Left ideals of Banach algebras and dual Banach algebras
aa r X i v : . [ m a t h . F A ] F e b LEFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACHALGEBRAS
JARED T. WHITE
Laboratoire de Mathématiques de BesançonUniversité de Franche-Comté16 Route de Gray25030 BesançonFrance
Abstract.
We investigate topologically left Noetherian Banach algebras. We show thatif G is a compact group, then L p G q is topologically left Noetherian if and only if G ismetrisable. We prove that, given a Banach space E such that E has BAP, the algebra ofcompact operators K p E q is topologically left Noetherian if and only if E is separable; it istopologically right Noetherian if and only if E is separable. We then give some examples ofdual Banach algebras which are topologically left Noetherian in the weak*-topology. Finallywe give a unified approach to classifying the weak*-closed left ideals of certain dual Banachalgebras that are also multiplier algebras, with applications to M p G q for G a compact group,and B p E q for E a reflexive Banach space with AP. Introduction
Those studying Banach algebras have long been interested in the interplay between abstractalgebra and abstract analysis. This motivates the comparison of the following two definitions:
Definition 1.1.
Let A be a Banach algebra, and let I be a closed left ideal of A .(i) Given n P N , we say that I is (algebraically) generated by x , . . . , x n P I if I “ A x ` ¨ ¨ ¨ ` A x n . When there exist such elements x , . . . , x n for some n P N we say that I is (alge-braically) finitely-generated .(ii) Given n P N , we say that I is topologically generated by x , . . . , x n P I if I “ A x ` ¨ ¨ ¨ ` A x n . When there exist such elements x , . . . , x n for some n P N we say that I is topologicallyfinitely-generated . E-mail address : [email protected] . Date : 2018.2010
Mathematics Subject Classification.
Primary: 16P40, 46H10; secondary: 43A10, 43A20, 47L10.
Key words and phrases.
Noetherian, Banach algebra, dual Banach algebra, Multiplier algebra, weak*-closedideal.
Here A denotes the unitisation of A .It seems both natural and obvious that Definition 1.1(ii) is the appropriate one for thenormed setting since it takes account of the topology. However, as Banach algebraists we wishto establish a more precise picture of exactly how these two definitions play out. One resultin this spirit is the beautiful theorem of Sinclair and Tullo from 1974 [32]: Theorem 1.2.
Let A be a Banach algebra which is (algebraically) left Noetherian. Then A isfinite-dimensional. In recent years a number of papers have appeared which further illustrate that algebraicfinite-generation of left ideals in Banach algebras is a very strong condition. This has beenmotivated by a conjecture of Dales and Żelazko [12] which states that a unital Banach algebrain which every maximal left ideal is finitely-generated is finite-dimensional. The conjecture isknown to hold for many classes of Banach algebras [12, 11, 4, 36]. For example, in [36] thepresent author verified the conjecture for many of the algebras coming from abstract harmonicanalysis, including the measure algebra of a locally compact group, as well as a large class ofBeurling algebras.In this article we aim to fill in another corner of this picture and contrast with the aboveresults by investigating topologically left Noetherian Banach algebras, which we define asfollows:
Definition 1.3.
