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Abstract and Applied Analysis
VolumeΒ 2008Β (2008), Article IDΒ 459310, 13 pages
Research Article

Noncoherence of a Causal Wiener Algebra Used in Control Theory

Mathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, UK

Received 18 March 2008; Accepted 13 June 2008

Academic Editor: Ülle Kotta

Copyright Β© 2008 Amol Sasane. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let β„‚β‰₯0∢={π‘ βˆˆβ„‚βˆ£Re(𝑠)β‰₯0}, and let 𝒲+ denote the ring of all functions π‘“βˆΆβ„‚β‰₯0β†’β„‚ such that 𝑓(𝑠)=π‘“π‘Ž(𝑠)+βˆ‘βˆžπ‘˜=0π‘“π‘˜π‘’βˆ’π‘ π‘‘π‘˜(π‘ βˆˆβ„‚β‰₯0), where π‘“π‘ŽβˆˆπΏ1(0,∞),(π‘“π‘˜)π‘˜β‰₯0βˆˆβ„“1, and 0=𝑑0<𝑑1<𝑑2<β‹― equipped with pointwise operations. (Here 0π‘₯0005𝑒⋅ denotes the Laplace transform.) It is shown that the ring 𝒲+ is not coherent, answering a question of Alban Quadrat. In fact, we present two principal ideals in the domain 𝒲+ whose intersection is not finitely generated.

1. Introduction

The aim of this paper is to show that the ring 𝒲+ (defined below) is not coherent.

We first recall the notion of a coherent ring.

Definition 1.1. Let 𝑅 be a commutative ring with identity element 1, and let π‘…π‘š=𝑅×⋯×𝑅 (π‘š times). Suppose that 𝑓=(𝑓1,…,π‘“π‘š)βˆˆπ‘…π‘š.
(1) An element (𝑔1,…,π‘”π‘š)βˆˆπ‘…π‘š is called a relation on   𝑓 if 𝑔1𝑓1+β‹―+π‘”π‘šπ‘“π‘š=0.(1.1)(2) Let π‘“βŸ‚ denote the set of all relations on π‘“βˆˆπ‘…π‘š. (Then π‘“βŸ‚ is an 𝑅-submodule of the 𝑅-module π‘…π‘š.)(3) The ring 𝑅 is called coherent if for all π‘šβˆˆβ„• and all π‘“βˆˆπ‘…π‘š, π‘“βŸ‚ is finitely generated, that is, there exists a π‘‘βˆˆβ„• and there exist π‘”π‘—βˆˆπ‘“βŸ‚, π‘—βˆˆ{1,…,𝑑}, such that for all π‘”βˆˆπ‘“βŸ‚, there exist π‘Ÿπ‘—βˆˆπ‘…, π‘—βˆˆ{1,…,𝑑} such that 𝑔=π‘Ÿ1𝑔1+β‹―+π‘Ÿπ‘‘π‘”π‘‘.

An integral domain is coherent if and only if the intersection of any two finitely generated ideals of the ring is again finitely generated; see [1, Theorem 2.3.2, page 45].

The coherence of some rings of analytic functions has been investigated in earlier works. For example, McVoy and Rubel [2] showed that the Hardy algebra 𝐻∞(𝔻) is coherent, while the disc algebra 𝐴(𝔻) is not. Mortini and von Renteln proved that the Wiener algebra π‘Š+(𝔻) (of all absolutely convergent Taylor series in the open unit disc) is not coherent [3]. In this article, we will show that the ring 𝒲+ (defined below, and which is useful in control theory) is not coherent.

Notation 1. Throughout the article, we will use the following notation:
β„‚β‰₯0∢={π‘ βˆˆβ„‚βˆ£Re(𝑠)β‰₯0}.(1.2)

Definition 1.2. Let 𝒲+ denote the Banach algebra𝒲+={π‘“βˆΆβ„‚β‰₯0β†’β„‚|𝑓(𝑠)=ξπ‘“π‘Ž(𝑠)+βˆžβˆ‘π‘˜=0π‘“π‘˜π‘’βˆ’π‘ π‘‘π‘˜(π‘ βˆˆβ„‚β‰₯0),π‘“π‘ŽβˆΆ(0,∞)β†’β„‚,π‘“π‘ŽβˆˆπΏ1(0,∞),βˆ€π‘˜β‰₯0,π‘“π‘˜βˆˆβ„‚,(π‘“π‘˜)π‘˜β‰₯0βˆˆβ„“1,βˆ€π‘˜β‰₯0,π‘‘π‘˜βˆˆβ„,0=𝑑0<𝑑1<𝑑2<β‹―}(1.3)equipped with pointwise operations and the norm‖𝑓‖𝒲+∢=β€–π‘“π‘Žβ€–πΏ1+β€–(π‘“π‘˜)π‘˜β‰₯0β€–β„“1.(1.4)Here ξπ‘“π‘Ž denotes the Laplace transform of π‘“π‘Ž, given byξπ‘“π‘Ž(𝑠)=ξ€œβˆž0π‘’βˆ’π‘ π‘‘π‘“π‘Ž(𝑑)𝑑𝑑,π‘ βˆˆβ„‚β‰₯0.(1.5)

The above algebra arises as a natural class of transfer functions of stable distributed parameter systems in control theory; see [4, 5].

Our main result is the following.

Theorem 1.3. The ring 𝒲+ is not coherent.

