A look into homomorphisms between uniform algebras over a Hilbert space
aa r X i v : . [ m a t h . F A ] F e b A LOOK INTO HOMOMORPHISMS BETWEEN UNIFORM ALGEBRAS OVER AHILBERT SPACE.
VER ´ONICA DIMANT AND JOAQU´IN SINGER
Abstract.
We study the vector-valued spectrum M u, ∞ ( B ℓ , B ℓ ) which is the set of nonzero algebrahomomorphisms from A u ( B ℓ ) (the algebra of uniformly continuous holomorphic functions on B ℓ ) to H ∞ ( B ℓ ) (the algebra of bounded holomorphic functions on B ℓ ). This set is naturally projected ontothe closed unit ball of H ∞ ( B ℓ , ℓ ) giving rise to an associated fibering. Extending the classical notionof cluster sets introduced by I. J. Schark (1961) to the vector-valued spectrum we define vector-valuedcluster sets. The aim of the article is to look at the relationship between fibers and cluster sets obtainingresults regarding the existence of analytic balls into these sets. Introduction
The discussions between eight renowned mathematicians at the Conference on Analytic Functions (In-stitute for Advanced Studies, 1957) gave rise to a common article published in 1961 under the pseudonymI. J. Schark [22]. The object of study was the maximal ideal space (or spectrum ) of the algebra H ∞ ( D )of bounded holomorphic functions over the complex unit disk. The results obtained, the questions raisedand the approach considered in that article have inspired the work of many researchers since then.The description of fibers and cluster sets of the spectrum proposed by I. J. Schark was later studied(by several authors) for algebras of bounded analytic functions on the unit ball of an infinite dimensional Banach space. Even if the infinite dimensional framework is linked to a more complex picture of thespectrum, the essence of I. J. Schark’s point of view has continued guiding the research. One more stepin this road has appeared at considering this perspective for a vector-valued spectrum ; and here is whereour contribution fits in. To be more precise we need to introduce some definitions and notation.Let X be a complex Banach space with open unit ball B X . We denote by A u ( B X ) the Banach algebraof holomorphic functions f : B X → C that are uniformly continuous on B X . This is a sub-algebra of H ∞ ( B X ), the Banach algebra of all bounded holomorphic mappings on B X with the supremum normon B X . The (scalar-valued) spectrum of this uniform algebra is the set M ∞ ( B X ) = { ϕ : H ∞ ( B X ) → C nonzero algebra homomorphisms } . Since X ∗ is contained in H ∞ ( B X ), there is a natural projection π : M ∞ ( B X ) → B X ∗∗ given by π ( ϕ ) = ϕ | X ∗ . Through this projection, M ∞ ( B X ) has a fibered structure:for each z ∈ B X ∗∗ the set π − ( z ) = { ϕ ∈ M ∞ ( B X ) : π ( ϕ ) = z } is called the fiber over z . The spectrum M u ( B X ) = { ϕ : A u ( B X ) → C nonzero algebra homomorphisms } is similarly projected onto B X ∗∗ andthe fibering over this set is consequently defined.Our particular interest is to study the homomorphisms from the uniform algebra A u ( B ℓ ) to H ∞ ( B ℓ )as a whole set by studying the vector-valued spectrum M u, ∞ ( B ℓ , B ℓ ) defined as M u, ∞ ( B ℓ , B ℓ ) = { Φ : A u ( B ℓ ) → H ∞ ( B ℓ ) non-zero algebra homomorphism } . The uniform algebras A u ( B X ) and H ∞ ( B X ) are typical settings in Infinite dimensional Holomorphy.Their corresponding spectrums M u ( B X ) and M ∞ ( B X ) are studied, with fibers [2, 10, 14, 8, 9] andcluster sets [5, 4, 1] being of particular interest. Besides the study of specific algebras and their spectrums,homomorphisms between uniform algebras have been a subject of research in both the finite [16, 18] andthe infinite dimensional setting [15, 3]. Through the vector-valued spectrum, originally defined in [11], weare able to study homomorphisms between uniform algebras as a whole set and analyze the structure of Mathematics Subject Classification.
Key words and phrases. spectrum, algebras of holomorphic functions, homomorphisms of algebras.Partially supported by Conicet PIP 11220130100483 and ANPCyT PICT 2015-2299 . this set analogously as it is done for the scalar-valued spectrum. Focusing in the infinite dimensional case,the infinite dimensional polydisk B c and the euclidean ball B ℓ appear as natural domains to extend thefinite-dimensional study of Banach algebras of holomorphic mappings over D n and the euclidean ball of C n . Having worked on the structure of the vector valued spectrum for X = c in [12] we now turn ourattention to ℓ . In contrast to c , where scalar-spectrum M u ( B c ) is simple (it can be identified with B ℓ ∞ ), here the spectrum M u ( B ℓ ) is more complex, which relates to the number of homomorphismsfrom A u ( B ℓ ) to H ∞ ( B ℓ ). As a result, M u, ∞ ( B ℓ , B ℓ ) not only provides some information abouthomomorphisms from H ∞ ( B ℓ ) to itself but also it has its own interest.In this paper we study fibers and cluster sets for the vector-valued spectrum M u, ∞ ( B ℓ , B ℓ ) and howthey are related. We present sufficient conditions for these sets to be either isolated or to contain analyticcopies of balls .1.1. Fibers.
As every function in A u ( B ℓ ) can be continuously extended to B ℓ , each function g ∈ B H ∞ ( B ℓ ,ℓ ) naturally produces a composition homomorphism C g ∈ M u, ∞ ( B ℓ , B ℓ ) given by C g ( f ) = f ◦ g. Conversely, as in [11] we can define a projection ξ : M u, ∞ ( B ℓ , B ℓ ) → H ∞ ( B ℓ , ℓ )Φ [ x (Φ( h· , e n i )( x )) n ] . Since ξ ( C g ) = g for all g ∈ B H ∞ ( B ℓ ,ℓ ) , the image of this projection is B H ∞ ( B ℓ ,ℓ ) . However, theremight be additional elements Φ ∈ M u, ∞ ( B ℓ , B ℓ ) such that ξ (Φ) = g . The set F ( g ) = ξ − ( g ) is calledthe fiber over g . Section 2 is devoted to the picture of such fibers. There we show that the fiber over anyfunction g ∈ B H ∞ ( B ℓ .ℓ ) is large, in fact it contains an analytical copy of a ball. The remaining fibers(i.e. over functions with norm 1) are more nuanced, so we present conditions under which we can stillfind an analytical copy of a ball and also we describe singleton fibers.1.2. Cluster sets.