Let A be a Banach algebra. We say that A is topologically left Noetherian ifevery closed left ideal is topologically finitely-generated.We shall demonstrate below that, in contrast to the Sinclair–Tullo Theorem, there aremany infinite-dimensional examples of topologically left Noetherian Banach algebras, andthat, moreover, the condition often picks out a nice property of some underlying group orBanach space.We note that versions of Noetherianity for topological rings and algebras are not at all new,and various different versions of this notion exist: see e.g. [7, 27]. Moreover, topological leftNoetherianity as in Definition 1.3 was the subject of a post by Kevin Casto on Mathoverflow[6]. His question is partially answered by our Theorem 1.6. We also mention that the fact thatseparable C*-algebras are topologically left Noetherian has been known since Prosser’s 1963memoir [30], in which it was shown that every closed left ideal of a separable C*-algebra istopologically principal [30, Corollary, pg. 26]. It should be noted that using modern techniquesa short proof of this fact can be obtained by using the correspondence between the closed leftideals of a C*-algebra and its hereditary C*-subalgebras, and the fact that, in the separablecase, hereditary C*-subalgebras of a C*-algebra A are all of the form xAx , for some positiveelement x P A .One natural condition that we could consider for Banach algebras that is different fromours would be an ascending chain condition on chains of closed left ideals as in [7, Proposition4.1]. However, we know of no infinite-dimensional examples of Banach algebras satisfyingthis condition. Indeed, many of the natural examples of Banach algebras satisfying Definition1.3 are easily seen to fail the ascending chain condition. These include C r , s , as well asthe infinite-dimensional examples in Theorems 1.4 and 1.5 below. One might be tempted totry to prove that there are no infinite-dimensional Banach algebras satisfying an ascendingchain condition for closed left ideals. However, if this were true, it would imply a negativesolution to the question of whether there exists an infinite-dimensional, topologically simple, EFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACH ALGEBRAS 3 commutative Banach algebra, which is a notorious open question. For these reasons we havechosen only to study Definition 1.3 in this article.We now outline the main results of the paper. In Section 2 we shall fix some notation andprove some general results that we shall require in the sequel. Section 3 is concerned withBanach algebras on groups, and our main result is the following:
Theorem 1.4.
Let G be a compact group. Then L p G q is topologically left Noetherian if andonly if G is metrisable. We also prove an analogous result for the Fourier algebra of certain discrete groups (Propo-sition 3.1).In Section 4 we turn our attention to algebras of operators on a Banach space. We denotethe set of compact operators on a Banach space E by K p E q . Our main result is the following: Theorem 1.5. (i)
Let E be a Banach space with AP. Then K p E q is topologically leftNoetherian if and only if E is separable. (ii) Let E be a Banach space such that E has BAP. Then K p E q is topologically rightNoetherian if and only if E is separable. This theorem actually follows from more general results about the algebra of approximableoperators A p E q , for a formally larger class of Banach spaces E (Theorem 4.5 and Theorem4.9).In Section 5 we consider weak*-topologically left Noetherian dual Banach algebras, believingthat this will be a more appropriate notion for the measure algebra M p G q of a locally compactgroup G , and for B p E q , the algebra of bounded linear operators on a reflexive Banach space E . We consider Banach algebras A for which the multiplier algebra M p A q is a dual Banachalgebra in a natural way, and prove a general result, Proposition 5.3, which says that M p A q is weak*-topologically left Noetherian whenever A is topologically left Noetherian. We thenapply this theorem to the algebras of the form M p G q and B p E q to get the following corollary: Corollary 1.6. (i)