The relevance of the coherence property in control theory can be found in [6, 7]. We will prove Theorem 1.3 following the same method as in the proof of the noncoherence of π‘Š+(𝔻) given by Mortini and von Renteln in [3].

In Section 3, we will give the proof of Theorem 1.3. But before doing that, in Section 2, we first prove a few technical results needed in the sequel.

2. Preliminaries

We first recall the definition of the Hardy algebra 𝐻∞ of the open right half plane.

Definition 2.1. Let 𝐻∞ denote the Hardy space of all bounded analytic functions in the open right half plane equipped with the normβ€–πœ‘β€–βˆžβˆΆ=supRe(𝑠)>0|πœ‘(𝑠)|,πœ‘βˆˆπ»βˆž.(2.1)

In order to prove our main result (Theorem 1.3), we will use the relation between the convergence in 𝐻∞ versus that in 𝒲+.

Lemma 2.2. If π‘“βˆˆπ’²+, then π‘“βˆˆπ»βˆž and β€–π‘“β€–βˆžβ‰€β€–π‘“β€–π’²+.

Proof. Let𝑓(𝑠)=ξπ‘“π‘Ž(𝑠)+βˆžξ“π‘˜=0π‘“π‘˜π‘’βˆ’π‘ π‘‘π‘˜(π‘ βˆˆβ„‚β‰₯0).(2.2)For π‘ βˆˆβ„‚β‰₯0, we have|ξπ‘“π‘Ž(𝑠)|=|ξ€œβˆž0π‘’βˆ’π‘ π‘‘π‘“π‘Ž(𝑑)𝑑𝑑|β‰€ξ€œβˆž0π‘’βˆ’Re(𝑠)𝑑|π‘“π‘Ž(𝑑)|π‘‘π‘‘β‰€ξ€œβˆž01β‹…|π‘“π‘Ž(𝑑)|𝑑𝑑=β€–π‘“π‘Žβ€–πΏ1,(2.3)and moreover,|βˆžξ“π‘˜=0π‘“π‘˜π‘’βˆ’π‘ π‘‘π‘˜|β‰€βˆžξ“π‘˜=0|π‘“π‘˜|π‘’βˆ’Re(𝑠)π‘‘π‘˜β‰€βˆžξ“π‘˜=0|π‘“π‘˜|β‹…1=β€–(π‘“π‘˜)π‘˜β€–β„“1.(2.4)So the result follows.

The maximal ideal π”ͺ0 (defined below) of 𝒲+ will play an important role in the remainder of this article.

Notation. Let π”ͺ0 denote the kernel of the complex algebra homomorphism 𝑓↦𝑓(0)βˆΆπ’²+β†’β„‚, that is,

Then π”ͺ0 is a maximal ideal of 𝒲+, and this maximal ideal plays an important role in the proof of our main result in the next section. We will prove a few technical results about π”ͺ0 in this section, which will be used in the sequel. The following result is analogous to [3, Lemma 1].

Lemma 2.3. Let 𝐿≠(0) be an ideal in 𝒲+ contained in the maximal ideal π”ͺ0. If 𝐿=𝐿π”ͺ0, that is, if every function π‘“βˆˆπΏ can be factorized in a product 𝑓=β„Žπ‘” of two functions β„ŽβˆˆπΏ and π‘”βˆˆπ”ͺ0, then 𝐿 cannot be finitely generated.

Proof. Suppose that𝐿=(𝑓1,…,𝑓𝑁)β‰ (0)(2.5)is a finitely generated ideal in 𝒲+ contained in the maximal ideal π”ͺ0. By our assumption, there are functions β„Žπ‘›βˆˆπΏ, π‘”π‘›βˆˆπ”ͺ0 with𝑓𝑛=β„Žπ‘›π‘”π‘›(𝑛=1,…,𝑁).(2.6)Since β„Žπ‘›βˆˆπΏ, there exist functions π‘ž(𝑛)π‘˜βˆˆπ’²+ withβ„Žπ‘›=π‘ξ“π‘˜=1π‘ž(𝑛)π‘˜π‘“π‘˜(𝑛=1,…,𝑁;π‘˜=1,…,𝑁).(2.7)From this it follows that𝑁𝑛=1|β„Žπ‘›|≀𝑁𝐢𝑁𝑛=1|𝑓𝑛|=𝑁𝐢𝑁𝑛=1|β„Žπ‘›π‘”π‘›|inβ„‚β‰₯0,(2.8)where 𝐢 is a constant chosen so thatβ€–π‘ž(𝑛)π‘˜β€–βˆžβ‰€πΆ,βˆ€π‘˜and𝑛.(2.9)(Here β€–β‹…β€–βˆž denotes the supnorm over β„‚β‰₯0.) This implies together with the Cauchy-Schwarz inequality thatπ‘βˆ‘π‘›=1|β„Žπ‘›|2≀(𝑁𝑛=1|β„Žπ‘›|)2≀𝑁2𝐢2(𝑁𝑛=1|β„Žπ‘›π‘”π‘›|)2≀𝑁2𝐢2(𝑁𝑛=1|β„Žπ‘›|2)(𝑁𝑛=1|𝑔𝑛|2).(2.10)This inequality holds for all π‘ βˆˆβ„‚β‰₯0. With π›ΏβˆΆ=1/(𝑁2𝐢2), we obtain the inequality𝛿≀𝑁𝑛=1|𝑔𝑛(𝑠)|2(2.11)for all points π‘ βˆˆπΈ, where𝐸∢={π‘ βˆˆβ„‚β‰₯0|𝑁𝑛=1|β„Žπ‘›(𝑠)|2>0}.(2.12)Since 𝐿≠(0), 𝐸 is a dense subset of β„‚β‰₯0 (for otherwise, if 𝑠0βˆˆβ„‚β‰₯0 is such that it has a neighbourhood 𝑉 in β„‚β‰₯0 where there is no point of 𝐸, then each β„Žπ‘› is identically zero in 𝑉, and by the identity theorem for holomorphic functions, each β„Žπ‘› is zero; consequently each 𝑓𝑛 is zero, and so 𝐿=(0), a contradiction). So by continuity, inequality (2.11) holds in β„‚β‰₯0. But this contradicts the fact that each 𝑔𝑛 vanishes at 0.

