Abstract

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 of measures 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 , our approach avoids the use of integral formulae for analytic functions, strict pseudoconvexity, or some other regularity of . We also investigate the relation between the algebra of bounded holomorphic functions on and its abstract counterpart—the * 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.

1. 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 or even on complex manifolds. Under certain regularity assumptions on , the abstract counterpart of 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 —the space of nonzero linear and multiplicative homomorphisms of with Gelfand topology. In this abstract setting the domain corresponds to a nontrivial Gleason part of and measures on absolutely continuous with respect to measures representing certain points of form a band of measures denoted by . The weak-star density of in the spectrum of some quotient algebra of is an equivalent formulation of the corona problem in 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 (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 closures with the Lebesgue-type band decompositions. As a corollary, we identify these decompositions as the Arens multiplications by some idempotent .

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 and “dual algebra property.”

Our results give also an abstract solution to the “A-measures 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 A-measure. In Theorem 19 we obtain a general result on A-measures. 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.

2. 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

A closed linear subspace in the space of regular complex Borel measures on is called a band of measures (cf. [3, 4] V§ 17), if along with any it contains all measures absolutely continuous with respect to . For the set of all measures singular to any is a band, and is the smallest band containing , referred to as the band generated by . In particular, we have . Any has the following Lebesgue-type decomposition: The space is a direct sum of and and for any pair , their total variation norms satisfy

Let be the set of all measures annihilating , that is, such that for any . We say that a band is called reducing, if for any in the above decomposition (2).

If is a nonzero linear and multiplicative functional (a fact denoted by ), we say that is a complex representing measure for if By representing measure we understand a nonnegative representing measure. For belonging to the same Gleason part (i.e., satisfying ) the bands generated by all measures representing (resp., ) are equal and we denote this band by . A form of an abstractversion of F. and M. Riesz theorem ([4] V 18.2, [2] II 7.6) asserts that is always a reducing band. It is also well known that Gleason parts are Borel sets in the Gelfand topology, as (with fixed) the above is a countable union of compact sets (namely, of ).

In the next section we need the following properties of three quotient norms of the equivalence classes of : the first one is the norm in (defined as the distance: ). The second one, the norm in , is denoted here simply as . These both turn out to be equal to the third one, in . (Note that each of the equivalence classes, denoted for simplicity as , is in a different quotient space.)

Lemma 1. If is a reducing band then for any and one has For the quotient space norm of   in satisfies also

Proof. Decompose with respect to as in (2). From (3) we obtain Since is a reducing band, and the nontrivial inequality in (5) follows. The opposite inequality in (5) is obvious, because .
Now using (3) for ,   we have since , . Hence for . Passing to the infimum over we get finally showing (6). After replacing with in (10), we obtain the nontrivial inequality “” in (7), since . The rest follows from the inclusion .

3. 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 weak-star 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 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 , we write for any set in the considered space. Since , the notation for this closure is justified when 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 of bounded nets and 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 the measure on obtained by lifting , that is, by representing the functional , 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 , we will take the closure in endowed with its topology . Then it is natural to ask whether this closure is still a subset of . To see that this is the case, first check that the canonical embedding satisfies . (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 , where , we verify the needed multiplicativity of .

Let us look now at the weak-star closure of a band of measures in . Note that if one identifies with , then one can write . If and , we denote by the measure on having the density with respect to .

Lemma 2. The following formula holds:

Proof. Let . Then there is a net such that and weak-star. For each we have , where , , and . We have and for each . Hence, by the Banach-Alaoglu theorem, the net has an adherent point and the net has an adherent point . After passing to a suitable subnet, if necessary, we can write and . Consequently , , and which completes the proof.

There is an important, yet easy to verify, relation between bands (which are ideals with respect to an order in ) and closed ideals in . The latter are of the form for some closed subset (see [2]). We also identify with a band in ; namely, , where for all Borel sets .

Proposition 3. If is an ideal in and is an ideal in , then (1)the annihilator, , is a band in ;(2)the preannihilator, , is a weak-star closed band in .
Conversely, if is a band in then the following holds.(3) is a closed ideal in and is a closed ideal in .(4) is a weak-star closed ideal in and is a weak-star closed ideal in .(5)For some closed sets with , one has (6)For some closed sets such that , the closures (of in , where and satisfy (7)If a band is weak-star closed (namely, -closed), then for some closed subset and in this case, the Lebesgue-type decomposition (2) has the simplest form: , .