Let G be a compact, metrisable group. Then M p G q is weak*-topo-logically left Noetherian. (ii) Let E be a separable, reflexive Banach space with AP. Then B p E q is weak*-topologicallyleft and right Noetherian. We then consider a more restricted class of Banach algebras, and we formulate an abstractapproach for relating the ideal structure of a Banach algebra A belonging to this class to theweak*-ideal structure of M p A q (Theorem 5.10). In Proposition 5.11 we use this to show that,for this class, weak*-topological left Noetherianity of M p A q is equivalent to a } ¨ } -topologicalcondition on A . In Section 6 we demonstrate how Theorem 5.10 gives a unified strategy forclassifying the weak*-closed left ideals of both M p G q , for G a compact group, and B p E q , for E a reflexive Banach space with AP. We then observe that this leads to classifications of theclosed right submodules of the predules.Finally, we mention that in [25] the author of the present work together with Niels Laustsenwill show that there is a certain reflexive Banach space E which has AP such that B p E q failsto be weak*-topologically left Noetherian. This is the only example of a dual Banach algebrathat we know of that fails to have this property. J. T. WHITE Preliminaries
We first fix some general notation. Given a locally compact group G , we denote by L p G q the Banach algebra of integrable functions on G , which we refer to as the group algebra of G .We write M p G q for the Banach algebra of complex, regular Borel measures on G , known hereas the measure algebra. We write A p G q for the Fourier algebra of G , as defined by Eymardin [13]. We write C p G q for the linear space of complex-valued functions on G , and C p G q for the subspace consisting of functions vanishing at infinity. Of course, C p G q is a Banachspace with the supremum norm; when G is compact C p G q “ C p G q , and we prefer the formernotation over the latter.By a representation of G we implicitly mean a continuous unitary representation of G ona Hilbert space. We write p G for the unitary dual of G , and a typical element of p G will berepresented as p π, H π q , where H π is a Hilbert space, and π is an irreducible representation of G on H π . Given an arbitrary representation p π, H π q of G , and vectors ξ, η P H π , we write ξ ˚ π η for the function on G defined by t ÞÑ x π p t q ξ, η y p t P G q . We shall denote the modularfunction on G by ∆ . We also use the notation q f p t q “ f p t ´ q p t P G q for f P L p G q .For a Banach algebra A we denote by A the (conditional) unitisation of A , and by M p A q the multiplier algebra of A . We write CLI p A q for the lattice of closed left ideals of A , and CRI p A q for the lattice of closed right ideals. We denote the left action of A on its dual by a ¨ λ for a P A and λ P A , and set A ¨ A “ t a ¨ λ : a P A, λ P A u . Usually A will have a boundedapproximate identity, in which case, by Cohen’s factorisation theorem, A ¨ A “ span p A ¨ A q .We use similar notation for the right action.Given a Banach space E , we shall denote its dual space by E . We write B p E q for the algebraof bounded linear operators E Ñ E . We write K p E q for the ideal of compact operators, A p E q for the approximable operators, and F p E q for the finite rank operators. We write SUB p E q for the lattice of closed linear subspaces of E . Given x P E and λ P E , we write x b λ for therank one operator y ÞÑ λ p y q x p y P E q .Given subsets X Ă E and Y Ă E , we use the notation X K “ t λ P E : x x, λ y “ p x P X qu , Y K “ t u P E : x u, ϕ y “ p ϕ P Y qu , and we recall the well-known formulae(2.1) p X K q K “ span p X q , p Y K q K “ span w ˚ p Y q . A Banach space E is said to have the approximation property , or simply AP , if, whenever F is another Banach space, we have A p F, E q “ K p F, E q . There is also an equivalent formulationof the approximation property which has some useful generalizations: a Banach space E hasAP if and only if, for every compact subset K Ă E and every ε ą , there exists T P F p E q suchthat } T x ´ x } ă ε p x P K q [26, Theorem 3.4.32]. We say that E has the bounded approximationproperty , or BAP , if there exists a constant C ą such that the operator T can be chosen tohave norm at most C . Clearly BAP implies AP. Moreover, a reflexive Banach space with APhas BAP [5, Theorem 3.7]. Many Banach spaces have the bounded approximation property:for instance any Banach space with a Schauder basis [26, Theorem 4.1.33] has BAP, and itcan be deduced from this that any Hilbert space has BAP. The Banach space B p H q , for H aninfinite-dimensional Hilbert space, does not even have AP [33].We recall from [31] that a dual Banach algebra is a Banach algebra A which is isomorphicallya dual Banach space, in such a way that the multiplication is separately