Remark 2.4. Lemma 2.3 can be proved purely algebraically using Nakayama's lemma. Indeed, it holds in the following more general algebraic situation: if 𝐼 is a nonzero ideal of a commutative domain 𝐷 contained in a maximal ideal 𝑀 and 𝐼=𝐼𝑀, then 𝐼 cannot be finitely generated. However, we have given an analytic proof in our special case above.

Since every maximal ideal is closed, π”ͺ0 is a commutative Banach subalgebra of 𝒲+, but obviously without identity element. But there is a substitute, namely the notion of the approximate identity, which turns out to be useful.

Definition 2.5. Let 𝑅 be a commutative Banach algebra (without identity element). One says that 𝑅 has an approximate identity if there exists a bounded sequence (𝑒𝑛)𝑛 of elements 𝑒𝑛 in 𝑅 such that for any π‘“βˆˆπ‘…,limπ‘›β†’βˆžβ€–π‘’π‘›π‘“βˆ’π‘“β€–=0.(2.13)

We will now prove the following result, which shows that the maximal ideal π”ͺ0 in 𝒲+ has an approximate identity.

Theorem 2.6. Let π‘’π‘›βˆΆ=𝑠𝑠+1/𝑛,π‘›βˆˆβ„•.(2.14)Then (𝑒𝑛)π‘›βˆˆβ„• is an approximate identity for π”ͺ0.

The existence of an approximate identity for the maximal ideal π”ͺ0 in 𝒲+ is not obvious. In order to prove Theorem 2.6, we will need the following lemma.

Lemma 2.7. Suppose Μ‚β€Œπ‘“βˆˆπ”ͺ0. Then, for all πœ–>0, there exists a Μ‚π‘βˆˆπ”ͺ0 such that 𝑝 has compact support in [0,∞), and β€–Μ‚β€Œπ‘“βˆ’Μ‚π‘β€–π’²+<πœ–.

Proof. Let πœ–>0 be given. Suppose that𝑓=π‘“π‘Ž+βˆžξ“π‘˜=0π‘“π‘˜π›Ώ(β‹…βˆ’π‘‘π‘˜),(2.15)where π‘“π‘ŽβˆˆπΏ1(0,∞), (π‘“π‘˜)π‘˜β‰₯0βˆˆβ„“1, and 0=𝑑0<𝑑1<𝑑2<β‹―. Since ∫∞0|π‘“π‘Ž(𝑑)|𝑑𝑑<∞, we can choose an 𝑀>0 large enough such thatξ€œβˆžπ‘€|π‘“π‘Ž(𝑑)|𝑑𝑑<πœ–4.(2.16)With π‘π‘Ž(𝑑)∢=π‘“π‘Ž(𝑑) if π‘‘βˆˆ[0,𝑀], and 0 otherwise, we have that π‘π‘ŽβˆˆπΏ1(0,∞) is compactly supported andβ€–π‘π‘Žβˆ’π‘“π‘Žβ€–πΏ1<πœ–4.(2.17)Furthermore, select π‘βˆˆβ„• such thatξ“π‘˜>𝑁|π‘“π‘˜|<πœ–4.(2.18)Now let π‘‡βˆˆ(0,∞) be any number satisfying 𝑑𝑁<𝑇<𝑑𝑁+1, and defineπ‘“π‘‡βˆΆ=βˆ’(ξ€œβˆž0π‘π‘Ž(𝑑)𝑑𝑑+0β‰€π‘˜β‰€π‘π‘“π‘˜).(2.19)Setπ‘βˆΆ=π‘π‘Ž+0β‰€π‘˜β‰€π‘π‘“π‘˜π›Ώ(β‹…βˆ’π‘‘π‘˜)+𝑓𝑇𝛿(β‹…βˆ’π‘‡).(2.20)Then Μ‚π‘βˆˆπ’²+ and̂𝑝(0)=ξ€œβˆž0𝑝(𝑑)𝑑𝑑=ξ€œβˆž0π‘π‘Ž(𝑑)𝑑𝑑+0β‰€π‘˜β‰€π‘π‘“π‘˜+𝑓𝑇=0.(2.21)So Μ‚π‘βˆˆπ”ͺ0. Clearly 𝑝 has compact support contained in [0,∞). We have|𝑓𝑇|=|ξ€œβˆž0π‘π‘Ž(𝑑)𝑑𝑑+0β‰€π‘˜β‰€π‘π‘“π‘˜|=|ξ€œβˆž0π‘“π‘Ž(𝑑)𝑑𝑑+βˆžξ“π‘˜=0π‘“π‘˜+ξ€œβˆž0(π‘π‘Ž(𝑑)βˆ’π‘“π‘Ž(𝑑))π‘‘π‘‘βˆ’ξ“π‘˜>π‘π‘“π‘˜|≀|ξ€œβˆž0𝑓(𝑑)𝑑𝑑|+β€–π‘π‘Žβˆ’π‘“π‘Žβ€–πΏ1+ξ“π‘˜>𝑁|π‘“π‘˜|=|Μ‚β€Œπ‘“(0)|+β€–π‘π‘Žβˆ’π‘“π‘Žβ€–πΏ1+ξ“π‘˜>𝑁|π‘“π‘˜|<0+πœ–4+πœ–4=πœ–2.(2.22)Thusβ€–Μ‚β€Œπ‘“βˆ’Μ‚π‘β€–π’²+=β€–π‘“π‘Žβˆ’π‘π‘Žβ€–πΏ1+ξ“π‘˜>𝑁|π‘“π‘˜|+|𝑓𝑇|<πœ–4+πœ–4+πœ–2=πœ–.(2.23)This completes the proof.