It was shown in [20] that H ∞ ( B ℓ ) is a dual space with a predual that we denotewith G ∞ ( B ℓ ), keeping the notation from [19]. The vector-valued spectrum M u, ∞ ( B ℓ , B ℓ ) can then beviewed as a subset of the unit sphere of L ( A u ( B ℓ ) , H ∞ ( B ℓ )) = (cid:0) A u ( B ℓ ) b ⊗ π G ∞ ( B ℓ ) (cid:1) ∗ . We endow M u, ∞ ( B ℓ , B ℓ ) with the the weak-star topology from (cid:0) A u ( B ℓ ) b ⊗ π G ∞ ( B ℓ ) (cid:1) ∗ that satisfiesΦ α → Φ whenever Φ α ( f )( x ) → Φ( f )( x ) for all x ∈ B ℓ and f ∈ A u ( B ℓ ).(1)It is then readily seen that M u, ∞ ( B ℓ , B ℓ ) is a weak-star closed (and thus compact) subset of the unitsphere of L ( A u ( B ℓ ) , H ∞ ( B ℓ )) = (cid:0) A u ( B ℓ ) b ⊗ π G ∞ ( B ℓ ) (cid:1) ∗ (see [11, Th. 11] or [13, p. 3]).For vector-valued cluster sets we focus on a particular case of the limits considered in (1): thoseobtained for a fixed function f whenever Φ α are composition homomorphisms.Let g ∈ B H ∞ ( B ℓ ,ℓ ) , f ∈ A u ( B ℓ ). We define the vector-valued cluster set of f over g as C ℓ M u, ∞ ( B ℓ ,B ℓ ) ( f, g ) = { h ∈ H ∞ ( B ℓ ) : ∃ ( g α ) ⊂ B H ∞ ( B ℓ ,ℓ ) , g α w ∗ −−→ g satisfying C g α ( f )( y ) → h ( y ) ∀ y ∈ B ℓ } . Here g α w ∗ −−→ g denotes the weak-star convergence associated to the identification H ∞ ( B ℓ , ℓ ) = L ( G ∞ ( B ℓ ) , ℓ ) = ( G ∞ ( B ℓ ) b ⊗ π ℓ ) ∗ , where the fist equality was proved in [19]. These sets are the vector-valued counterpart to the (scalar-valued) cluster sets Cℓ B X ( f, z ) for f ∈ H ∞ ( B X ) and z ∈ B X defined as Cℓ B X ( f, z ) = { λ ∈ C : ∃ ( z α ) ⊂ B ℓ , z α w ∗ −−→ z satisfying f ( z α ) → λ } . OMOMORPHISM BETWEEN UNIFORM ALGEBRAS OVER A HILBERT SPACE. 3
The study of cluster sets and cluster values for the complex unit disk D was initiated in [22]. This worklinked the set of cluster points at a boundary point of D for a function f ∈ H ∞ ( D ) with the set ofevaluations ϕ ( f ) for ϕ in the scalar-valued spectrum. This topic in the infinite dimensional setting havebeen recently developed in [4, 5, 17, 21, 1]. In section 3 we analyze the vector-valued version of clustersets with particular interest in the size and analytical structure of such sets.2. Fibers in M u, ∞ ( B ℓ , B ℓ )Recall from the introduction that M u ( B ℓ ) is the scalar-valued spectrum M u ( B ℓ ) = { ϕ : A u ( B ℓ ) → C continuous algebra homomorphism } \ { } . In this setting, the projection π can be computed as π : M u ( B ℓ ) → ℓ π ( ϕ ) = ( ϕ ( h· , e n i )) n . (2)A lot of progress has been made towards describing the structure of the fibers of this spectrum. It wasshown in [14] (see also [4]) that the fibers over elements z ∈ S ℓ are singleton sets. For fibers over elements z ∈ B ℓ the situation is completely different. Cole Gamelin and Johnson proved in [10] that the fiberover 0 contains an analytical copy of B ℓ ∞ . This result was later extended to every z ∈ B ℓ in [9]. Withthese results in mind we focus on the vector-valued spectrum M u, ∞ ( B ℓ , B ℓ ) to provide a descriptionof its fibered structure over B H ∞ ( B ℓ ,ℓ ) .When, in [13], we tackled the study of the fibers for the vector-valued spectrum M ∞ ( B X , B Y ) (nonzeroalgebra homomorphisms from H ∞ ( B X ) to H ∞ ( B Y )), we proposed a division into what we called interior,middle and edge fibers . This was based on both the norm of the function over which we were projectingand whether or not the corresponding composition homomorphism could be defined. In our currentsetting of M u, ∞ ( B ℓ , B ℓ ) we noted in the introduction that composition homomorphisms C g can bedefined for every g ∈ B H ∞ ( B ℓ ,B ℓ ) . It is, however, still worth to keep a distinction of interior, middle and edge fibers due to the different behaviour of elements belonging to each of these classes. They aredescribed as follows • Interior fibers: Fibers over functions g with k g k < g ( B ℓ ) ⊂ B ℓ ). • Middle fibers: Fibers over functions g with k g k = 1 and g ( B ℓ ) ⊂ B ℓ . • Edge fibers: Fibers over constant functions g with | g ( x ) | = 1 for all x ∈ B ℓ .Note that for a holomorphic function g ∈ B H ∞ ( B ℓ ,ℓ ) if g ( x ) ∈ S ℓ then g ( B ℓ ) ⊂ S ℓ . Also, since ℓ is strictly convex, this implies that g is a constant function.The logic behind this split is that we want to obtain new homomorphisms in the fiber over g by takinglimit of composition homomorphisms that are in some sense close to C g (that is, such that the projectionof the limit is still g ). Because of the way k · k works, moving from interior to edge fibers we get lessroom to work and thus this distinction is then reflected in the results regarding the size of the fibers foreach case. As we see next, in one extreme we can inject a ball in any interior fiber while the opposite istrue for edge fibers, we show that they are always singleton sets. Middle fibers, being a transition fromthe interior to the edge, are the most complex case. They can be isolated, they can contain an analyticcopy of a ball (we exhibit conditions for both situations) but we do not know if there could be a thirdpossibility.2.1. Interior Fibers.