weak*-continuous.Equivalently, a Banach algebra with (isomorphic) predual X is a dual Banach algebra if and EFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACH ALGEBRAS 5 only if X may be identified with a closed A -submodule of A . Examples include the measurealgebra M p G q of a locally compact group G , with predual C p G q , as well as B p E q for a reflexiveBanach space E , with predual given by E p b E , where p b denotes the projective tensor productof Banach spaces.Recall that a semi-topological algebra is a pair p A, τ q , where A is an algebra, and τ is atopology on A such that p A, ` , τ q is a topological vector space, and such that multiplicationon A is separately continuous. For example, a dual Banach algebra with its weak*-topologyis a semi-topological algebra.Let p A, τ q be a semi-topological algebra. Let I be a closed left ideal of A , and let n P N . Wesay that I is τ -topologically generated by elements x , . . . , x n P I if I “ A x ` ¨ ¨ ¨ ` A x n . We say that I is τ -topologically finitely-generated if there exist n P N and x , . . . , x n P I which τ -topologically generate I . We say that I is τ -topologically principal if I is of the form A x ,for some x P I . We say that A is τ -topologically left Noetherian if every closed left ideal of A is τ -topologically finitely-generated. For example, we shall often discuss weak*-topologicallyleft Noetherian dual Banach algebras. When the topology is the norm topology on a Banachalgebra we may simply speak of “topologically finitely-generated left ideals” et cetera.Analogously we may define τ -topologically finitely-generated right ideals , as well as τ -topologically right Noetherian algebras . If the algebra in question is commutative we usuallydrop the words “left” and “right”.We note that when a semi-topological algebra A has a left approximate identity we have A x ` ¨ ¨ ¨ ` A x n “ Ax ` ¨ ¨ ¨ ` Ax n , for each n P N , and each x , . . . , x n P A . When this is the case we usually drop the unitisationsin order to ease notation. For example, in the proof of Theorem 3.4 below, we shall write L p G q ˚ g in place of L p G q ˚ g , for G a locally compact group and g P L p G q .The following lemma will be invaluable throughout this article. Lemma 2.1.
Let A be a semi-topological algebra with a left approximate identity. Let J be adense right ideal of A . Then J intersects every closed left ideal of A densely.Proof. Let p e α q be a left approximate identity for A , which we may assume belongs to J (ifnot, then for each open neighbourhood of the origin U, and each index α , choose f α,U P J such that e α ´ f α,U P U . Then p f α,U q is easily seen to be a left approximate identity for A .)Let I be a closed left ideal of A and let a P I . Then for every index α we have e α a P J X I .Since a “ lim α e α a P J X I , and a was arbitrary, it follows that J X I “ I , as required. (cid:3) Next we show that τ -topological left Noetherianity is stable under taking quotients andextensions. For this lemma only we drop the τ s and write “topologically left Noetherian” etcetera, even though we are not necessarily talking about a topology induced by a norm. Lemma 2.2.
Let A be a semi-topological algebra, and let I be a closed (two-sided) ideal of A . (i) If A is topologically left Noetherian then so is A { I . (ii) Suppose that both I and A { I are topologically left Noetherian. Then so is A . (iii) A is topologically left Noetherian if and only if A is topologically left Noetherian.Proof. Parts (i) and (ii) follow from routine arguments. For part (iii) we may suppose that A is non-unital for otherwise the result is trivial. If A is topologically left Noetherian then, since J. T. WHITE A { A – C is topologically left Noetherian, it follows from (ii) that A is also. The conversefollows from the fact that every closed left ideal of A is also a closed left ideal of A . (cid:3) Examples From Abstract Harmonic Analysis
In this section we shall prove Theorem 1.4. It is surely easiest to determine whether or nota Banach algebra is topologically left Noetherian when we know what its closed left idealsare. Fortunately, this is the case for the group algebra of a compact group, as well as for theFourier algebra of certain discrete groups, including all amenable groups. As a sort of warmup for the proof of Theorem 1.4 we shall show that, for such groups, the Fourier algebra A p G q is topologically Noetherian if and only if G is countable. Both proofs involve similar ideas. Proposition 3.1.
