Dual Algebras and A-Measures

Weak-star closures of Gleason parts in the spectrum of a function algebra are studied. These closures relate to the bidual algebra and turn out both closed and open subsets of a compact hyperstonean space. Moreover, weak-star closures of the corresponding bands ofmeasures are reducing. Among the applications we have a complete solution of an abstract version of the problem, whether the set of nonnegative A-measures (called also Henkin measures) is closed with respect to the absolute continuity. When applied to the classical case of analytic functions on a domain of holomorphy Ω ⊂ C, our approach avoids the use of integral formulae for analytic functions, strict pseudoconvexity, or some other regularity ofΩ. We also investigate the relation between the algebra of bounded holomorphic functions onΩ and its abstract counterpart—the w∗ closure of a function algebra A in the dual of the band of measures generated by one of Gleason parts of the spectrum of A.


Introduction
The idea of using second duals for algebras of analytic function goes back to the works of Brian Cole and Theodore W. Gamelin.It turned out to be a powerful tool for studying some spectral and approximation properties of algebras of analytic functions over domains Ω or even on complex manifolds.Under certain regularity assumptions on Ω ⊂ C  , the abstract counterpart of  ∞ (Ω) obtained by duality techniques corresponds isomorphically to the classical algebra of bounded analytic functions, allowing the use of functional analysis tools for dual spaces.
Let us recall that a uniform algebra (or function algebra) over a compact Hausdorff space  is a closed unital subalgebra  of () separating the points of .Then we can assume that  is a uniform algebra on its spectrum Sp()-the space of nonzero linear and multiplicative homomorphisms of  with Gelfand topology.In this abstract setting the domain Ω corresponds to a nontrivial Gleason part  of Sp() and measures on  absolutely continuous with respect to measures representing certain points of  form a band of measures denoted by M  .The weak-star density of  in the spectrum of some quotient algebra of  * * is an equivalent formulation of the corona problem in  ∞ (Ω) settled 50 years ago by Lennart Carleson in the unit disc case and still open for the higher dimensional balls or polydiscs.This alone justifies the need for better understanding of the nature of weak-star closures of nontrivial Gleason parts  canonically embedded in the dual space for M  (or in some other bidual spaces).
In Section 2 we formulate some preliminary observations.Quite simple distance estimates for quotient norms in dual spaces (modulo the set  ⊥ of annihilating measures) are obtained from the corresponding band decompositions.
Section 3 focuses on dualities and the related weak-star topologies (abbreviated as - * ), namely, on the - * closures.Some natural relations are shown to hold between bands of measures on a compact space , closed ideals in () and - * closed ideals in its bidual space () * * , exploring the fact that the latter is of the form () for a compact space .
Our main result concerns Gleason parts in the spectrum of a uniform algebra  ⊂ ().Due to the canonical embeddings, we consider such parts  ⊂  * as subsets of certain bidual spaces.Theorem 6 describes some unexpected property of their - * closures, relating them also to the - * closures of the bands of measures generated by .The obtained relations imply the compatibility of such 2 Journal of Function Spaces closures with the Lebesgue-type band decompositions.As a corollary, we identify these decompositions as the Arens multiplications by some idempotent  0 ∈  * * .
One of the consequences of Theorem 6 presented in Section 4 is that the representing measures are supported by - * closures of the corresponding Gleason parts.
In Section 5 we apply the results of Section 3 relating an abstract algebra of  ∞ -type to the algebra of bounded analytic functions on a star-like domain.As a result, we obtain an alternative representation of a predual space to  ∞ (Ω) and "dual algebra property." Our results give also an abstract solution to the "Ameasures problem."The A-measures appearing in Section 6 (an abbreviation for analytic measures) were introduced in [1] under the name "L-measures" in the case of  = (), the algebra of analytic functions on a domain , continuous at its Euclidean closure .(Some authors call them Henkin measures; the notation () can also be met for the algebra ().)These are the measures  on , for which the integrals of the pointwise convergent to 0 on , bounded sequences in , converge to zero.The problem was to verify whether a measure on , which is absolutely continuous with respect to some representing measure, is itself an Ameasure.In Theorem 19 we obtain a general result on Ameasures.Here the domain  is replaced by a Gleason part of  and the measures are supported by .When applied to concrete algebras (), this solves the A-measures problem for a wide range of domains, extending the previously known results.