We have shown in [13, 12] that under certain circumstances we can constructanalytical injections for the vector-valued spectrum by building upon analytical injections for the scalarvalued case. With that in mind let us take a moment to recall what is known for the scalar-valuedanalogue of interior fibers, that is, scalar-valued fibers over elements in the interior of the unit ball B ℓ .In that direction we have the following result from Cole, Gamelin and Johnson. Theorem 2.1. [10, Th. 6.1]
Suppose X has a normalized basis { x j } which is shrinking, that is, whoseassociated coefficient functionals { L j } have linear span dense in X ∗ . Suppose furthermore that there is V. DIMANT AND J. SINGER an integer N ≥ such that ∞ X j =1 | L j ( x ) | N < ∞ for all x = P ∞ j =1 L j ( x ) x j in X . Then there is an analytic injection of the polydisk B ℓ ∞ into the fiberover 0 in M ∞ ( B X ) . While not in the original statement, it is readily seen that the proof of this result applies withoutmodification to the spectrum M u ( B ℓ ). Further, we know from [7, Ex. 1.8] that via composition withthe mapping β x ( y ) = 11 + p − k x k (cid:28) x − y − h y, x i , x (cid:29) x + p − k x k x − y − h y, x i we can obtain an isometric isomorphism (in the metric given by being a subset of A u ( B ℓ ) ∗ ) betweenthe fibers π − ( x ) and π − (0) for all x ∈ B ℓ . As a consequence, the fiber over each x ∈ B ℓ containsan analytic copy of B ℓ ∞ . However, we can not take advantage of this construction for the vector-valuedcase. The main reason being that the mappings β x are not “smooth” on x . Still, we can manage toget analytic injections in the vector-valued case. For that, we first prove a new embedding result for thescalar-valued spectrum in the spirit of [10, Th. 6.7]. Then, we make use of this to show how the definingproperty of the interior fibers ( k g k <
1) ensures the injection of an analytic copy of the ball B H ∞ ( B ℓ ,ℓ ) .As usual, for the scalar-valued spectrum M u ( B ℓ ) we denote by δ z the evaluation mapping given by δ z ( f ) = f ( z ) for f ∈ A u ( B ℓ ) and z ∈ B ℓ . Lemma 2.2.
For any < r < there exists an analytic injection Φ r : rB ℓ × B ℓ → M u ( B ℓ ) such thatthe image of Φ r ( w, · ) is contained in the fiber π − ( w ) .Proof. For each k ∈ N we define s k : rB ℓ × B ℓ → B ℓ as s k ( w, z ) = k X j =1 h w, e j i e j + (1 − r ) ∞ X j =1 h z, e j i e k + j and Φ k : rB ℓ × B ℓ → M u ( B ℓ ) as Φ k ( w, z ) = δ s k ( w,z ) .We now fix a free ultrafilter U containing the sets { k : k ≥ k } and define the mapping Φ r : rB ℓ × B ℓ → M u ( B ℓ ) where, for ( w, z ) ∈ rB ℓ × B ℓ , Φ r ( w, z ) is the weak star limit along U of(Φ k ( w, z )) k . Note that the limit along the ultrafilter exists because M u ( B ℓ ) is a weak-star compactsubset of ( A u ( B ℓ )) ∗ . Then, we have, for ( w, z ) ∈ rB ℓ × B ℓ and f ∈ A u ( B ℓ ),Φ r ( w, z )( f ) = lim U Φ k ( w, z )( f ) . We now move on to show that Φ r : rB ℓ × B ℓ → M u ( B ℓ ) is an analytic injection, mapping { w } × B ℓ into the fiber over w .To see that the image of Φ r ( w, · ) is contained in π − ( w ) we fix z ∈ B ℓ , n ∈ N . We then have forevery k ≥ n the equality Φ k ( w, z )( h· , e n i ) = w n . This implies that Φ r ( w, z )( h· , e n i ) = w n for all n , concluding that Φ r ( w, z ) ∈ π − ( w ).Let us now see that Φ r is analytic. For such purpose, we fix f ∈ A u ( B ℓ ) and we have to prove thatthe mapping b f ◦ Φ r : rB ℓ × B ℓ → C given by [( w, z ) Φ r ( w, z )( f )] is holomorphic.Since the sequence ( b f ◦ Φ k ) k is contained in the weak-star compact set k f k B H ∞ ( rB ℓ × B ℓ ) it shouldhave a weak-star limit along the ultrafilter U . This limit should coincide with b f ◦ Φ r , meaning that thismapping belongs to k f k B H ∞ ( rB ℓ × B ℓ ) and so b f ◦ Φ r is analytic.To verify the injectivity of Φ r , note first that when w = w ′ we have Φ r ( w, z ) = Φ r ( w ′ , z ) since theybelong to different fibers. Now, inspired by [9, Th. 3.1], we consider for each λ ∈ D the test functions OMOMORPHISM BETWEEN UNIFORM ALGEBRAS OVER A HILBERT SPACE. 5 f λ ( x ) = P ∞ j =1 λ j x j , so thatΦ k ( w, z )( f λ ) = k X j =1 λ j h w, e j i + λ k ∞ X j =1 λ j ((1 − r ) h z, e j i ) . Let χ ( λ ) = lim U λ k which exists for all λ ∈ D . We can then computeΦ r ( w, z )( f λ ) = ∞ X j =1 λ j h w, e j i + χ ( λ ) ∞ X j =1 λ j ((1 − r ) h z, e j i ) . (3)As a consequence, for λ ∈ ∂ D we get that Φ r ( w, z )( f λ ) = Φ r ( w, z ′ )( f λ ) if and only if ∞ X j =1 λ j ((1 − r ) h z, e j i ) = ∞ X j =1 λ j ((1 − r ) h z ′ , e j i ) . Since F ( λ ) = P ∞ j =1 λ j ((1 − r ) h z, e j i ) is a holomorphic function in D , continuous in D and coincides with F ′ ( λ ) = P ∞ j =1 λ j ((1 − r ) h z ′ , e j i ) over ∂ D , the Taylor expansion for both functions must coincide andthus ( h z, e j i ) = ( h z ′ , e j i ) for all j ∈ N . Repeating the argument with P ∞ j =1 λ j x j yields the equality z = z ′ , showing that Φ r is in fact an analytic injection. (cid:3) Theorem 2.3.
For every g ∈ B H ∞ ( B ℓ ,ℓ ) there is an analytic injection Ψ g : B H ∞ ( B ℓ ,ℓ ) → F ( g ) . Proof.