Let G be a discrete group such that f P A p G q f for all f P A p G q . Then A p G q is topologically Noetherian if and only if G is countable.Proof. Given E Ă G , write I p E q “ t f P A p G q : f p x q “ , x P E u . By [23, Proposition 2.2]the closed ideals of A p G q are all of the from I p E q for some subset E of G .Suppose first that G is countable and let I Ÿ A p G q be closed. Let E Ă G be such that I “ I p E q , and enumerate G z E “ t x , x , . . . , u . Define g “ ř n “ n δ x n P A p G q . It is clearthat supp g “ G z E , and hence that ! x P G : f p x q “ for every f P A p G q g ) “ E. It follows from the classification of the closed ideals of A p G q given above that I “ A p G q g .As I was arbitrary we conclude that A p G q is topologically Noetherian.Now suppose that A p G q is topologically Noetherian. Then there exist n P N and h , . . . , h n P A p G q such that A p G q “ A p G q h ` ¨ ¨ ¨ ` A p G q h n . Since A p G q Ă c p G q , every function in A p G q must have countable support. Hence S : “ Ť ni “ supp h i is a countable set. Every f P A p G q h ` ¨ ¨ ¨ ` A p G q h n has supp f Ă S , and of course, after taking closures, we see thatthis must hold for every f P A p G q . This clearly forces S “ G , so that G must be countable. (cid:3) Remark.
The hypothesis of the previous proposition is satisfied by any discrete, amenablegroup, since in this case A p G q has a bounded approximate identity by Leptin’s Theorem, aswell as many other groups including the free group on n generators for each n P N [34]. Thequestion of whether there are any locally compact groups which do not satisfy f P f A p G q forevery f P A p G q is considered a difficult open problem.We now recall some facts about compact groups. Firstly, for G a compact group the closedleft ideals of L p G q have the following characterisation [21, Theorem 38.13]: Theorem 3.2.
Let G be a compact group, and let I be a closed left ideal of L p G q . Thenthere exist linear subspaces E π Ă H π p π P p G q such that I “ ! f P L p G q : π p f qp E π q “ , π P p G ) . Let G be a compact group. Given π P p G we write T π p G q “ span t ξ ˚ π η : ξ, η P H π u , and wewrite T p G q “ span t ξ ˚ π η : ξ, η P H π , π P p G u . We recall the following facts about these spacesfrom [20, 21]:
Theorem 3.3.
Let G be a compact group. (i) Let σ, π P p G with σ ‰ π . Then σ p ξ ˚ π η q “ . EFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACH ALGEBRAS 7 (ii)
The linear space T p G q is a dense ideal in L p G q . (iii) For each π P p G the space T π p G q is an ideal in L p G q , and as an algebra T π p G q – M d π p G q , where d π denotes the dimension of H π .Proof. The formula x σ p f q ζ , ζ y “ ż G f p t qx σ p t q ζ , ζ y d t, for f P L p G q , σ P p G and ζ , ζ P H σ is well-known. Part (i) follows from this and theorthogonality relations [21, Theorem 27.20 (iii)]. Part (ii) follows from [21, Theorem 27.20,Lemma 31.4], and part (iii) follows from [21, Theorem 27.21]. (cid:3) We now prove the main theorem of this section, Theorem 3.4. Observe that Theorem 1.4is simply “(a) if and only if (c)”. The equivalence of conditions (b) and (c) has surely beennoticed before, but we include the proof to make our argument more transparent.
Theorem 3.4.