Preliminaries
In what follows  will denote a uniform algebra on some compact Hausdorff space , meaning a closed subalgebra of () containing the constants and separating the points of .We may assume additionally (see [2], Chapter II) that  is equal to the spectrum of  (denoted by Sp ()) . ( A closed linear subspace M in the space () of regular complex Borel measures on  is called a band of measures (cf.[3,4] V § 17), if along with any  ∈ M it contains all measures ] absolutely continuous with respect to .For E ⊂ () the set E  of all measures singular to any ] ∈ E is a band, and E  := (E  )  is the smallest band containing E, referred to as the band generated by E. In particular, we have M = M  .Any  ∈ () has the following Lebesgue-type decomposition: ( The space () is a direct sum of M and M  and for any pair Let  ⊥ be the set of all measures  ∈ () annihilating , that is, such that ∫   = 0 for any  ∈ .We say that a band is called reducing, if  M ∈  ⊥ for any  ∈  ⊥ in the above decomposition (2).

Second Duals
If the dual  * of a Banach space is contained isometrically as a subspace of some Banach space B, one way of representing the second dual,  * * (see, e.g., [3]), is to consider the weakstar closure in  * of , denoted here by   .In some cases this - * topology will be denoted more precisely as ( * , ), to avoid possible ambiguity.
By the bipolar theorem applied to linear subspaces , we have   = ( ⊥ ) ⊥ , where ⊥  = { ∈  : () = 0 for any  ∈ } is the preannihilator of .The symbol  used for all canonical embeddings in the second duals will (almost always) be suppressed, while taking the - * closures: instead of (Δ)