Given g ∈ B H ∞ ( B ℓ ,ℓ ) there exists r > k g k < r <
1. Considering Φ r the mappingconstructed in the previous lemma, we define Ψ g : B H ∞ ( B ℓ ,ℓ ) → F ( g ) byΨ g ( h )( f )( x ) = Φ r ( g ( x ) , h ( x ))( f )for each h ∈ B H ∞ ( B ℓ ,ℓ ) , f ∈ A u ( B ℓ ), x ∈ B ℓ .To see that this is well defined we need to check that Ψ g ( h ) is an algebra homomorphism from A u ( B ℓ )to H ∞ ( B ℓ ) and that ξ (Ψ g ( h )) = g . Note that from Lemma 2.2 we can derive that the mapping from B ℓ to M u ( B ℓ ) given by [ x Φ r ( g ( x ) , h ( x ))] is analytic. As a consequence we have that Ψ g ( h )( f ) isholomorphic. Also, it is clear that is bounded resulting that Ψ g ( h )( f ) ∈ H ∞ ( B ℓ ). The fact that Ψ g ( h )is an algebra homomorphism follows easily from Lemma 2.2. Finally, we appeal again to Lemma 2.2 toproduce the identity ξ (Ψ g ( h )) ( x ) = π (Φ r ( g ( x ) , h ( x ))) = g ( x )which shows that Ψ g ( h ) ∈ F ( g ).The injectivity of Ψ g is also an easy consequence of the injectivity of Φ r . We finish by showing that Ψ g is analytic (i. e. each mapping [ h Ψ g ( h )( f )( x )] from B H ∞ ( B ℓ ,ℓ ) to C is analytic). This is obtainedthrough the following composition of analytic mappings (the second one due to Lemma 2.2): h ( g ( x ) , h ( x )) Φ r ( g ( x ) , h ( x ))( f ) = Ψ g ( h )( f )( x ) . (cid:3) Remark 2.4.
Note that Ψ g (0) = C g . For any h g along with the fact that ξ (Ψ g ( h )) = g tell us that Ψ g ( h ) could not be a composition homomorphism.2.2. Edge Fibers.
As we mentioned at the beginning of the section, edge fibers are the polar oppositeto interior fibers . By relating the behaviour of the scalar-valued spectrum, particularly fibers over S ℓ , tothat of edge fibers of the vector-valued spectrum we can show that the fiber over any constant function g ∈ S H ∞ ( B ℓ ,ℓ ) consists only of the corresponding composition homomorphism C g . With that goal inmind we begin with the following remark giving another description of composition homomorphisms (cf[13, Rmk. 3.1]). Remark 2.5.
Let g ∈ B H ∞ ( B ℓ ,ℓ ) and Ψ ∈ M u, ∞ ( B ℓ , B ℓ ). Then Ψ = C g if and only if δ x ◦ Ψ = [ f Ψ( f )( x )] coincides with δ g ( x ) for all x ∈ B ℓ . V. DIMANT AND J. SINGER
To complete our picture for the edge fibers we need the following well-known result regarding fibersfor the scalar spectrum M u ( B ℓ ). Lemma 2.6. [14]
For every z ∈ S ℓ the scalar-valued fiber π − ( z ) ⊂ M u ( B ℓ ) consists only of theevaluation homomorphism δ z . Now we have all the necessary ingredients to prove our main result regarding edge fibers . Proposition 2.7.
Let g ∈ S H ∞ ( B ℓ ,ℓ ) be a constant function. Then the fiber F ( g ) consists only of thecomposition homomorphism C g .Proof. Let Ψ ∈ M u, ∞ ( B ℓ , B ℓ ) such that ξ (Ψ) = g . For all x ∈ B ℓ we consider the mapping δ x ◦ Ψ ∈M u ( B ℓ ). This mapping verifies π ( δ x ◦ Ψ) = ( δ x ◦ Ψ( h· , e n i )) n = (Ψ( h· , e n i )) n ( x ) = ξ (Φ)( x ) = g ( x ) . By Lemma 2.6, the previous equality implies that δ x ◦ Ψ = δ g ( x ) for every x ∈ B ℓ . Recalling Remark 2.5we conclude that Ψ = C g . (cid:3) Middle Fibers.
Middle fibers are somewhat a transition case as we can find examples for bothsingleton middle fibers and middle fibers containing an analytic copy of a ball. To give some context toour results we present some guiding examples.Let Id : B ℓ → B ℓ be the identity function. Clearly the fiber over Id is a middle fiber. However,the identity is singular in many ways. One of them being that Id is continuously extendable to B ℓ andthe extension verifies Id ( S ℓ ) ⊂ S ℓ . Recall that Lemma 2 . z ∈ S ℓ has a singletonfiber for the scalar-valued spectrum, so the condition Id ( S ℓ ) ⊂ S ℓ turns out to be quite restrictive. Wewill see that the only homomorphism in M u, ∞ ( B ℓ , B ℓ ) lying in the fiber over Id is the correspondingcomposition mapping (which of course is the identity mapping).Before we go into further detail regarding the fiber over Id we need to introduce some additionalnotation.We denote with M ∞ ( B ℓ ) the scalar-valued spectrum M ∞ ( B ℓ ) = { ϕ : H ∞ ( B ℓ ) → C continuous algebra homomorphism } \ { } . As it happens to M u ( B ℓ ), this spectrum also has an associated projection π : M ∞ ( B ℓ ) → ℓ π ( ϕ ) = ( ϕ ( h· , e n i )) n . (4)To prevent confusion, we will write π ∞ to denote the projection defined in (4) and π u to denote theprojection defined in (2), whenever both are involved in the same computation.Also, we recall the following result (which was used to derive Lemma 2.6). Lemma 2.8. [14, Lem. 4.4]
Let B be the ball of a uniformly convex Banach space. Then f ∈ H ∞ ( B ) iscontinuously extendable to a point x of the unit sphere if and only if f is constant on the fiber π − ∞ ( x ) . We can now focus on the previously mentioned example.
Example 2.9.