Let G be a compact group. Then the following are equivalent: (a) L p G q is topologically left Noetherian; (b) p G is countable; (c) G is metrisable.Proof. We first demonstrate that (b) implies (c). Our method is to show that G is first-countable, which will imply that G is metrisable by [20, Theorem 8.3]. Indeed, it followsfrom Tannaka–Krein duality [22] that the topology on G is the initial topology induced byits irreducible continuous unitary representations, and as such has a base given by sets of theform U p π , . . . , π n ; ε ; t q : “ t s P G : } π i p t q ´ π i p s q} ă ε, i “ , . . . , n u , where ε ą , t P G, and p π , H q , . . . , p π n , H n q P p G . Hence, if p G is countable, for every t P G the sets U p π , . . . , π n ; 1 { m ; t q p m P N , π , . . . , π n P p G q form a countable neighbourhood baseat t , and so G is first-countable.Now suppose instead that G is metrisable. Then C p G q is separable. Since the infinity normdominates the L -norm for a compact space, and since C p G q is dense in L p G q , it followsthat L p G q is separable. By [21, Theorem 27.40] L p G q – à π P p G H ‘ dim H π π , which is clearly separable only if p G is countable. Hence (c) implies (b).Next we show that (b) implies (a). Suppose that p G is countable. By Theorem 3.3(ii) T p G q is a dense ideal in L p G q so that, by Lemma 2.1, I X T p G q “ I for every closed left ideal I in L p G q .Fix a closed left ideal I in L p G q . By Theorem 3.2 there exist linear subspaces E π Ă H π p π P p G q such that I “ ! f P L p G q : π p f qp E π q “ , π P p G ) . By Theorem 3.3(iii), for each π P p G we have T π p G q – M d π p C q , where d π is the dimension of H π , and since I X T π p G q is a left ideal in T π p G q there must be an idempotent P π P T π p G q J. T. WHITE such that I X T π p G q “ T π p G q ˚ P π . Set α π “ } P π } ´ if P π ‰ , and set α π “ otherwise.Enumerate p G “ t π , π , . . . u , and define g “ ÿ i “ i α π i P π i P L p G q , which belongs to I because each P π i does, and I is closed.We claim that I “ L p G q ˚ g . Indeed, I Ą L p G q ˚ g because g P I . For the reverseinclusion we show that, for j P N and ξ P H π j , we have π j p f qp ξ q “ for all f P L p G q ˚ g ifand only if ξ P E π j . The claim then follows from Theorem 3.2. Indeed, if f P L p G q ˚ g then π j p f qp ξ q “ because f P I . On the other hand if ξ P H π j z E π j then π j p P π j qp ξ q ‰ , whereas π i p P π j q “ for i ‰ j by Theorem 3.3(i), which implies that π j p g q ξ “ j α π j π j p P π j qp ξ q ‰ .This establishes the claim.Finally we show that (a) implies (b). Assume that L p G q is topologically left Noetherian.Then there exist r P N and g , . . . , g r P L p G q such that L p G q “ L p G q ˚ g ` ¨ ¨ ¨ ` L p G q ˚ g r . For each n P N there exist t p i q n P T p G q p i “ , . . . , r q such that } t p i q n ´ g i } ă n p i “ , . . . , r q . Let S be the set S “ ! π P p G : there exist i, n P N such that π ´ t p i q n ¯ ‰ ) . We see that S is countable because, by Theorem 3.3(i), each function t p i q n satisfies π p t p i q n q ‰ for at most finitely many π P p G . We shall show that S “ p G .Assume instead that there exists some π P p G z S , and let u be the identity element of T π p G q .For σ P p G zt π u we have σ p u q “ , whereas π ´ t p i q n ¯ “ for every n P N and every i “ , . . . , r .Hence σ ´ t p i q n ˚ u ¯ “ p σ P p G, n P N , i P t , . . . , r uq , which implies that t p i q n ˚ u “ for every n P N and i “ , . . . , r .By taking the limit as n goes to infinity, this shows that g i ˚ u “ p i “ , . . . , r q , andhence that f ˚ u “ for every f P L p G q . However, since u was chosen to be an identity u ˚ u “ u ‰ . This contradiction implies that p G “ S , as claimed. (cid:3) The next proposition suggests to us that weak*-topological Noetherianity is a more in-teresting notion for the measure algebra of a locally compact group G than } ¨ } -topologicalNoetherianity, and we explore this in the next section. Given a discrete group G we write ℓ p G q “ L p G q , and write ℓ p G q for its augmentation ideal, i.e. the maximal ideal consistingof those f P ℓ p G q such that ř t P G f p t q “ . Proposition 3.5.