𝑤𝑠
, we write Δ  for any set Δ in the considered space.Since ⊥ (Δ) = Δ ⊥ , the notation Δ ⊥⊥ for this closure Δ  is justified when Δ is a subspace.Recall that by Goldstine's theorem the unit ball of  is weak-star dense in the unit ball of  * * .
The second dual of a uniform algebra  has a multiplication extending that in , called the Arens product.Actually, there are two Arens products (they correspond to different orders of iteration mentioned below), but for subalgebras  of commutative  * -algebras they both coincide (a fact known as the Arens regularity of ).Moreover, () * * endowed with the Arens multiplication is a commutative  * -algebra; hence it is isometrically isomorphic to () for some extremally disconnected space , called the hyperstonean envelope of .In other words, The Arens product of ,  ∈  * * can also be interpreted as an iterated - * limit of the product   ℎ  of bounded nets   and ℎ  in , weak-star converging to  (resp., to ), and the reverse order of iteration yields the same result, due to the Arens regularity of uniform algebras.We refer to [5] for the details on the Arens product and to [3] for the isometric identification of  ⊥⊥ with  * * .Let us just note that, as () = () * , the isomorphism from ()/ ⊥ onto  * is obtained through the factorisation of the restriction of  ∈ () to ; that is, Next, denote by μ the measure on  obtained by lifting , that is, by representing the functional () ∋ ℎ → ∫ ℎ , extended to ().Given  ∈ () = () * and [] ∈ ()/ ⊥ , we have the natural duality formulae well-defined (independent of the choice of the representative of []) if and only if  ∈  ⊥⊥ .On the other hand, for certain subsets  of the algebra's spectrum Sp() ⊂  * , we will take the closure   in  * * * endowed with its  * topology ( * * * ,  * * ).Then it is natural to ask whether this closure is still a subset of Sp( * * ).To see that this is the case, first check that the canonical embedding  :  * →  * * * satisfies (Sp()) ⊂ Sp( * * ).(Here one can either invoke Lemma 3.6 in [6] or use the iterated limits representation of the Arens product  and the  *continuity on  * * of () for  ∈  * , deducing that () = ()() in this case.)Then passing to  * -limits in  * * * of the form Φ = lim (  ), where   ∈ , we verify the needed multiplicativity of Φ.
Conversely, if M is a band in () then the following holds.
(3) ⊥ M is a closed ideal in () and ( ⊥ M) ∩  is a closed ideal in .
(6) Here the proof is quite analogous to that of (5).
The last claim comes from the description of closed ideals in (), giving ⊥ M the form of some   and from the equality M = ( ⊥ M) ⊥ , which holds in this case.
Let us recall from [2], II.12, that a peak set for  is a subset  ⊂  on which some  ∈  equals 1, while satisfying |()| < 1 for any point  ∈ \.Sets arising as intersections of a family of peak sets are called p-sets.Let M be a reducing band for our uniform algebra.By Glicksberg's theorem ( [2], 12.7) we get the following.
and M   is a reducing band for  * * .
Let  ∈ .Then ‖ 0 − ‖ < 2. Since the norm in  * with respect to  is the same as the norm with respect to  * * ,  0 and  (canonically embedded in  * * * ) are also in the same Gleason part in .Thus we can find a pair of mutually absolutely continuous Borel regular measures  0 and  on  such that  0 (resp., ) is a representing measure for  0 (resp.for ).The equality 1 = ⟨ 0 ,  0 ⟩ = ∫   0  0 implies that  0 as an element of () equals 1 [ 0 ]-almost everywhere.Consequently it is also equal to 1 , almost everywhere, and hence ⟨ 0 , ⟩ = ∫   0  = 1.Since  ∈  was chosen arbitrarily, we have ⟨ 0 , ⟩ = 1 for all  ∈ , and consequently  0 ≡ 1 on   , which is the ( * * * , which implies We have had  0 ≡ 1 on   and  0 ≡ 0 on  \   .So the sets   and  \   must be disjoint and closed-open.We have ⟨ 0 , ⟩ = 1 for any  ∈ , and hence  0 ≡ 1 almost everywhere []  ] for all its representing measures ]  ∈ ().This equality almost everywhere to 1 occurs also with respect to any linear combination of such ]  's, generating the band M  .Hence  0 = 1 almost everywhere with respect to any  from the band M  .Passing to - * limits in the equalities ⟨ 0 , ⟩ = 1 valid for probabilistic measures  ∈ M  , we extend it to any probabilistic  ∈ M   .
Since it must be a reducing band.
From the above proof we also obtain the following.Its spectrum consists of one nontrivial Gleason part Ω and the union of singleton Gleason parts filling in its boundary.The fact that the whole Ω is one Gleason part follows from the existence of mutually absolutely continuous representing measures for any pair of its points.Such a pair of representing measures is given by suitable Poisson kernels (see [7], Theorem 3.3.2).
We have the following decomposition: where S = M  Ω .The measures in S are called totally singular; in the  = 1 case they are simply the measures singular to the Lebesgue measure on the unit circle.It is known (see [7], Chapter 9) that S ∩ (Ω) ⊥ = {0}.Hence S/ (Ω) ⊥ ∩S = S, and using Lemma 1 we get where the last term is isometrically isomorphic to the - * closure of (Ω) in M * Ω .By Theorem 17 (cf.below) we get where  ∞ (Ω) * denotes the predual of  ∞ (Ω).By the equality of Corollary 7 we now obtain Hence the measures on the hyperstonian envelope of the ball Ω decompose onto those  * -approximable by absolutely continuous (resp., totally singular) ones.