Let Φ ∈ M u, ∞ ( B ℓ , B ℓ ) such that ξ (Φ) = Id and let C ∗ Φ denote the composition mapping C ∗ Φ : M ∞ ( B ℓ ) → M u ( B ℓ ) C ∗ Φ ( ϕ ) = ϕ ◦ Φ . Since for every n ∈ N we have C ∗ Φ ( ϕ )( h· , e n i ) = ϕ (Φ( h· , e n i )) = ϕ ( h· , e n i ) it follows that π u ( C ∗ Φ ( ϕ )) = π ∞ ( ϕ ). As a consequence, for every z ∈ S ℓ and ϕ ∈ π − ∞ ( z ) the composition homomorphism C ∗ Φ maps ϕ to the only homomorphism in π − u ( z ), namely δ z . Now for every f ∈ A u ( B ℓ ) we have thatΦ( f ) ∈ H ∞ ( B ℓ ) is constant on the fiber over any z ∈ S ℓ (taking the value f ( z )). By Lemma 2.8,Φ( f ) has a continuous extension to S ℓ where it coincides with f . By the maximum modulus principle itfollows that Φ( f ) = f , showing that the identity function has a singleton fiber. OMOMORPHISM BETWEEN UNIFORM ALGEBRAS OVER A HILBERT SPACE. 7
Now, isolating some of the core elements of the previous example we obtain the following statement.
Lemma 2.10.
Let g ∈ H ∞ ( B ℓ , ℓ ) , continuously extendable to S ℓ such that g ( S ℓ ) ⊂ S ℓ . Then thefiber F ( g ) coincides with { Φ g } . Proof.
Let z ∈ S ℓ and take ϕ ∈ π − ∞ ( z ). For Φ ∈ F ( g ) we have that C ∗ Φ ( ϕ )( h· , e n i ) = ϕ (Φ( h· , e n i )) = ϕ ( h g, e n i ) . Since ϕ lies in the fiber over z ∈ S ℓ and g is continuously extendable to the unit sphere, by Lemma 2.8we conclude that ϕ ( h g, e n i ) = h g, e n i ( z ). It follows that π u ( C ∗ Φ ( ϕ )) = g ( z ) ∈ S ℓ . We can now computefor f ∈ A u ( B ℓ ) ϕ (Φ( f )) = C ∗ Φ ( ϕ )( f ) = δ g ( z ) ( f ) . As this equality holds for any ϕ in the fiber over z we obtain that Φ( f ) is constant in π − ∞ ( z ) so byLemma 2.8 it is continuously extendable to z where it coincides with f ◦ g ( z ). This in turn implies thatΦ( f ) is in H ∞ ( B ℓ ) and is continuously extendable to S ℓ where it coincides with f ◦ g so we must havethat Φ( f ) = C g ( f ). (cid:3) Through the previous result we see that any function g ∈ H ∞ ( B ℓ , ℓ ) with k g k = 1 and g ( B ℓ ) ⊂ B ℓ that can be continuously extendable to S ℓ and such that g ( S ℓ ) ⊂ S ℓ (as we mentioned, Id is an exampleof this) has a singleton middle fiber. Further, any linear isometry T : ℓ → ℓ (not necessarily onto) whenrestricted to the unit ball satisfies the above conditions (e.g. the shift operator S ( x , x , x , . . . ) =(0 , x , x , x , . . . )) and thus has a singleton fiber.A, perhaps more general, restatement of the previous lemma can be obtained by noting that thecondition g ( S ℓ ) ⊂ S ℓ is used to ensure that whenever Φ lies in the fiber over g , the correspondingmapping C ∗ Φ maps any ϕ in π − ∞ ( z ) to the same homomorphism in M u ( B ℓ ) whenever z ∈ S ℓ . Lemma 2.11.
Let Φ ∈ M u, ∞ ( B ℓ , B ℓ ) such that there exists a continuous function h : S ℓ → B ℓ satisfying C ∗ Φ ( π − ∞ ( z )) = { δ h ( z ) } for all z ∈ S ℓ . Then Φ coincides with the homomorphism C ξ (Φ) . We do not know, however, if there are homomorphisms Φ in M u, ∞ ( B ℓ , B ℓ ) satisfying the conditionsof Lemma 2.11 which are not in fibers of functions fulfilling the conditions of Lemma 2.10.So far we have shown middle fibers consisting only of the corresponding composition homomorphism.We can say that they behave like edge fibers. As we have anticipated, there are other middle fiberscontaining a ball (so, acting as interior fibers). Again we begin with a prototype example. Example 2.12.
Let g ∈ H ∞ ( B ℓ , ℓ ), given by g ( x ) = x e . Then g satisfies k g k = 1, g ( B ℓ ) ⊂ B ℓ andthere exists an analytic injection Ψ g : B H ∞ ( B ℓ ,ℓ ) → F ( g ).For every h ∈ B H ∞ ( B ℓ ,ℓ ) , k ∈ N let s h,k ∈ H ∞ ( B ℓ , ℓ ) be defined by s h,k ( x ) = x e + x ∞ X j =1 h h ( x ) , e j i e k + j , for all x ∈ B ℓ . Note that k s h,k ( x ) k = | x | + | x | k h ( x ) k ≤ | x | + | x | < . (5)Thus we see that s h,k ( B ℓ ) ⊂ B ℓ . We now define for each k ∈ N the composition homomorphismΨ k : B H ∞ ( B ℓ ,ℓ ) → M u, ∞ ( B ℓ , B ℓ ) as Ψ k ( h )( f ) = C s h,k ( f ) = f ◦ s h,k .Take Ψ g to be a weak-star limit along a ultrafilter of the sequence (Ψ k ) k . Now, proceeding as inTheorem 2.3 we can see that Ψ g is an analytic injection whose image is contained in F ( g ).The arguments in the previous example can be slightly modified to reach more cases as we see inthe following theorem. The hypothesis go in line with the idea that we need conditions that ensure wehave some ‘room’ to work with to obtain an analytic injection. This is done by ensuring (perhaps aftera change of coordinates) that g does not depend on at least one variable. Additionally, as we want toextend these ideas beyond functions g with g (0) = 0, we include the condition k g (0) k + k g − g (0) k ≤ .
12 for g − g (0). V. DIMANT AND J. SINGER
Theorem 2.13.
Let g ∈ S H ∞ ( B ℓ ,ℓ ) with g ( B ℓ ) ⊂ B ℓ . If g satisfies:(1) k g (0) k + k g − g (0) k ≤ . (2) There is a linear transformation P : ℓ → ℓ with k P k ≤ and non-trivial kernel such that g = g ◦ P .Then there exists an analytic injection Ψ g : B H ∞ ( B ℓ ,ℓ ) → F ( g ) . Proof.
