The cluster value problem in spaces of continuous functions

We study the cluster value problem for certain Banach algebras of holomorphic functions defined on the unit ball of a complex Banach space X. The main results are for spaces of the form X = C(K).

In an effort to research the corona problem, we investigate conditions that guarantee the simpler cluster value theorem for a Banach algebra of analytic functions defined on the unit ball of a complex Banach space X. In particular, we generalize some of the results in [4]. Our main results are for the spaces of the form X = C(K), including a translation result of a cluster value problem, from any point in B * * C(K) to the origin.
We thank Richard Aron and Manuel Maestre for their communications.
In [4], the authors obtain a cluster value theorem at the origin for Banach spaces with shrinking 1-unconditional bases for the algebra H = A u (B) of bounded analytic functions on B that are also uniformly continuous. Slight modifications of their arguments in Section 3 of [4] yield the following: Proposition 1. Let S be a finite rank operator on X, so that P = I − S has norm one. If φ ∈ M 0 (B), thenf (φ) = f • P (φ), for all f ∈ A u (B).
Proposition 2. Suppose that for each finite dimensional subspace E of X * and ǫ > 0 there exists a finite rank operator S on X so that ||(I − S * )| E || < ǫ and ||I − S|| = 1. Then the cluster value theorem holds for A u (B) at 0.
Since Proposition 2 builds on Proposition 1, one naturally wonders if Proposition 1 can be extended to the larger algebra H ∞ (B) of all bounded analytic functions on B. The answer is no in general, as shown by the following example of Aron.
Clearly S is a finite rank operator and P = I − S has norm one.
Let (r j ) and (s j ) be sequences of positive real numbers, such that (r j ) ↓ 0 and (s j ) ↑ 1 in such a way that each r 2 j + s 2 j < 1 and r 2 j + s 2 j → 1 − . For each j = 1, 2, 3, · · · , let δ r j e 1 +s j e j be the usual point evaluation homomorphism where the square root is taken with respect to the usual logarithm branch. Then When a Banach space has a shrinking reverse monotone finite dimensional decomposition (FDD), that is, a shrinking FDD so that the natural projections are at distance one from the identity operator, we have that the condition in Proposition 2 holds, and therefore we obtain a cluster value theorem: Corollary 1. If X is a Banach space with a shrinking reverse monotone FDD, then the cluster value theorem holds for A u (B) at 0.
The operators P considered in Propositions 1 and 2 have finite-codimensional rank, which suggests that the cluster value problem at the origin of a Banach space can be studied by considering the same problem in its finitecodimensional subspaces. As we conjectured, this turns out to be the case: Proof. A u (B) coincides with the uniform limits onB of continuous polynomials on X (see Theorem 7.13 in [11] and p. 56 in [3]), where polynomials are finite linear combinations of symmetric m-linear mappings restricted to the diagonal. Thus, by passing to the uniform limit onB, we may assume f is an m-homogeneous polynomial, with associated symmetric m-linear functional F. Let (x α ) be a weakly null net in B such that f (x α ) → λ.
Finally, since lim sup ||y α || ≤ 1, we can take a sequence of scalars (t α ) such that ||t α y α || < 1 for all α and t α → 1, and consequently, As a consequence we obtain that the cluster sets of an element f of A u (B) at 0 can be described in terms of the Gelfand transforms of f | B Y as Y ranges over finite-codimensional subspaces of X: Proof. From Proposition 3 and the inclusion in (1), for every finite-codimensional subspace Y of X, For the reverse inclusion, suppose 0 / ∈ Cl B (f, 0). Then there are ǫ > 0 and a weak neighborhood Going back to Proposition 2, we see that having the cluster value property at 0 only requires the existence of a certain type of finite rank operators at distance one from the identity operator. However simple this condition may seem, it is impossible in the case of the Banach space c of continuous functions on ω, also seen as the subspace of l ∞ of convergent sequences: Proof. For each k ∈ N, consider L k ∈ B c * given by The reader may check that the condition is also impossible for L p , 1 ≤ p = 2 < ∞.
However, note that since c 0 is one-codimensional in c, Proposition 3 implies that for all f ∈ A u (B c ), Also, Propositions 1.59 and 2.8 of [8] imply that all functions in A u (B c 0 ) can be uniformly approximated on B by polynomials in the functions in X * , which in turn implies that each fiber at x ∈B * * consists only of x, so the cluster value theorem for A u (B c 0 ) holds, and in particular Note that an inclusion is evident: The reverse inclusion is unclear. However, the space c also has the property of being isomorphic to c 0 , which implies, as we will see, that c has the cluster value property too.
Let P (X) denote the continuous polynomials on X, P f (X) the polynomials in the functions of X * (known as finite type polynomials), and A(B X ) the uniform algebra of uniform limits of elements in P f (X).
Proof. Let T : Y → X be the Banach space isomorphism between Y and X. Let f ∈ A u (B Y ). Then there exist a sequence of polynomials P n ∈ P(Y ) such that ||P n − f || B Y ≤ 1 n , ∀n ∈ N. For each n ∈ N, P n • T −1 ∈ P(X), so there exists a polynomial Q n ∈ P f (X) 3 Cluster value problem in C(K) ≇ c.
Bessaga and Pe lczyński proved in [6] that, when α ≥ ω ω is a countable ordinal, C(α) is not isomorphic to c = C(ω). Therefore we no longer can use Lemma 1 to obtain a cluster value theorem on such spaces of continuous functions.