Let G be a locally compact group such that M p G q is topologically left Noe-therian. Then G is countable. If, in addition, G is either compact or abelian, then G isfinite.Proof. Suppose that M p G q is topologically left Noetherian. Then, by Lemma 2.2 (i), so areits quotients, whence ℓ p G d q is topologically left Noetherian, where G d denotes the group G EFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACH ALGEBRAS 9 with the discrete topology. It follows that ℓ p G d q “ ℓ p G d q ˚ g ` ¨ ¨ ¨ ` ℓ p G d q ˚ g n , for some n P N and some g , . . . , g n P ℓ p G d q . Let H be the subgroup of G generated by thesupports of the functions g , . . . , g n . This is a countable set. Define σ : ℓ p G d q Ñ C by σ : f ÞÑ ÿ x P H f p x q p f P ℓ p G d qq . Then, by the calculation performed in [36, Lemma 3.6], σ p f q “ p f P ℓ p G d q ˚ g ` ¨ ¨ ¨ ` ℓ p G d q ˚ g n q , and hence, since σ is clearly bounded, σ p f q “ for every f P ℓ p G d q . This forces G “ H .Hence G is countable.A countable locally compact group is always discrete, so that if it is also compact it mustbe finite. If G is abelian, then the fact that ℓ p G d q is topologically Noetherian implies that G is finite by [2, Theorem 1.1]. (cid:3) Left and Right Ideals of Approximable Operators on a Banach Space
In this section we shall prove Theorem 1.5. This will follow as a corollary of the formally moregeneral Theorems 4.5 and 4.9 below. Along the way we shall give a characterisation of theclosed right ideals of A p E q , for E any Banach space such that A p E q has as right approximateidentity. This is analogous to the characterisation given by Grønbæk in [16, Proposition 7.3]of the closed left ideals of K p E q , for a Banach space E with the approximation property. Weshall observe below that Grønbæk’s proof actually goes through, with A p E q in place of K p E q ,under the formally weaker hypothesis that A p E q has a left approximate identity.Let E be a Banach space, and X Ă B p E q . Then we write E ˝ X : “ t λ ˝ T : T P X u “ ď T P X im T . Let A be a closed subalgebra of B p E q . Given closed linear subspaces F Ă E and D Ă E wedefine(4.1) L A p F q “ t T P A : im T Ă F u and(4.2) R A p D q “ t T P A : im T Ă D u . These define families of closed left and right ideals respectively. We also define a family ofclosed left ideals by(4.3) I A p D q “ t T P A : ker T Ą D u , where D is a closed linear subspace of E . When the ambient algebra A is unambiguous weshall often drop the subscript and simply write L p F q , R p D q , and I p D q . Usually A willbe either A p E q or B p E q . We shall show that when A p E q has a right approximate identityevery closed right ideal of A p E q has the form R p D q , for some closed linear subspace D of E (Theorem 4.6). We can restate Grønbæk’s result in a similar fashion: Theorem 4.1.
Let E be a Banach space such that A p E q has a left approximate identity. Thenthe map p SUB p E q , Ăq Ñ p
CLI p A p E qq , Ăq , F ÞÑ L p F q is a lattice isomorphism, with inverse given by I ÞÑ E ˝ I, p I P CLI p A p E qqq . Proof.
Observe that, by Lemma 2.1, every closed left ideal of A p E q intersects densely withthe finite rank operators. It follows from this that Grønbæk’s proof [16, Proposition 7.3] goesthrough under our hypothesis on E . We claim that, in Grønbæk’s notation, Φ p F q “ L p F q and Ψ p I q “ E ˝ I : showing each inclusion is routine, except Φ p F q Ą L p F q . For this, we againuse the fact that F p E q X L p F q is dense in L p F q to see that it is sufficient to check that, for afinite rank operator T “ ř ni “ x i b λ i P L p F q , we have T P span t x b λ : x P E, λ P F u . Indeed,we may assume that x , . . . , x n are linearly independent, and choose η j P E p j “ , . . . , n q such that x η j , x i y “ δ ij p i, j “ , . . . , n q . It then follows that T p η i q “ λ i P F p i “ , . . . , n q , sothat T has the required form. (cid:3) We begin by addressing the topological left Noetherianity question for A p E q . Lemma 4.2.