We first prove the result with the additional hypothesis g (0) = 0. Fix w ∈ S ℓ ∩ Ker ( P ). Wedefine for each k ∈ N , h ∈ B H ∞ ( B ℓ ,ℓ ) the function s h,k ∈ H ∞ ( B ℓ , ℓ ) as s h,k ( x ) = k X j =1 h g ( x ) , e j i e j + h x, w i ∞ X j =1 h h ( x ) , e j i e k + j . Note that for every x ∈ B ℓ we have that(6) k s h,k ( x ) k ≤ k X j =1 |h g ( x ) , e j i| + |h x, w i| ≤ k g ( x ) k + |h x, w i| . Now, since g (0) = 0 and k g k ≤ g = g ◦ P and obtain k g ( x ) k + |h x, w i| ≤ k P ( x ) k + |h x, w i| = k P ( x − h x, w i w ) k + |h x, w i| ≤ k x − h x, w i w k + |h x, w i| < . (7)Joining both inequalities we obtain that s h,k ( B ℓ ) ⊂ B ℓ . For k ∈ N let Ψ k : B H ∞ ( B ℓ ,ℓ ) → M u, ∞ ( B ℓ , B ℓ )be given by Ψ k ( h )( f ) = C s h,k ( f ) = f ◦ s h,k . The proof of this case finishes as usual, defining Ψ g to be aweak-star limit along a ultrafilter of the sequence (Ψ k ) k and repeating the arguments used in Theorem2.3.For the general case, let e g = g − g (0) and set for every k ∈ N the function s h,k ∈ H ∞ ( B ℓ , ℓ ) as s h,k ( x ) = k X j =1 h g ( x ) , e j i e j + k e g kh x, w i ∞ X j =1 h h ( x ) , e j i e k + j . Repeating the computation done in (6) and (7) for k X j =1 h e g ( x ) , e j i e j + k e g kh x, w i ∞ X j =1 h h ( x ) , e j i e k + j yields that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X j =1 h e g ( x ) , e j i e j + k e g kh x, w i ∞ X j =1 h h ( x ) , e j i e k + j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < k e g k , for all x ∈ B ℓ . Now the hypothesis k g (0) k + k g − g (0) k ≤ s h,k ( B ℓ ) ⊂ ( B ℓ ). Theargument ends by proceeding as in the first case. (cid:3) Example 2.14.
Fix m ∈ N , Q : ℓ → ℓ an m − homogeneous polynomial with k Q k B ℓ = 1 and a linearprojection P : ℓ → ℓ . Define for z ∈ S ℓ and 0 < α < g : B ℓ → ℓ by g ( x ) = α ( Q ◦ P )( x ) + (1 − α ) z. Note that g (0) = (1 − α ) z so for every x ∈ B ℓ we have the inequality k g (0) k + k g ( x ) − g (0) k = (1 − α ) k z k + α k Q ◦ P ( x ) k < . Further, since P satisfies P ◦ P = P , it follows that g = g ◦ P . As a result g satisfies the conditions ofTheorem 2.13 and thus there is an analytic copy of a ball in the fiber F ( g ). OMOMORPHISM BETWEEN UNIFORM ALGEBRAS OVER A HILBERT SPACE. 9
Throughout this section we have presented conditions for middle fibers to be either isolated or tocontain a ball. However, as we do not know the complete description of the spectrum, it remains openif these are the only possible alternatives for the remaining fibers. For example we do not have a clue ofwhat happen with the (middle) fiber over the function g ∈ H ∞ ( B ℓ , ℓ ) given by g ( x ) = ( x n ) n ∈ N . Thisfunction is clearly extended to S ℓ but the extension do not map S ℓ into S ℓ . Also, since g ( x ) = 0 if andonly if x = 0 it can not exist a projection P with non trivial kernel such that g ◦ P = g . Thus, g does notsatisfy the conditions of Lemma 2.10 nor those of Theorem 2.13. There are, of course, many functionsnot satisfying the conditions of the referred Lemma and Theorem but this mapping g is a very naturalelement of H ∞ ( B ℓ , ℓ ) so it might be interesting to have a description of its fiber.3. Vector-Valued Cluster Sets
Before focusing on the specific case of ℓ we present some general results regarding vector-valued clustersets for X, Y
Banach spaces.Let g ∈ B H ∞ ( B Y ,X ∗∗ ) , f ∈ A u ( B X ). We define the vector-valued cluster set of f over g as C ℓ M u, ∞ ( B X ,B Y ) ( f, g ) = { h ∈ H ∞ ( B Y ) : ∃ ( g α ) ⊂ B H ∞ ( B Y ,X ∗∗ ) , g α w ∗ −−→ g satisfying C g α ( f )( y ) → h ( y ) ∀ y ∈ B Y } . Here g α w ∗ −−→ g denotes the weak-star convergence associated to the identification(8) H ∞ ( B Y , X ∗∗ ) = L ( G ∞ ( B Y ) , X ∗∗ ) = ( G ∞ ( B Y ) b ⊗ π X ∗ ) ∗ . In this topology, g α w ∗ −−→ g whenever g α ( y )( x ∗ ) → g ( y )( x ∗ ) for all y ∈ B Y , x ∗ ∈ X ∗ . We direct the readerto [19] for the proof of the first identification and further reference. Whenever there is no ambiguityregarding the vector-valued spectrum referenced we simply write C ℓ ( f, g ) to denote the cluster set of f over g .We mentioned in the introduction that the scalar-valued cluster set was first considered by I.J. Scharkin [22] for the unit disk D . There it is shown that the cluster set C ℓ D ( f, z ) is a compact subset of b f ( π − ( z )), where b f ( ϕ ) = ϕ ( f ) . The inclusion C ℓ B X ( f, z ) ⊂ b f ( π − ( z ))(9)still holds for a general Banach space (see for instance [4, Lem. 2.2]). Whether or not the equality in(9) holds is known as the Cluster Problem and remains open. There are, however, positive results forinstance in [17] and [4] where the equality in (9) is proved for every f ∈ A u ( B ℓ ) and every z ∈ B ℓ .The vector-valued cluster set C ℓ M u, ∞ ( B X ,B Y ) ( f, g ) is analogously related to the evaluations Φ( f ) = b f (Φ) for Φ ∈ F ( g ) as we show in the following Lemma: Lemma 3.1.