We are now ready to prove the existence of an approximate identity for the maximal ideal π”ͺ0 in 𝒲+.

Proof of Theorem 2.6. We have𝑒𝑛=𝑠𝑠+1/𝑛=𝑠+1/π‘›βˆ’1/𝑛𝑠+1/𝑛=1βˆ’1𝑛1𝑠+1/𝑛=1+ξ„žβˆ’1π‘›π‘’βˆ’π‘‘/𝑛.(2.24)Thus for an π‘›βˆˆβ„•,‖𝑒𝑛‖𝒲+=β€–βˆ’1π‘›π‘’βˆ’π‘‘/𝑛‖𝐿1+|1|=1+1=2.(2.25)Given Μ‚β€Œπ‘“βˆˆπ’²+, and πœ–>0 arbitrarily small, in view of Lemma 2.7, we can find a Μ‚π‘βˆˆπ”ͺ0 such that 𝑝 has compact support and β€–Μ‚β€Œπ‘“βˆ’Μ‚π‘β€–π’²+<πœ–. Thenβ€–π‘’π‘›Μ‚β€Œπ‘“βˆ’Μ‚β€Œπ‘“β€–π’²+β‰€β€–π‘’π‘›Μ‚π‘βˆ’Μ‚π‘β€–π’²++‖𝑒𝑛‖𝒲+β€–Μ‚β€Œπ‘“βˆ’Μ‚π‘β€–π’²++β€–Μ‚β€Œπ‘“βˆ’Μ‚π‘β€–π’²+.(2.26)So it is enough to prove thatlimπ‘›β†’βˆžβ€–π‘’π‘›Μ‚π‘βˆ’Μ‚π‘β€–π’²+=0(2.27)for all Μ‚π‘βˆˆπ”ͺ0 such that 𝑝 has compact support in [0,∞). We do this below.
We haveπ‘’π‘›Μ‚π‘βˆ’Μ‚π‘=𝑠+1/π‘›βˆ’1/𝑛𝑠+1/π‘›Μ‚π‘βˆ’Μ‚π‘=βˆ’1𝑛1𝑠+1/𝑛̂𝑝=βˆ’1𝑛(ξ„žπ‘’βˆ’π‘‘/π‘›βˆ—π‘).(2.28)Let 𝐢 denote the convolution π‘’βˆ’π‘‘/π‘›βˆ—π‘:𝐢(𝑑)∢=ξ€œπ‘‘0π‘’βˆ’(π‘‘βˆ’πœ)/𝑛𝑝(𝜏)π‘‘πœ.(2.29)We note that 𝐢∈𝐿1(0,∞), since 𝐿1(0,∞) is an ideal in 𝒲+. Let 𝑇>0 be such thatsupp(𝑝)βŠ‚[0,𝑇].(2.30)We haveβ€–π‘’π‘›Μ‚π‘βˆ’Μ‚π‘β€–π’²+=1𝑛‖𝐢‖𝐿1=1π‘›ξ€œβˆž0|𝐢(𝑑)|𝑑𝑑=1π‘›ξ€œπ‘‡0|𝐢(𝑑)|π‘‘π‘‘ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œ(𝐼)+1π‘›ξ€œβˆžπ‘‡|𝐢(𝑑)|π‘‘π‘‘ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œ(𝐼𝐼).(2.31)We estimate (𝐼) as follows:(𝐼)=1π‘›ξ€œπ‘‡0|𝐢(𝑑)|𝑑𝑑=1π‘›ξ€œπ‘‡0|ξ€œπ‘‘0π‘’βˆ’(π‘‘βˆ’πœ)/𝑛𝑝(𝜏)π‘‘πœ|𝑑𝑑≀1π‘›ξ€œπ‘‡0ξ€œπ‘‘0π‘’βˆ’(π‘‘βˆ’πœ)/𝑛|𝑝(𝜏)|π‘‘πœπ‘‘π‘‘β‰€1𝑛(ξ€œπ‘‡0ξ€œπ‘‘01β‹…|𝑝(𝜏)|π‘‘πœπ‘‘π‘‘)ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œ(𝐼𝐼𝐼).(2.32)Since the integral (𝐼𝐼𝐼) does not depend on 𝑛, we obtain thatlimπ‘›β†’βˆž1π‘›ξ€œπ‘‡0|𝐢(𝑑)|𝑑𝑑=0.(2.33)Furthermore,(𝐼𝐼)=1π‘›ξ€œβˆžπ‘‡|𝐢(𝑑)|𝑑𝑑=1π‘›ξ€œβˆžπ‘‡π‘’βˆ’π‘‘/𝑛|ξ€œπ‘‘0π‘’πœ/𝑛𝑝(𝜏)π‘‘πœ|𝑑𝑑=1π‘›ξ€œβˆžπ‘‡π‘’βˆ’π‘‘/𝑛|ξ€œβˆž0π‘’πœ/𝑛𝑝(𝜏)π‘‘πœ|𝑑𝑑(sincesupp(𝑝)βŠ‚[0,𝑇])=1π‘›ξ€œβˆžπ‘‡π‘’βˆ’π‘‘/𝑛|̂𝑝(βˆ’1𝑛)|𝑑𝑑.(2.34)Since 𝑝 has compact support in [0,𝑇], ̂𝑝 is an entire function by the Payley-Wiener theorem (see, e.g., [8, Theorem 7.2.3, page 122]). Consequently,(𝐼𝐼)=1π‘›ξ€œβˆžπ‘‡π‘’βˆ’π‘‘/𝑛|̂𝑝(βˆ’1𝑛)|𝑑𝑑=π‘’βˆ’π‘‡/𝑛|̂𝑝(βˆ’1𝑛)|π‘›β†’βˆžβ†’1β‹…|̂𝑝(0)|=1β‹…0=0.(2.35)This completes the proof.

