Poncelet.sciences.univ-metz.fr

Abstract. We show that there do not exist finitely generated,non-principal ideals of denominators in the disk-algebra A(D). Ourproof involves a new factorization theorem for A(D) that is basedon Treil’s determination of the Bass stable rank for H∞.
Let H∞ be the uniform algebra of bounded analytic functions on the open unit disk D and let A(D) denote the disk-algebra; that is thesubalgebra of all functions in H∞ that admit a continuous extensionto the Euclidean closure D = {z ∈ C : |z| ≤ 1} of D.
Let γ = n/d be a quotient of two functions n and d in A = H∞ or A = A(D). It is well known that every ideal of denominators in A = H∞ is a principal ideal, since H∞ is a pseudo-B´ latter means that each pair of functions in H∞ has a greatest commondivisor (see [11]). The situation in A(D) is completely different, dueto the fact that A(D) does not enjoy the property of being a pseudo-B´ ezout ring. For example 1 − z and (1 − z) exp(− 1+z ) do not have a greatest common divisor. Answering several questions of Frank Forelli[3, 4], the first author could prove in his Habilitationsschrift [8] thatany closed ideal in A(D) is an ideal of denominators; that an ideal ofdenominators is closed if and only if γ ∈ L∞(T); that the complementinside D of the zero set of D(γ) is the set of points a in D for which there exists a neighborhoodU in D such that |γ| admits a continuous extension to U ; and that every 1991 Mathematics Subject Classification. 46J15, 30D50.
Key words and phrases. disk-algebra, ideals of denominators.
ideal of denominators in A(D) contains a function f whose zero setequals the zero set of the ideal (one then says that D(γ) has the Forelli-property.) The proof of this last result was based on the approximationtheorem of Carleman (see [5, p. 135 ]).
In the present note we shall be concerned with the question whether finitely generated, but non-principal ideals in A(D) can be representedas ideals of denominators. It turns out that this is not the case. Ourproof uses as main ingredient a deep result of Treil [14] that tells usthat H∞ has the Bass stable rank one. This is a generalization goingfar beyond the corona theorem and tells us that whenever (f, g) is acorona pair in H∞, that is whenever |f | + |g| ≥ δ > 0 in D, thenthere exists h ∈ H∞ such that f + hg is invertible in H∞. We actuallyneed an extension of this found by the second author of this paper toalgebras of the form H∞ = {f ∈ H∞ : f extends continously to where E is a closed subset of the unit circle T. That result on theBass stable rank of H∞ will be used to prove a factorization property of functions in A(D), which will be fundamental to achieve our maingoal of characterizing the finitely generated ideals of denominators inA(D).
¿From the applications point of view, there is also a control the- oretic motivation for considering the question of finding out whetherthere are ideals of denominators which are finitely generated, but notprincipal. Indeed, [10, Theorem 1, p.30] implies that if a plant is in-ternally stabilizable, then the corresponding ideal of denominators isgenerated by at most two elements, and moreover, if an ideal of de-nominators corresponding to a plant is principal, then the plant has aweak coprime factorization. In light of these two results, our result onthe nonexistence of nonprincipal finitely generated ideal of denomina-tors in the disk-algebra implies that every internally stabilizable plantover the disk-algebra has a weak coprime factorization. Finally, sincethe disk-algebra is pre-B´ ezout [12], it also follows that every plant hav- ing a weak coprime factorization, possesses a coprime factorization [10,Proposition, p. 54]. Consequently, every internally stabilizable plantover the disk-algebra has a coprime factorization.
Cohen’s factorization theorem for commutative, non-unital Banach algebras X tells us that if X has a bounded approximate identity, thenevery f ∈ X factors as f = gh, where both factors are in X (see e.g.
[1, p.76]). For A(D) this may be applied to every closed ideal of the form X = I(E, A(D)) := {f ∈ A(D) : f |E ≡ 0}, whenever E is a closedsubset of T of Lebesgue measure zero (note that (en) with en = 1 − pnEis such a bounded approximate identity, where pE is a peak functionin A(D) associated with E; see [7, p. 80] for a proof of the existenceof pE). In the present paragraph we address the following question:Let f ∈ A(D). Suppose that f vanishes on E ⊆ T and that E canbe written as E = E1 ∪ E2, where the Ej are closed, not necessarilydisjoint.