Closures of Parts and Representing Measures
Using the results of Section 3 we can localise supports of measures ]  representing the points  ∈ Sp().These sets supp ]  are "not too far" from the respective Gleason parts  (such that  ∈ ).In fact, they are contained in the Gelfandtopology closures .In the special case of the algebra () for a compact set  ⊂ C such a statement appears already in Theorem 3.3 in Chapter VI of [2].Before stating the result, it is convenient to establish one lemma relevant for any nontrivial Gleason part  ⊂ Sp().At the beginning of Section 3 we have noted that the second conjugate to () is identified with ().Hence we have the canonical embedding  : () → ().The canonical surjection restricted to  ( meaning here {  :  ∈ }, a subset of () * ) yields a canonical map Π  :  → .In fact, the point mass at  is linear and multiplicative on (); hence its restriction to () (which is precisely Π(  )) belongs to Sp(()) = .Similar restriction of the other canonical embedding  : () * → () * results in   :  → .
From the compactness of   (which is a simplified notation for the weak-star closure of   () in ) and by the - * continuity of Π  , we get the closedness of Π  (  ) resulting in one inclusion in the following lemma.(The other inclusion, namely, (⊂), results just from the continuity of Π  .)Lemma 9.The canonical projection Π  maps this weak-star closure of   () onto the Gelfand closure of : Theorem 10.If ]  is a representing measure for  ∈  then supp ]  ⊂ .

Algebras of 𝐻 ∞ -Type
One of the approaches to study properties of  ∞ (Ω), the algebra of bounded analytic functions on a given domain Ω ⊂ C  , is to consider its abstract counterpart, the algebra  ∞ (M  ) corresponding to a nontrivial Gleason part  for a function algebra .The band of measures M  corresponding to  is now considered as a Banach space, so that its dual space M *  carries its weak-star topology.34) is isometric and surjective.By Nagasawa's theorem (see [9] V.31), it is also isomorphic (preserves both linear and multiplicative structure).Corollary 14.  is a subset of the spectrum of  ∞ (M  ).
The following proposition (for its proof and details, see Proposition 2.8 of [10]) is a consequence of the Hahn-Banach theorem and Theorem 12.
Proposition 15.The band M  is equal to the norm closed linear span of all representing measures for points in , taken in the quotient space ()/ ⊥ .
Note that for  ∈  ∞ (M  ) and  ∈  we can define () as the value of  on a representing measure ]  for .By the weak-star density of  in  ∞ (M  ), this value () does not depend on the choice of representing measure.So the elements of  ∞ (M  ) can be regarded as functions on .Similarly as in Proposition 3.6 of [11], we can show the following.
Proposition 16.If  is a bounded domain in C  and  ∈  ∞ (M  ) then the defined above mapping  ∋  → () is a bounded analytic function of  ∈ .
Proof.Let us consider an arbitrary point  0 ∈  and a small open polydisc Δ with the center at  0 , included in .Without the loss of generality we can assume  0 = 0. Denote by  the normalized Lebesgue measure on the Shilov boundary of Δ and by   the n-dimensional Cauchy kernel for .Then  is a representing measure for  0 (with respect to the algebra ).The measure    is absolutely continuous with respect to  and consequently is in M  .
In the next theorem we consider a bounded domain of holomorphy  ⊂ C  such that its closure, , is the spectrum of (), which plays the role of our initial uniform algebra .For this it suffices to assume either that  is an intersection of a sequence of domains of holomorphy, that is,  has a Stein neighbourhoods basis [12], or that it has a smooth boundary [13].
For the purpose of the proof we assume additionally that  is a star-shaped domain.
Theorem 17.If  is a domain in C  satisfying the above conditions, then the algebras  ∞ () and  ∞ (M  ) are isometrically isomorphic.
Note that this provides also a direct representation of a predual to  ∞ () and since the multiplication is weak-star continuous, it shows that  ∞ () is a dual algebra.For the relevance of dual algebras to operator theory see [14].