We will also need the following lemma, which is basically a repetition of key steps from Browder's proof of Cohen's factorization theorem; see [9, Theorem 1.6.5, page 74]. We will need this version since in our application in the proof of Theorem 1.3, we are not able to use Cohen's factorization theorem directly.

Lemma 2.8. Let 𝑓1,𝑓2∈π”ͺ0, and 𝛿>0. Let π‘ˆ(𝒲+) denote the set of all invertible elements in 𝒲+. Then there exists a sequence (𝑔𝑛)π‘›βˆˆβ„• in 𝒲+ such that
(1) for all π‘›βˆˆβ„•, π‘”π‘›βˆˆπ‘ˆ(𝒲+);(2)(𝑔𝑛)π‘›βˆˆβ„• is convergent in 𝒲+ to a limit π‘”βˆˆπ”ͺ0;(3) for all π‘›βˆˆβ„•, β€–π‘”βˆ’1π‘›π‘“π‘–βˆ’π‘”βˆ’1𝑛+1𝑓𝑖‖𝒲+≀𝛿/2𝑛, 𝑖=1,2.

Proof. We will first prove two general results in steps (A) and (B), which we will use in the rest of the proof.
(A) Let π‘’βˆˆπ”ͺ0 and ‖𝑒‖𝒲+≀𝐾, where 𝐾>1. Then 1βˆ’π‘+π‘π‘’βˆˆπ‘ˆ(𝒲+), where 𝑐 is a number chosen such that0<𝑐<14𝐾<14.(2.36)Indeed,β€–π‘π‘βˆ’1𝑒‖𝒲+<1/(4𝐾)3/4⋅𝐾=13<1,(2.37)and so(1βˆ’π‘+𝑐𝑒)βˆ’1=11βˆ’π‘βˆžξ“π‘˜=0(π‘π‘βˆ’1)π‘˜π‘’π‘˜.(2.38)
(B) Furthermore, under the same assumptions and notation as in (A) above, we now show that if β€–π‘’πΉβˆ’πΉβ€–π’²+ is small for some 𝐹, then so is β€–πΈπΉβˆ’πΉβ€–π’²+, where 𝐸∢=(1βˆ’π‘+𝑐𝑒)βˆ’1. Since1=11βˆ’π‘βˆžξ“π‘˜=0(π‘π‘βˆ’1)π‘˜,(2.39)we haveβ€–πΈπΉβˆ’πΉβ€–π’²+=β€–11βˆ’π‘βˆžξ“π‘˜=0(π‘π‘βˆ’1)π‘˜(π‘’π‘˜πΉβˆ’πΉ)‖𝒲+≀11βˆ’π‘βˆžβˆ‘π‘˜=0(𝑐1βˆ’π‘)π‘˜β€–π‘’π‘˜πΉβˆ’πΉβ€–π’²+.(2.40)Butβ€–π‘’π‘˜πΉβˆ’πΉβ€–π’²+=β€–π‘˜βˆ’1βˆ‘π‘—=0(𝑒𝑗+1πΉβˆ’π‘’π‘—πΉ)‖𝒲+β‰€π‘˜βˆ’1𝑗=0‖𝑒𝑗‖𝒲+β€–π‘’πΉβˆ’πΉβ€–π’²+β‰€β€–π‘’πΉβˆ’πΉβ€–π’²+π‘˜βˆ’1𝑗=0‖𝑒‖𝑗𝒲+<β€–π‘’πΉβˆ’πΉβ€–π’²+πΎπ‘˜πΎβˆ’1.(2.41)Henceβ€–πΈπΉβˆ’πΉβ€–π’²+<β€–π‘’πΉβˆ’πΉβ€–π’²+11βˆ’π‘βˆžξ“π‘˜=01πΎβˆ’1(14(1βˆ’π‘))π‘˜<2πΎβˆ’1β€–π‘’πΉβˆ’πΉβ€–π’²+.(2.42)This estimate will be used in constructing the sequence of 𝑔𝑛's.
Let (𝑒𝑛)π‘›βˆˆβ„• denote the approximate identity for π”ͺ0 from Theorem 2.6. Let 𝐾>1 be such that ‖𝑒𝑛‖𝒲+≀𝐾 for all π‘›βˆˆβ„•. Choose 𝑐 such that0<𝑐<14𝐾<14.(2.43)
We will inductively define a sequence (π‘’π‘šπ‘˜)π‘˜βˆˆβ„• with terms from the approximate identity for π”ͺ0 such that ifπ‘”π‘›βˆΆ=π‘π‘›ξ“π‘˜=1(1βˆ’π‘)π‘˜βˆ’1π‘’π‘šπ‘˜+(1βˆ’π‘)𝑛,(2.44)then we have β€–π‘“π‘–βˆ’π‘”βˆ’11𝑓𝑖‖𝒲+<𝛿/2, 𝑖=1,2, and