Proof. (1) Obviously, is a closed subspace in . It suffices to check that for any absolutely continuous measure (which is of the form with ) is also in . Since is regular, is dense in . For some sequence of we have in the norm of . For we have , since is an ideal. Now converge to . Hence .
(2) The only difference from (1) is that given , , and as above, we have , where the first equality follows from the convergence in . The second one follows since and , while , by the definition of the Arens product in .
(3) and (4) are obvious and follow from the absolute continuity .
(5) The ideals , in are closed. Hence , for some closed subsets . Obviously, the annihilators of the latter ideals are identified with and , respectively, showing that and .
As , the equality follows. Indeed, given an arbitrary , decompose any measure , like in (2). Then , showing that . Taking the annihilators we get by Lemma 2.
(6) Here the proof is quite analogous to that of (5).
The last claim comes from the description of closed ideals in , giving the form of some and from the equality , 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 for any point . Sets arising as intersections of a family of peak sets are called p-sets. Let be a reducing band for our uniform algebra. By Glicksberg’s theorem ([2], 12.7) we get the following.

Corollary 4. If is a weak-star closed band in , where , then is a p-set if and only if the band is reducing.

Example 5. When is the classical disc algebra and is the closed unit disc, take a Cantor-type subset of the unit circle of zero arc length. In the case of bidisc algebra as we may take the set for some . 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 is not necessarily a direct sum as the related sets and may intersect. However, in the most important case, the situation is much better: in what follows, we fix a Gleason part and denotes the band generated by representing measures for the points , while is the set of all regular complex Borel measures on .

Theorem 6. If is a Gleason part of then (the closure of in the -topology) is a closed-open subset of . Moreover and is a reducing band for .

Proof. Let . Since is isometrically identified with , we have for each representing measure of . By (7) of Lemma 1, we therefore get There exists having norm 1 and such that and . In fact, the norm-attaining (at ) functional on composed with the canonical surjection assigning to its equivalence class does the job.
Let . Then . Since the norm in with respect to is the same as the norm with respect to , 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 and on such that (respectively, ) is a representing measure for (resp. for ). The equality implies that as an element of equals 1 -almost everywhere. Consequently it is also equal to 1 , almost everywhere, and hence . Since was chosen arbitrarily, we have for all , and consequently on , which is the -closure, since .
Similarly, as , we have on and on , by continuity. But by Lemma 2 applied to the band . Since , we have . On the other hand, , which implies that it is a weak-star closed band, and hence . Similarly , whence which implies We have had on and on . So the sets and must be disjoint and closed-open.
We have for any , and hence 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 . Hence almost everywhere with respect to any from the band . Passing to - limits in the equalities valid for probabilistic measures , we extend it to any probabilistic .
Consequently, for all measures . Since (yielding ) and since as an idempotent element has norm 1, we have It follows from the last inequality that the bands and are mutually singular. The latter combined with Lemma 2 gives the equality . By Proposition 3, there are closed subsets of such that and . Hence are disjoint and, by Lemma 2, we have . But it must be ; hence because is weak-star closed. Since on (see (21)) and for all probabilistic measures , we must have . Hence , and by the properties of it must be a reducing band.

From the above proof we also obtain the following.

Corollary 7. There exists a characteristic function vanishing exactly on . The projection associated with the decomposition into mutually singular components is the Arens multiplication by . (Metrically, this sum is of -type.)

Example 8. Let be an open unit ball in . Consider the algebra of all continuous functions on which are analytic in . Its spectrum consists of one nontrivial Gleason part and the union of singleton Gleason parts filling in its boundary. The fact that the whole 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. ).
We have the following decomposition: where . The measures in are called totally singular; in the case they are simply the measures singular to the Lebesgue measure on the unit circle. It is known (see [7], Chapter 9) that . Hence , and using Lemma 1 we get Now , where the last term is isometrically isomorphic to the - closure of in . By Theorem 17 (cf. below) we get where denotes the predual of . By the equality of Corollary 7 we now obtain Hence the measures on the hyperstonian envelope of the ball decompose onto those -approximable by absolutely continuous (resp., totally singular) ones.

4. Closures of Parts and Representing Measures

Using the results of Section 3 we can localise supports of measures representing the points . These sets are “not too far” from the respective Gleason parts (such that ). In fact, they are contained in the Gelfand-topology closures . In the special case of the algebra for a compact set 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 . 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 . In fact, the point mass at is linear and multiplicative on ; hence its restriction to (which is precisely )) belongs to . 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 , we get the closedness of resulting in one inclusion in the following lemma. (The other inclusion, namely, , results just from the continuity of .)