(1) Do there exist factors fj of f such that f = f1f2 and such that (2) Do there exist factors fj of f such that f = f1f2 and such that f1 vanishes only on E1 and f2 has the same zero set as f ? We will first answer question (2) above. The proof works along the model of [8, Proposition 2.3]. It uses the following lemma that is basedon the approximation theorem of Carleman (see [5]): Lemma 1.1. [8, Lemma 1.1] Let I be an open interval. Then for everycontinuous function u and every positive, continuous error functionε(x) > 0 on I there exists a C1-function v on I such that |u − v| < εon I.
We shall also give an answer to a variant of question (1) whenever the sets Ej are disjoint closed subsets in D. That result will be themain new ingredient to prove our result on the ideals of denominators.
In the sequel, let Z(f ) denote the zero set of a function.
Theorem 1.2. Let E be closed subset of T and suppose that f |E ≡ 0for some f ∈ A(D), f ≡ 0. Then there exists a factor g of f thatvanishes exactly on E. Moreover, g can be taken so that the quotientf /g vanishes everywhere where f does.
Proof. We shall construct an outer function g ∈ A(D) with Z(g) = Esuch that |f | ≤ |g|2 on T. Then, by the extremal properties for outerfunctions (see [6]), |f | ≤ |g|2 on D. Hence |f /g| ≤ |g| on D \ E. Clearlythis quotient has a continuous extension (with value 0) at every pointin E. Thus f = gh for some h ∈ A(D). To construct g, we write T \ Eas a countable union of open arcs In. Note that f vanishes at the two(or in case E is a singleton, a single) boundary points of In. Let pEbe a peak function associated with E. Consider on T the continuousfunction q = |f | + |1 − pE|. Then |q| > 0 on In, Z(q) = E and q = 0on the boundary points of In. If the outer function associated with qwould be in A(D), we were done. But we are not able to prove that.
So we need to proceed as in [8, p. 22]. Let In =]an, bn[. Using Lemma1.1, there exists functions un ∈ C1(In) so that Let u : T → R be defined by u = un on In, n = 1, 2, . . . , and u = 0elsewhere on T. Then u ∈ C(T), u ≥ 0, and by the left inequality in(1.1), log u ∈ L1(T). Since u ∈ C1(T \ Z(q)) and u|Z(q) ≡ 0, the outerfunction belongs by [12, p.52] to A(D). It is clear that on T we have |g|2 = 2u ≥|q| ≥ |f | and that g vanishes only on E. Moreover, |f |/|g| ≤ |g| showsthat f /g ∈ A(D) and that Z(f /g) = Z(f ).
The following H∞, A(D) -multiplier type result will yield our final factorization result (Theorem 1.4), that will be central to our study ofideals of denominators.
Theorem 1.3. Let E be a closed subset of Lebesgue measure zero in Tand let f ∈ H∞ be a function that has a continuous extension to T \ E;i.e f ∈ H∞. Suppose that 0 does not belong to the cluster set of f at each point in E. Then there exists a function h ∈ H∞, invertible in Proof. Consider a peak function pE ∈ A(D) associated with E. Byassumption, the ideal I generated by f and 1 − pE in H∞ is proper.
(Here we have used the corona theorem for H∞ [2].) Since H∞ has the stable rank one ([13, Theorem 5.2]), there exist h invertible in H∞ and g ∈ H∞ such that hf + g(1 − p the only points of discontinuity of g are located on E, we see thatg(1 − pE) ∈ A(D). Thus hf ∈ A(D).
Theorem 1.4. Let f ∈ A(D). Suppose that Z(f ) = E1 ∪ E2, wherethe Ej are two disjoint closed sets in D. Then there exist factors fj off in A(D) such that f = f1f2 and Z(fj) = Ej.
Proof. By assumption, 2ε := dist (E1, E2) > 0. Choose around eachpoint α ∈ E1 ∩T a symmetric open arc A ⊆ T with center α and lengthε. Due to compactness, there are finitely many of these arcs whoseunion covers E1 ∩ T. Let V be the union of these arcs. By combiningtwo adjacent arcs, we may assume that V writes as V = ∪N ]α the closures of the arcs Ij :=]αj, βj[ being pairwise disjoint. We alsohave that V ∩ E2 = ∅ as well as E1 ∩ ∂V = ∅.
We first consider the outer factor F of f . Note that Consider the factorization F = F1F2, where Then the Fj have continuous extensions to every point in T \ ∂V . AlsoZ(Fj) = Ej ∩ T. Applying Theorem 1.3, there exists an invertiblefunction h ∈ H∞ so that G we obtain that outside the zeros of F1, that is outside E1, the function 2 is continuous. Note that h is continuous on E1 as well as F2 itself.