Let g ∈ B H ∞ ( B Y ,X ∗∗ ) , f ∈ A u ( B X ) . Then the vector-valued cluster set C ℓ ( f, g ) is aweak-star compact subset of b f ( F ( g )) .Proof. Let h ∈ C ℓ ( f, g ). Then there exists a net g α in B H ∞ ( B Y ,X ∗∗ ) such that g α w ∗ −−→ g and C g α ( f )( y ) → h ( y ) for all y ∈ B Y . Since ( C g α ) α is a subset of the compact set M u, ∞ ( B X , B Y ) we can find a convergingsub-net ( C g αβ ) β with limit Φ . Then Φ( f )( y ) = lim β C g αβ ( f )( y ) = h ( y ).Additionally, since g α w ∗ −−→ g means that g α ( y )( x ∗ ) → g ( y )( x ∗ ) for all y ∈ B Y , x ∗ ∈ X ∗ , we have ξ (Φ)( y )( x ∗ ) = Φ( x ∗ )( y ) = lim β C g αβ ( x ∗ )( y ) = lim α g α β ( y )( x ∗ ) = g ( y )( x ∗ ) . This proves the inclusion C ℓ ( f, g ) ⊂ b f ( F ( g )).Now we move on to show that the cluster C ℓ ( f, g ) is weak-star compact. It is readily seen that C ℓ ( f, g ) ⊂ k f k B H ∞ ( B Y ) so it only remains to show that C ℓ ( f, g ) is weak-star closed.Let U denote the set of all weak-star neighborhoods U of g in H ∞ ( B Y , X ∗∗ ). To simplify the notationwe put e U = U ∩ B H ∞ ( B Y ,X ∗∗ ) for each U ∈ U . We are going to prove the equality C ℓ ( f, g ) = \ U ∈U b f ( j ( e U )) w ∗ , where j : H ∞ ( B Y , X ∗∗ ) → M u, ∞ ( B x , B Y ) is the mapping defined by j ( u ) = C u .Let h ∈ T U ∈U b f ( j ( e U )) w ∗ . Then for every U ∈ U and any weak-star neighborhood V of h there exists afunction g U,V such that(1) g U,V ∈ U ∩ B H ∞ ( B Y ) .(2) C g U,V ( f ) ∈ V .. We consider all possible pairs ( U, V ) with the order given by the reverse inclusion (that is, ( U ′ , V ′ ) ≥ ( U, V ) whenever both U ′ ⊂ U and V ′ ⊂ V ). As a result, we have obtained a net ( g U,V ) ( U,V ) indexed inthe pairs of weak-star neighborhoods of g and h respectively. It is clear from construction that this netsatisfies(1) g U,V w ∗ −−→ g .(2) C g U,V ( f ) w ∗ −−→ h .We conclude that h ∈ C ℓ ( f, g ) and thus the inclusion T U ∈U b f ( j ( e U )) w ∗ ⊂ C ℓ ( f, g ) holds. To check theconverse, let h ∈ C ℓ ( f, g ) satisfying(1) g α w ∗ −−→ g .(2) C g α ( f ) w ∗ −−→ h .Given U ∈ U , let α such that g α ∈ U for all α ≥ α . Since h ∈ ( C g α ) ( f ) w ∗ α ≥ α it follows that h ∈ b f ( j ( e U )) w ∗ . Being the neighborhood U arbitrary the desired equality is proved. (cid:3) Cluster sets for M u, ∞ ( B ℓ , B ℓ ) . Back in Section 2 we presented results regarding the size of thefibers.
A priori , this is directly related to the size of the sets b f ( F ( g )). However, we show that the toolsdeveloped for Section 2 can be adapted for a set of functions to gain insight on the size of cluster side ofthe inclusion C ℓ ( f, g ) ⊂ b f ( F ( g )) . This also goes in line with the philosophy of [1] of looking for sets of functions with large cluster sets.The study of cluster sizes for different functions f differs from that of the fibers in the previous sectionin many aspects, but mainly it is worth remarking that we now lack the possibility to try different “testfunctions” as the function is fixed when we consider a certain cluster. Consequently we can not take fulladvantage of the previous results for the fibers as the proofs require a set of test functions instead of asingle one. Additionally, it is worth noting that the inclusion C ℓ ( f, g ) ⊂ b f ( F ( g )) proved in Lemma 3.1tells us that we can only look for non-singleton cluster sets for functions g with non-singleton fibers.Also, if f can be approximated by finite type polynomials (i. e. polynomials that are linear combina-tions of products of linear functionals) then Φ( f ) = C ξ (Φ) ( f ). Thus in this case for any function g thecluster set is singleton: C ℓ ( f, g ) = b f ( F ( g )) = { C ξ (Φ) ( f ) } .By [6, Lem. 6.2], a function f ∈ A u ( B ℓ ) can be approximated by finite type polynomials if and onlyif it is weakly continuous at every x ∈ B ℓ .Hence, the cluster set C ℓ ( f, g ) could be large whether f is not weakly continuous at some point and g is an inner or middle function with non-singleton fiber.We begin by showing that if g is an inner function and f is not weakly continuous at some point ofthe image of g then the cluster set Cℓ ( f, g ) contains an analytic copy of the complex disk. The proof isinspired by [13, Th. 3.3] (see also [8, Th. 3.1]). OMOMORPHISM BETWEEN UNIFORM ALGEBRAS OVER A HILBERT SPACE. 11
Proposition 3.2.