Nevertheless, for α a countable ordinal, the intervals [1, α] are always compact, Hausdorff and dispersed (they contain no perfect non-void subset). The compact, Hausdorff and dispersed sets K satisfy, from the Main Theorem in [12], that X = C(K) contains no isomorphic copy of l 1 . Moreover, from Theorem 5.4.5 in [1], X = C(K) has the Dunford-Pettis property. Therefore, for dispersed K, the continuous polynomials on X = C(K) are weakly (uniformly) continuous on bounded sets by Corollary 2.37 in [8].
Moreover, since X * = l 1 (K) has the approximation property, Proposition 2.8 in [8] now yields that all continuous polynomials on X can be uniformly approximated, on bounded sets, by polynomials of finite type. Thus the elements of A u (B) can be approximated, uniformly on B, by polynomials of finite type. Hence A u (B) = A(B), so each fiber at x ∈B * * is the singleton {x}, and then X satisfies the cluster value theorem for the algebra A u (B).
We now consider the cluster value problem on X for the algebra of all bounded analytic functions H ∞ (B). Following the line of proof of Theorem 5.1 in [4], we still get a cluster value theorem: Theorem 1. If X is the Banach space C(K), for K compact, Hausdorff and dispersed, then the cluster value theorem holds for H ∞ (B) at every x ∈B * * .
Since 0 is not a cluster value of f at w, there exists a weak-star neighborhood for some ǫ > 0 and x * 1 , · · · , x * n ∈ X * = l 1 (K). We have that x * i = (x * i (t)) t∈K has countably many nonzero coordinates {x * i (t)} t∈F i for i = 1, · · · , n. Thus, In summary, there exist c > 0, δ > 0 and a finite set F ⊂ K such that if z ∈ B satisfies |z t − w t | < δ for t ∈ F then |f (z)| ≥ c. Relabel the indices in F as t 1 , · · · , t m , where m = |F |. Then proceed as in the proof of Theorem 5.1 in [4]: Note that 1/f is bounded and analytic on U 0 .
We claim that for each k, 1 ≤ k ≤ m, there are functions g k and h k,j , Once this claim is established, the proof is easily completed as follows. The functions g m and h mj belong to H ∞ (B) and satisfy Since each z t j − w t j vanishes on M w (by the definition of M w ), we obtain f g m = 1 on M w , and consequently f does not vanish on M w , as required.
Just as in [4], the claim is established by induction on k. The first step, the construction of g 1 and h 11 , is as follows. We regard 1/f ((z t ) t∈K ) as a bounded analytic function of z t 1 for |z t 1 | < 1 and |z t 1 − w t 1 | < δ, with z t , t ∈ K − {t 1 }, as analytic parameters in the range |z t | < 1 for t ∈ K − {t 1 }, and |z t j − w t j | < δ for 2 ≤ j ≤ m. According to lemma 5.3 in [4], we can so that (2) is valid for k = 1. Note that h 11 = −hf on U 0 . Consequently h 11 is bounded and analytic on U 0 . The defining formula then shows that h 11 is analytic on all of U 1 , and since |z t 1 − w t 1 | ≥ δ on U 1 − U 0 , h 11 is bounded on U 1 . Now suppose that 2 ≤ k ≤ m, and that there are functions g k−1 and h k−1,j (1 ≤ j ≤ k − 1) that satisfy (2) and are appropriately analytic. We apply lemma 5.3 in [4] to these as functions of z t k , with the other variables regarded as analytic parameters, to obtain decompositions and where g k and the h kj 's are in H ∞ (U k ), and G k and the H kj 's are in H ∞ (U k−1 ). From the identity (2), with k replaced with k − 1, we obtain Then (2) is valid. On U k−1 we have so that h kk is bounded and analytic on U k−1 .
we see from the defining formula that h kk ∈ H ∞ (U k ). This establishes the induction step, and the proof is complete.
We do not know the answer to the cluster value problem for other spaces C(K).
The previous problem seems to be highly nontrivial. Since for every infinite compact Hausdorff space K, C(K) contains a subspace Y isometric to c 0 (Proposition 4.3.11 in [1]), the fiber M 0 (B C(K) ) is huge (and from Lemma 3, also each fiber M f 0 (B C(K) ) for f 0 ∈ B C(K) ). Indeed, according to Theorem 6.6 in [7], there is a family of distinct characters {τ α } α∈B ℓ∞ , such that each τ α : , which is clearly a homomorphism. Note that the characters {τ α • R} α∈B ℓ∞ are all distinct due to Theorem 1.1 in [2], because ℓ ∞ is an isometrically injective space (Proposition 2.5.2 in [1]), so there exists a norm-one linear map S : We prove in Corollary 3 that if the latter cluster value problem has an affirmative answer at some point of B C(K) , then it has an affirmative answer at all points of B C(K) . For that let us first establish the following lemmas, the first of which is a folklore result mentioned e.g. in [14] and [5], but inasmuch there seems to be no proof in the literature we will sketch the proof.
Lemma 2. Let f 0 ∈ B = B C(K) . T : B → B given by is biholomorphic.
Let us now show that T is also holomorphic, or equivalently, C-differentiable. (1−f 0 f ) 2 h satisfies that, for h = 0 small enough, which goes to zero as h → 0. Thus T is holomorphic.
Since T clearly has a necessarily holomorphic inverse (S(f ) = f +f 0 1+f 0 ·f ), we have that T is a biholomorphic function on B that sends f 0 to the function identically zero. Proof. Note that T is a Lipschitz function. Indeed, if f, g ∈ B, The reader can easily check that the previous mappingT is actually a homeomorphism.