(P1) for all π‘›βˆˆβ„•, π‘”π‘›βˆˆπ‘ˆ(𝒲+),(P2)for all π‘›βˆˆβ„•, β€–π‘”βˆ’1π‘›π‘“π‘–βˆ’π‘”βˆ’1𝑛+1𝑓𝑖‖𝒲+<𝛿/2𝑛, 𝑖=1,2. Since (𝑒𝑛)π‘›βˆˆβ„• is an approximate identity for π”ͺ0, we can choose π‘š1 such thatβ€–π‘’π‘š1π‘“π‘–βˆ’π‘“π‘–β€–π’²+<𝛿4(πΎβˆ’1),𝑖=1,2.(2.45)Define 𝑔1=π‘π‘’π‘š1+1βˆ’π‘. So by (A), 𝑔1βˆˆπ‘ˆ(𝒲+) and using the calculation in (B), we see thatβ€–π‘“π‘–βˆ’π‘”βˆ’11𝑓𝑖‖𝒲+<𝛿2,𝑖=1,2.(2.46)Suppose that π‘’π‘š1,…,π‘’π‘šπ‘› have been constructed, so that 𝑔𝑛 defined by (2.44) satisfies (P1) and (P2). We assert that if we choose π‘’π‘šπ‘›+1 such thatβ€–π‘’π‘šπ‘›+1π‘“π‘–βˆ’π‘“π‘–β€–π’²+(𝑖=1,2),β€–π‘’π‘šπ‘›+1π‘’π‘šπ‘˜βˆ’π‘’π‘šπ‘˜β€–π’²+(1β‰€π‘˜β‰€π‘›)(2.47)are sufficiently small, then 𝑔𝑛+1 defined by (2.44) satisfies (P1) and (P2), completing the induction step.
Indeed, if 𝐸∢=(1βˆ’π‘+π‘π‘’π‘šπ‘›+1)βˆ’1, we have𝑔𝑛=πΈβˆ’1π‘π‘›ξ“π‘˜=1(1βˆ’π‘)π‘˜βˆ’1πΈπ‘’π‘šπ‘˜+(1βˆ’π‘)𝑛,𝑔𝑛+1=πΈβˆ’1(π‘π‘›ξ“π‘˜=1(1βˆ’π‘)π‘˜βˆ’1πΈπ‘’π‘šπ‘˜+(1βˆ’π‘)𝑛).(2.48)Let 𝐺𝑛 be defined by𝐺𝑛=π‘π‘›ξ“π‘˜=1(1βˆ’π‘)π‘˜βˆ’1πΈπ‘’π‘šπ‘˜+(1βˆ’π‘)𝑛.(2.49)Then we haveβ€–πΊπ‘›βˆ’π‘”π‘›β€–π’²+<π‘π‘›βˆ‘π‘˜=1(1βˆ’π‘)π‘˜βˆ’1β€–πΈπ‘’π‘šπ‘˜βˆ’π‘’π‘šπ‘˜β€–π’²+<max1β‰€π‘˜β‰€π‘›β€–πΈπ‘’π‘šπ‘˜βˆ’π‘’π‘šπ‘˜β€–π’²+<2πΎβˆ’1max1β‰€π‘˜β‰€π‘›β€–π‘’π‘šπ‘›+1π‘’π‘šπ‘˜βˆ’π‘’π‘šπ‘˜β€–π’²+.(2.50)Hence πΊπ‘›βˆˆπ‘ˆ(𝒲+) and moreover β€–πΊβˆ’1π‘›βˆ’π‘”βˆ’1𝑛‖𝒲+ is small, provided only that β€–π‘’π‘šπ‘›+1π‘’π‘šπ‘˜βˆ’π‘’π‘šπ‘˜β€–π’²+ is small for π‘˜=1,…,𝑛. (Indeed, this is because π‘ˆ(𝒲+) is an open set in 𝒲+.)
Since 𝑔𝑛+1=πΈβˆ’1𝐺𝑛, we have then 𝑔𝑛+1βˆˆπ‘ˆ(𝒲+), π‘”βˆ’1𝑛+1=πΊβˆ’1𝑛𝐸, and so for 𝑖=1,2,β€–π‘”βˆ’1𝑛+1π‘“π‘–βˆ’π‘”βˆ’1𝑛𝑓𝑖‖𝒲+=β€–πΊβˆ’1π‘›πΈπ‘“π‘–βˆ’π‘”βˆ’1𝑛𝑓𝑖‖𝒲+β‰€β€–πΊβˆ’1π‘›πΈπ‘“π‘–βˆ’π‘”βˆ’1𝑛𝐸𝑓𝑖‖𝒲++β€–π‘”βˆ’1π‘›πΈπ‘“π‘–βˆ’π‘”βˆ’1𝑛𝑓𝑖‖𝒲+β‰€β€–πΊβˆ’1π‘›βˆ’π‘”βˆ’1𝑛‖𝒲+‖𝐸𝑓𝑖‖𝒲++β€–π‘”βˆ’1𝑛‖𝒲+β€–πΈπ‘“π‘–βˆ’π‘“π‘–β€–π’²+.(2.51)Moreover, recall that by (B), we know thatβ€–πΈπ‘“π‘–βˆ’π‘“π‘–β€–π’²+≀2πΎβˆ’1β€–π‘’π‘šπ‘›+1π‘“π‘–βˆ’π‘“π‘–β€–π’²+,𝑖=1,2.(2.52)Thus if β€–π‘’π‘šπ‘›+1π‘“π‘–βˆ’π‘“π‘–β€–π’²+ (𝑖=1,2) and β€–π‘’π‘šπ‘›+1π‘’π‘šπ‘˜βˆ’π‘’π‘šπ‘˜β€–π’²+ (1β‰€π‘˜β‰€π‘›) are sufficiently small, we will have β€–π‘”βˆ’1𝑛+1π‘“π‘–βˆ’π‘”βˆ’1𝑛𝑓𝑖‖𝒲+ as small as we please. This completes the induction step.
Since β€–π‘’π‘šπ‘˜β€–π’²+≀𝐾, 0<1βˆ’π‘<1, and 𝒲+ is a Banach algebra, it follows thatπ‘”π‘›β†’π‘βˆžξ“π‘˜=1(1βˆ’π‘)π‘˜βˆ’1π‘’π‘šπ‘˜=βˆΆπ‘”βˆˆπ”ͺ0,(2.53)and the proof is completed.