A-Measures
Let  be a Borel subset of  which is a union of some Gleason parts of : We say that a measure  ∈ () is an analytic measure for the algebra  at the points of  or, shortly, an A-measure at , if ∫    → 0 whenever {  } ∞ =1 ⊂  is a bounded sequence converging to 0 pointwise on .The notion was introduced under the name "L-measures" by Henkin in [1] for  = (),  = .This concept was useful not only in studying the isomorphisms between algebras of analytic functions over various domains , in approximation theory, but also in operator theory in the construction of analytic functional calculus in a given -tuple of commuting operators [15,16].
All representing measures for points in  are trivially A-measures.Our formulation of the so-called A-measures problem for the algebra  at the points of  is as follows.
( †) Does the absolute continuity of a measure  on  with respect to some representing measure of a point  ∈  imply that  is an -measure at ?By the "classical case" we mean the situation when  is a domain in C  and  is either the algebra () of complex continuous functions on , analytic on , or its subalgebra () generated by the rational functions having no singularities on .
The problem was solved positively (by advanced complex analysis methods) for two special cases: for the algebra () with  =  by Henkin in [1] and by Cole with Range (see [17]) on strictly pseudoconvex bounded domains in C  (resp., on domains in complex manifolds) with  2 boundaries and by Bekken [18] (and [15]) in the case of polydomains (cartesian product of planar domains).Bekken's results hold also for  equal to () on compact product sets  =  1 × ⋅ ⋅ ⋅ ×   with   ⊂ C. All these previous results will be covered by Theorem 19 below.In the latter case one needs to know that there are only countably many nontrivial Gleason parts of (  ) [2] VI 3.2.Proposition 18.Any nonnegative A-measure at  belongs to the band generated by .
Proof.By decomposing such a measure  with respect to M  we may assume that  belongs to M   .Applying equality (7) of Lemma 1 to M   in place of M, we obtain dist(, M  +  ⊥ ) = ‖[]‖ = ‖‖.The last equality holds since  is nonnegative.Now for some ℎ ∈ () of norm 1 annihilated by M  +  ⊥ we have ∫ ℎ  close to 1.But ℎ ∈ , since it is annihilated by  ⊥ .Taking the constant sequence ℎ  = ℎ vanishing on , we get a contradiction to the assumption on  being an Ameasure.
Note that usually the A-measures problem is formulated in a slightly different way.
( ‡) Is any measure which is absolutely continuous with respect to a nonnegative A-measure itself an Ameasure?

Corollary 4 .Theorem 6 .
If M = () is a weak-star closed band in (), where  ⊂ , then  is a p-set if and only if the band () is reducing.Example 5.When  = (D) is the classical disc algebra and  = D is the closed unit disc, take a Cantor-type subset  of the unit circle D of zero arc length.In the case of bidisc algebra (D 2 ) as  we may take the set { 0 } × D for some  0 ∈ D. In both cases peak functions for  can be constructed directly.Hence we obtain specific examples of reducing bands of the form () for these algebras.Note that M  + (M  )  is not necessarily a direct sum as the related sets Ẽ and K may intersect.However, in the most important case, the situation is much better: in what follows, we fix a Gleason part  ⊂ Sp() and M  denotes the band generated by representing measures for the points  ∈ , while (  ) = {   :  ∈ ()} is the set of all regular complex Borel measures on   .If  is a Gleason part of  then   (the closure of () in the ( * * * ,  * * )-topology) is a closed-open subset of .Moreover

Corollary 7 .Example 8 .
There exists a characteristic function  0 ∈  * * vanishing exactly on \  .The projection associated with the decomposition into mutually singular components  () = M   ⊕ M    (23) is the Arens multiplication by  0 .(Metrically, this sum is of  1type.)Let Ω be an open unit ball in C  .Consider the algebra (Ω) of all continuous functions on Ω which are analytic in Ω.