2 ∈ A(D). Thus F = G1G2 satisfies Z (Gj ) = Ej ∩ T.
Now suppose that f has an inner factor Θ = BSµ. Let σ(Θ) = {σ ∈ D : lim inf |Θ(z)| = 0} be the support of Θ. Note that σ(Θ) ⊆ Z(f ) = E1 ∪ E2. Now wesplit the support of Θ into the corresponding parts Ξ1 := σ(Θ) ∩ E1and Ξ2 := σ(Θ) ∩ E2 and write Θ as Θ1Θ2. Then fj = ΘjGj gives thedesired factorization.
Notation: Let A be a commutative unital algebra. For fj ∈ A, let denote the ideal generated by the functions fj (j = 1, . . . , N ). We alsodenote the principal ideal I(f ) by f A.
If γ = n/d is a quotient of two elements n, d in A \ {0}, then is the ideal of denominators generated by γ. If γ ∈ A, then it is easyto see that D(γ) = A.
Finally, if I is an ideal in A(D), then Z(I) = The following two Lemmas are well known (see [9]) and work for quite general function algebras. For the reader’s convenience we presentsimple proofs.
Lemma 2.1. Let I be an ideal in A(D) and let M be a maximal idealcontaining I. Suppose that I = IM . Then I is not finitely generated.
Proof. Suppose that I = (f1m1, . . . , fN mN ) for some fj ∈ I and mj ∈M . Then j |2 ≥ 1/C on D \ Z (I ). Since Z (I ) is nowhere dense, we get this estimate to hold true on D. But this is a contradiction, sinceall the mj vanish at some point.
Lemma 2.2. Let I be an ideal in A(D). Suppose that Z(I) ⊆ D. ThenI is generated by a finite Blaschke product.
Proof. Due to compactness of Z(I), we know that Z(I) is finite (orempty). Let Z(I) = {a1, · · · , aN } and let mn be the highest multiplic-ity of the zero an at which all functions in I vanish. We claim that I isgenerated by the Blaschke product B associated with these (an, mn).
In fact, the inclusion I ⊆ I(B) is trivial, since B divides every functionin I. By construction, are finitely many functions fj ∈ I so that corona theorem for A(D), we have that 1 ∈ I(f1/B, . . . , fn/B). ThusB ∈ I.
The following works for every commutative unital ring.
Lemma 2.3. Let n, d be two functions in A such that I(n, d) = A.
Then D(n/d) = dA.
Proof. Let x, y ∈ A be such that 1 = xn + yd. Then f = x(f n) + (f y)d.
Now let f ∈ D(n/d). Hence f n = ad implies that f = x(ad) + (f y)d ∈dA. The reverse inclusion is trivial, since d ∈ D(n/d).
Lemma 2.3 applies in particular to A = A(D) if we assume that Corollary 2.4. Suppose that the greatest common divisor of two ele-ments n and d in A(D) is a unit. Then D(n/d) is a principal ideal.
Proof. Since A(D) is a Pre-B´ezout ring (see [12]) we have that I(n, d) =A(D). The rest follows from Lemma 2.3 above.
Proposition 2.5. Let B be a finite Blaschke product and let f ∈ A(D),f ≡ 0. Then D(B/f ) is a principal ideal generated by a specific factorof f .
Proof. Let b be the Blaschke product formed with the common zerosof B and f (multiplicities included). Consider the function F = f /band B∗ = B/b. We claim that D(B/f ) = I(F ). In fact,we obviouslyhave that D(B/f ) = D(B∗/F ). But F does not vanish at the zeros ofB∗; so, by the corona theorem for A(D), I(B∗, F ) = A(D). By Lemma2.3, we get that D(B/f ) = D(B∗/F ) = I(F ).
Observation 2.6. Let I be an ideal in A(D). Suppose that f ∈ I andthat f = gh, where g, h ∈ A(D) and Z(g) ∩ Z(I) = ∅. Then h ∈ I.
This follows from the fact that the maximal ideal space is D: indeed, the assumption implies that the ideal generated by g and I is thewhole algebra; hence 1 = ag + r where a ∈ A(D) and r ∈ I. Thush = a(gh) + hr ∈ I.
Proposition 2.7. Let I = D(n/d) and J = D(d/n) be ideals of de-nominators in A(D). Suppose that Z(J) ⊆ D. Then J and I areprincipal ideals.