Let g ∈ B H ∞ ( B ℓ ,ℓ ) and let f ∈ A u ( B ℓ ) which is not weakly continuous at g ( x ) forcertain x ∈ B ℓ . Then, the cluster set Cℓ ( f , g ) contains an analytic copy of the complex disk D .Proof. Let k g k < r <
1. If f is not weakly continuous at g ( x ) it should exist a net ( y α ) α ⊂ B ℓ suchthat y α w ∗ → y = g ( x ) and | f ( y α ) − f ( y ) | > ε , for certain ε >
0. Note that since the argument in [6,Lem. 6.2] can be repeated in any ball we can chose that the net ( y α ) α is contained in the ball B − r ( y ).This is important for the well definition (for each k ∈ N ) of the following mapping:Ψ k : D → M u, ∞ ( B ℓ , B ℓ )Ψ k ( λ )( f )( x ) = f ( g ( x ) + λ ( y α − y )) . The inclusion M u, ∞ ( B ℓ , B ℓ ) ⊂ L ( A u ( B ℓ ) , H ∞ ( B ℓ )) = ( A u ( B ℓ ) b ⊗ π G ∞ ( B ℓ )) ∗ and the fact thateach Ψ k is a bounded analytic mapping, allow us to see the sequence (Ψ k ) k contained in the ball of H ∞ ( D , L ( A u ( B ℓ ) , H ∞ ( B ℓ ))) of radius k f k . Recall from (8) that H ∞ ( D , L ( A u ( B ℓ ) , H ∞ ( B ℓ ))) is adual space. Thus, the weak-star compactness of the ball assures the existence of a net (Ψ k α ) α weak-starconvergent to an element Ψ g ∈ H ∞ ( D , L ( A u ( B ℓ ) , H ∞ ( B ℓ ))).It is easy to see that for any λ ∈ D , Ψ g ( λ ) is multiplicative. Hence, Ψ g ∈ H ∞ ( D , M u, ∞ ( B ℓ , B ℓ )).Also, note that each Ψ k ( λ ) is a composition homomorphism. Thus, we have, for any λ ∈ D , x ∈ B ℓ ,Ψ g ( λ )( f )( x ) = lim α Ψ k α ( λ )( f )( x ) = lim α C ξ (Ψ kα ( λ )) ( f )( x ) . Since ξ (Ψ k α ( λ )) is weak star convergent to g we obtain that Ψ g ( λ )( f ) ∈ Cℓ ( f , g ).For the injectivity, first, observe that Ψ g can be continuously extended to D . Second, we knowΨ g (0)( f )( x ) = f ( g ( x )) = f ( y ) and Ψ g (1)( f )( x ) = lim α f ( y α ) . Therefore, | Ψ g (0)( f )( x ) − Ψ g (1)( f )( x | ≥ ε and so the mapping Ψ g ( · )( f )( x ) belongs to the diskalgebra A ( D ) and it is not constant. Hence, it should exist λ ∈ D and r > D ( λ , r ).This implies that the mapping Ψ g ( · )( f ) : D ( λ , r ) → Cℓ ( f , g ) is injective. Finally, composing withthe mapping from D to D ( λ , r ) given by [ λ r λ + λ ] we obtain that Cℓ ( f , g ) contains an analyticcopy of D . (cid:3) Now, if we consider very special functions f we can prove that the cluster set Cℓ ( f, g ) contains aninfinite dimensional ball for any inner function g ∈ B H ∞ ( B ℓ ,ℓ ) . Proposition 3.3.
Let f ( x ) = P ∞ j =1 x Nj with N ∈ N ≥ . Then for every g ∈ B H ∞ ( B ℓ ,ℓ ) the cluster Cℓ ( f , g ) contains an analytic copy of the ball B H ∞ ( B ℓ ) .Proof. Fix g ∈ B H ∞ ( B ℓ ,ℓ ) , k g k < r <
1. We define for each k ∈ N the mapping Ψ k : B H ∞ ( B ℓ ) →M u, ∞ ( B ℓ , B ℓ ) given byΨ k ( h )( f )( x ) = f k X j =1 h g ( x ) , e j i e j + (1 − r ) h ( x ) e k +1 , for h ∈ B H ∞ ( B ℓ ) , f ∈ A u ( B ℓ ) , x ∈ B ℓ . Clearly, Ψ k is a bounded analytic mapping and for any h ,Ψ k ( h ) is a composition homomorphism.Argumenting as in the previous proposition we know that there is a net (Ψ k α ) α weak-star convergentto an element Ψ g ∈ H ∞ ( B H ∞ ( B ℓ ) , L ( A u ( B ℓ ) , H ∞ ( B ℓ ))).By definition Ψ g ( h )( f )( x ) = lim α Ψ k α ( h )( f )( x ) = lim α C ξ (Ψ kα ( h )) ( f )( x ) . Since ξ (Ψ k α ( h )) is weak star convergent to g we obtain that Ψ g ( h )( f ) ∈ Cℓ ( f , g ). Further, for every k ∈ N , h ∈ B H ∞ ( B ℓ ) we haveΨ k ( h )( f )( x ) = k X j =1 h g ( x ) , e j i N + (1 − r ) N h ( x ) N . Taking the limit as k → ∞ we obtainΨ g ( h )( f )( x ) = ∞ X j =1 h g ( x ) , e j i N + (1 − r ) N h ( x ) N . As a result, if Ψ g ( h )( f ) = Ψ g ( h ′ )( f ) we obtain h ( x ) N = h ′ ( x ) N for all x ∈ B ℓ , so by the identity principlethere exists ζ ∈ C satisfying ζ N = 1 such that h ′ = ζh . Finally note that the equality k h − ζh k = | − ζ |k h k shows that if we put τ = min {| − ζ | : ζ N = 1 , ζ = 1 } then for any fixed h ∈ B H ∞ ( B ℓ ) satisfying k h k = we have that Ψ g is injective when restricted to B s ( h ) for s = τ τ +1) . In other words, we have seen thatthe cluster Cℓ ( f , g ) contains an analytic copy of a ball B s ( h ) ⊂ H ∞ ( B ℓ ).Finally, composing with the canonical mapping χ : B H ∞ ( B ℓ ) → B s ( h ) given by χ ( u ) = su + h weconclude that Ψ g ◦ χ : B H ∞ ( B ℓ ) → Cℓ ( f , g ) is an analytic injection. (cid:3) The same result can be obtained for functions g satisfying the hypothesis of Theorem 2.13. Proposition 3.4.
Let f ( x ) = P ∞ j =1 x Nj with N ∈ N ≥ . Then for every g ∈ S H ∞ ( B ℓ ,ℓ ) with g ( B ℓ ) ⊂ B ℓ such that(1) k g (0) k + k g − g (0) k ≤ . (2) There exists a linear transformation P : ℓ → ℓ with k P k ≤ and non-trivial kernel such that g = g ◦ P ,the cluster Cℓ ( f , g ) contains an analytic copy of the ball B H ∞ ( B ℓ ) .Proof. Let e g = g − g (0) and define for each k ∈ N the mapping Ψ k : B H ∞ ( B ℓ ) → M u, ∞ ( B ℓ , B ℓ ) givenby Ψ k ( h )( f )( x ) = f k X j =1 h g ( x ) , e j i e j + k e g kh x, w i h ( x ) e k +1 . Now, the result is obtained following the steps of the proof of Proposition 3.3 combined with somearguments from the proof of Theorem 2.13. (cid:3)
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