to the supremum of |h(z)| on G. ∞ ∞ We need to check whether H (G) ⊂ H (MG) in the sense of isometric embedding. Without loss of generality we can assume that G is strongly starlike with ∞ respect to 0. For f ∈ H (G) and 0 spectral radius and the latter is the sup-norm of f on any dense subset G0 of the spectrum ∞ of H (MG). By Theorems 9.2 and 10.1 we can take G0 = G. Finally, we obtain our main result: Theorem 11.4 (Corona Theorem). If a strongly starlike domain G ⊂ Cd is either strictly pseudoconvex or a polydomain then Corona Theorem holds true in H∞(G): its spectrum is the Gelfand closure of (the canonical image of) G.

∞ ∞ Proof. By Theorem 11.3 the spectra of algebras H (G) and H (MG) are homeo- morphic by a mapping preserving the points of G (sending x ∈ G to its canonical ∞ image in Sp(H (MG)) to be more precise). By Theorem 10.1, Gelfand and norm topologies are equal on G and the assumption (8.1) is satisﬁed. Now it remains to apply Theorem 9.2.

Additional Remarks The results from complex analysis employed in the present work come mostly from the late 1970’s. General theory of uniform algebras have been developed around the same period, with major contributions of that time by Gamelin, Cole, Glicksberg, Seever Hoﬀman and Rossi. We are also using analysis of bands of measures related to Gleason parts and their weak star closures developed in the ﬁrst part of [16]. Unfortunately, there was a gap in the proof of Theorem 6 in that paper -noticed and kindly communicated to us by Garth Dales. Theorem 6 in [16] and its corollaries do not hold in the claimed generality, in particular- for the algebras of continuous functions. The present work is using only the results of [16] preceding Theorem 6 - with one exception -when in the proof of Proposition 11.2 we invoke a method used in the proof of [16, Proposition 16], which is however quite routine.

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