3. Noncoherence of 𝒲+

Proof of Theorem 1.3. We will use the characterization that an integral domain is coherent if and only if the intersection of any two finitely generated ideals of the ring is again finitely generated; see [1, Theorem 2.3.2, page 45]. In fact, we present two finitely generated ideals 𝐼 and 𝐽 such that 𝐼∩𝐽 is not finitely generated.
Let 𝑝,𝑆 be given by𝑝=(1βˆ’π‘’βˆ’π‘ )3,𝑆=π‘’βˆ’(1+π‘’βˆ’π‘ )/(1βˆ’π‘’βˆ’π‘ ).(3.1)Clearly we have π‘βˆˆπ”ͺ0.
It is known (see, e.g., [3, Remark after Theorem 1, page 224]) that(1βˆ’π‘§)3π‘’βˆ’(1+𝑧)/(1βˆ’π‘§)βˆˆπ‘Š+(𝔻)∢={𝑓(𝑧)=βˆžξ“π‘›=0π‘Žπ‘›π‘§π‘›(π‘§βˆˆπ”»)|βˆžξ“π‘›=0|π‘Žπ‘›|<∞}.(3.2)Here π”»βˆΆ={π‘§βˆˆβ„‚βˆ£|𝑧|≀1}. So if π‘Žπ‘›'s are defined via(1βˆ’π‘§)3π‘’βˆ’(1+𝑧)/(1βˆ’π‘§)=π‘Ž0+π‘Ž1𝑧+π‘Ž2𝑧2+π‘Ž3𝑧3+β‹―,π‘§βˆˆπ”»,(3.3)then we haveβˆžξ“π‘˜=0|π‘Žπ‘˜|<∞.(3.4)If Re(𝑠)>0, then π‘’βˆ’π‘ βˆˆπ”», and so from (3.3), we have𝑝𝑆=π‘Ž0+π‘Ž1π‘’βˆ’π‘ +π‘Ž2π‘’βˆ’2𝑠+π‘Ž3π‘’βˆ’3𝑠+β‹―,Re(𝑠)>0.(3.5)Since βˆ‘βˆžπ‘˜=0|π‘Žπ‘˜|<∞, the right-hand side in (3.5) belongs to 𝒲+. So π‘π‘†βˆˆπ’²+.
We define the ideals 𝐼=(𝑝) and 𝐽=(𝑝𝑆) of 𝒲+.
Let𝐾∢={π‘π‘†π‘“βˆ£π‘“βˆˆπ’²+andπ‘†π‘“βˆˆπ’²+}.(3.6)We claim that 𝐾=𝐼∩𝐽. Trivially πΎβŠ‚πΌβˆ©π½. To prove the reverse inclusion, let π‘”βˆˆπΌβˆ©π½. Then there exist two functions 𝑓 and β„Ž in 𝒲+ such that 𝑔=π‘β„Ž=𝑝𝑆𝑓. Since 𝑝≠0 and 𝒲+ is an integral domain, we obtain that 𝑆𝑓=β„Žβˆˆπ’²+. So π‘”βˆˆπΎ.
Let 𝐿 denote the ideal𝐿∢={π‘“βˆˆπ’²+βˆ£π‘†π‘“βˆˆπ’²+}.(3.7)Then 𝐾∢=𝑝𝑆𝐿. Since 𝑆 has a singularity at 𝑠=0, it follows that πΏβŠ‚π”ͺ0. We will show that 𝐿=𝐿π”ͺ0. Let π‘“βˆˆπΏ. We would like to factor 𝑓=β„Žπ‘” with β„ŽβˆˆπΏ and π‘”βˆˆπ”ͺ0. Applying Lemma 2.8 with 𝑓1∢=π‘“βˆˆπ”ͺ0 and 𝑓2∢=π‘†π‘“βˆˆπ”ͺ0, for any 𝛿>0, there exists a sequence (𝑔𝑛)π‘›βˆˆβ„• in 𝒲+ such that
(1) for all π‘›βˆˆβ„•, π‘”π‘›βˆˆπ‘ˆ(𝒲+);(2)(𝑔𝑛)π‘›βˆˆβ„• is convergent in 𝒲+ to a limit π‘”βˆˆπ”ͺ0;(3) for all π‘›βˆˆβ„•,β€–π‘”βˆ’1π‘›π‘“βˆ’π‘”βˆ’1𝑛+1𝑓‖𝒲+≀𝛿2𝑛,β€–π‘”βˆ’1π‘›π‘†π‘“βˆ’π‘”βˆ’1𝑛+1𝑆𝑓‖𝒲+≀𝛿2𝑛.(3.8) Putβ„Žπ‘›βˆΆ=π‘”βˆ’1𝑛𝑓,π»π‘›βˆΆ=π‘”βˆ’1𝑛𝑆𝑓.