Proof. If Z(J ) ⊆ D, then, by Lemma 2.2, J is a principal ideal gen-erated by a finite Blaschke product B.
is a principal ideal, too. In fact, let γ = n/d. Suppose that J =D(d/n) = BA(D). Since n ∈ J, we have that n = BN for someN ∈ A(D). Since B ∈ J, Bd = kn = kBN; so d = kN . Thusγ = (BN )/(kN ) = B/k. Note that k and B have no common zerosinside D, otherwise J = D(k/B) would contain a factor of B. ThusI(B, k) = A(D). Hence, by Lemma 2.3, I = kA(D).
Applying Theorem 1.4, we obtain the following Proposition 2.8. Let I = D(n/d) be an ideal of denominators inA(D). Suppose that Z(I) ∩ Z(n) = ∅. Then I is a principal ideal.
Proof. Let I = D(n/d). Without loss of generality we may assumethat n and d have no common zeros (otherwise we split of the jointBlaschke product and use the fact that A(D) has the F -property; thatis that uf ∈ A(D) implies that f ∈ A(D) for any inner function u.
Note that by our assumption, Z(I) ⊆ Z(d) ⊆ Z(n) ∪ Z(I), and that this union is disjoint. By Theorem 1.4 we may factor d as d = d1d2,where Z(d1) = Z(I) and Z(d2) ∩ Z(I) = ∅. We claim that I = I1 :=D(n/d1). In fact, let f ∈ I1. Then f n = gd1 for some g ∈ A(D).
Then (d2f )n = g(d1d2) = gd, and hence d2f ∈ D(n/d) = I. ButZ(d2) ∩ Z(I) = ∅. Thus by the observation 2.6 above, we have thatf ∈ I. So D(n/d1) ⊆ D(n/d).
To prove the reverse inclusion, let f ∈ D(n/d). Then f n = hd for some h ∈ A(D). Hence f n = (hd2)d1. So f ∈ D(n/d1). We concludethat D(n/d1) = D(n/d). Since Z(d1) ∩ Z(n) = ∅, we obtain fromLemma 2.3 that I1(= I) is a principal ideal.
Recall that for α ∈ D, M (α) = {f ∈ A(D) : f (α) = 0} is the Using Theorem 1.4 and its companion Proposition 2.8, we are now ready to prove our main result on the structure of finitely generatedideals of denominators in A(D). We note that the result would holdfor the Wiener algebra W + of all absolutely convergent power series in D as well, if Theorem 1.4 and Proposition 2.8 could be proven for W +.
Theorem 2.9. Let γ = n/d be a quotient in A(D). Then the ideal ofdenominators, D(γ) = {f ∈ A(D) : f γ ∈ A(D)}, is either a principalideal or not finitely generated.
Proof. Associate with I := D(γ) the set J = {γf : f ∈ D(γ)}. Thenit is straightforward to check that J is an ideal in A, too. In fact,J = D(1/γ).
Suppose that J is not proper; then Z(J ) := compactness, there exist finitely many fj ∈ D(γ) so that j (γfj ) for some gj ∈ A(D). Then 1/γ ∈ A(D); hence γ = 1/a for some a ∈ A(D). Then D(γ) = aA(D), the principalideal generated by a.
Case 1. Z(J ) ∩ Z(I) = ∅. Let α ∈ Z(I) ∩ Z(J ). Consider any f ∈ I.
Then f n = gd for some g ∈ J .
If α ∈ D, then f = (z − α)F and g = (z − α)G. Hence F n = Gd and so F ∈ I. Thus I = I · M (α).
If α ∈ T, then we use the fact that the maximal ideal M (α) contains an approximate unit and hence by the Cohen-Varopoulos factorizationtheorem [15], for any f, g ∈ M (α), there is a joint factor h ∈ M (α) off and g, say f = hF and g = hG for F, G ∈ A(D). Hence F n = Gdand again F ∈ I. Thus, also in this case, I = I · M (α).
By Lemma 2.1 above, I cannot be finitely generated.
Case 2. Z(I)∩Z(J ) = ∅. Then there exist f, g ∈ I such that 1 = f + n g.
Hence d = df + ng and so d(1 − f ) = ng. Thus γ = n = 1−f . Without loss of generality, we may assume that I is proper. Let α ∈ Z(I). Sinceg ∈ I, we have that Z(I) ⊆ Z(g). Hence 0 = g(α) and (since f ∈ I),f (α) = 0, too. So Z(I) ∩ Z(1 − f ) = ∅. By Proposition 2.8, I is aprincipal ideal.
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Verlaine, Ile du Saulcy, F-57045 Metz, France E-mail address: [email protected] Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, U.K.

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