(3.9)Then β„Žπ‘›βˆˆπ”ͺ0. Also π»π‘›βˆˆπ”ͺ0, since |𝑆| is bounded by 1 on Re(𝑠)>0 and 𝑓(0)=0. The estimates above imply that (β„Žπ‘›)π‘›βˆˆβ„• and (𝐻𝑛)π‘›βˆˆβ„• are Cauchy sequences in 𝒲+. Since π”ͺ0 is closed, they converge to elements β„Ž and 𝐻, respectively, in π”ͺ0, that is, β„Žπ‘›=π‘”βˆ’1π‘›π‘“β†’β„Ž and 𝐻𝑛=π‘”βˆ’1𝑛𝑆𝑓=π‘†β„Žπ‘›β†’π». Since convergence in 𝒲+ implies convergence in 𝐻∞ (Lemma 2.2), it follows thatβ„Žπ‘›β†’π»βˆžβ„Ž(sinceβ„Žπ‘›β†’π’²+β„Ž),π‘†β„Žπ‘›β†’π»βˆžπ‘†β„Ž(sinceβ„Žπ‘›β†’π»βˆžβ„Ž,π‘†βˆˆπ»βˆž),π‘†β„Žπ‘›β†’π»βˆžπ»(since𝐻𝑛→𝒲+𝐻)(3.10)and so by the uniqueness of the limit of the sequence (π‘†β„Žπ‘›)π‘›βˆˆβ„• in 𝐻∞, we have π‘†β„Ž=𝐻. Also, in 𝒲+-norm we have𝑓=limπ‘›β†’βˆžβ„Žπ‘›π‘”π‘›=β„Žπ‘”,(3.11)since multiplication is continuous in the Banach algebra 𝒲+. Since β„Ž and π‘†β„Ž=𝐻 belong to π”ͺ0βŠ‚π’²+, we see that β„ŽβˆˆπΏ. Moreover, as π‘”βˆˆπ”ͺ0, we have got the desired factorization and 𝐿=𝐿π”ͺ0.
But 𝐿≠(0), since π‘βˆˆπΏ. By Lemma 2.3, it follows that 𝐿 cannot be finitely generated. Therefore, 𝑝𝑆𝐿=𝐼∩𝐽 is not finitely generated.

Remark 3.1. The ideal 𝐿 in the above proof can be interpreted as an ideal of denominators; see [10, page 396]. Using the fact that π‘π‘†βˆˆπ’²+, we have π‘†βˆˆπ‘„(𝒲+), where 𝑄(𝒲+) denotes the field of fractions of 𝒲+. We can then consider the fractional ideal π‘€βˆΆ=𝒲++𝒲+𝑆 of 𝒲+ (see [11, page 19]) and the ideal of denominators 𝐿 of 𝑆, namely 𝐿=𝒲+βˆΆπ‘€={π‘‘βˆˆπ’²+βˆ£π‘‘π‘†βˆˆπ’²+}.
Based on the results in [12, Theorem 3, Example 3], it follows that π‘†βˆˆπ‘„(𝒲+) does not admit a weak coprime factorization, since 𝐿 is not a principal ideal of 𝒲+. In particular, 𝑆 does not admit a coprime factorization, that is, there do not exist 𝑑,π‘₯,𝑦,π‘›βˆˆπ’²+ such that 𝑑≠0, 𝑆=𝑛/𝑑, and 𝑑π‘₯βˆ’π‘›π‘¦=1. Moreover, 𝑆 is not internally stabilizable, since otherwise 𝐿 would be generated by two elements. Finally, the fact that 𝐿 is not finitely generated implies that 𝒲+ is not a greatest common divisor domain: indeed, were it the case that 𝒲+ is a greatest common divisor domain, then by [12, Corollary 3], every element in 𝑄(𝒲+) would admit a weak coprime factorization.


The author thanks all the referees for their careful review, and in particular, two of the referees for the Remarks 2.4 and 3.1.


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