Let E be a Banach space. Let n P N , let T , . . . , T n P A p E q , and let I “ A p E q T ` ¨ ¨ ¨ ` A p E q T n . Then E ˝ I “ im T ` ¨ ¨ ¨ ` im T n . Proof. As E ˝ I “ Ť T P I im T we have E ˝ I Ą im T i p i “ , . . . n q . Since E ˝ I is a closedlinear subspace, it follows that E ˝ I Ą im T ` ¨ ¨ ¨ ` im T n .For the reverse inclusion, let S P I and let λ P E . There are sequences p R p j q q j , . . . , p R p j q n q j Ă A p E q ` C id E such that S “ lim j Ñ8 ´ R p j q ˝ T ` ¨ ¨ ¨ ` R p j q n ˝ T n ¯ . Then λ ˝ S “ lim j Ñ8 ´ λ ˝ p R p j q ˝ T q ` ¨ ¨ ¨ ` λ ˝ p R p j q n ˝ T n q ¯ “ lim j Ñ8 ´ T p λ ˝ R p j q q ` ¨ ¨ ¨ ` T n p λ ˝ R p j q n q ¯ P im T ` ¨ ¨ ¨ ` im T n . As λ and S were arbitrary, this concludes the proof. (cid:3) The next lemma gives a partial characterisation of when A p E q is topologically left Noether-ian. The full characterisation will be given in Theorem 4.5. Lemma 4.3.
Let E be a Banach space such that A p E q has a left approximate identity. (i) Let F Ă E be a closed linear subspace. Then L p F q is topologically generated by T , . . . , T n P A p E q if and only if (4.4) F “ im T ` ¨ ¨ ¨ ` im T n . (ii) The algebra A p E q is topologically left Noetherian if and only if every closed linearsubspace of E has the form (4.4) , for some n P N and T , . . . , T n P A p E q . EFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACH ALGEBRAS 11
Proof. (i) Suppose that L p F q “ A p E q T ` ¨ ¨ ¨ ` A p E q T n , for some T , . . . , T n P A p E q . Thenby Lemma 4.2 E ˝ L p F q “ im T ` ¨ ¨ ¨ ` im T n , so that, by Theorem 4.1, F “ im T ` ¨ ¨ ¨ ` im T n . Conversely, suppose that there are maps T , . . . , T n P A p E q such that F has the form (4.4).Consider the left ideal I “ A p E q T ` ¨ ¨ ¨ ` A p E q T n . By Lemma 4.2 we have E ˝ I “ F, and so by Theorem 4.1 we have I “ L p E ˝ I q “ L p F q .Hence L p F q “ A p E q T ` ¨ ¨ ¨ ` A p E q T n , as required.(ii) This is clear from (i) and Theorem 4.1. (cid:3) In the proof of the next lemma we use the fact that every infinite-dimensional Banach spacecontains a basic sequence [26, Theorem 4.1.30].
Lemma 4.4.
Let E be a Banach space, and let F Ă E be a closed, separable linear subspace.Then there exists T P A p E q such that im T “ F .Proof. We may suppose that E is infinite-dimensional, since otherwise the lemma follows fromroutine linear algebra. Let t λ n : n P N u be a dense subset of the unit ball of F , and let p b n q be a normalised basic sequence in E . Let p β n q Ă E satisfy x b i , β j y “ δ ij p i, j P N q . Define T “ ř n “ ´ n b n b λ n . The operator T is a limit of finite-rank operators and T ϕ “ ÿ n “ ´ n ϕ p b n q λ n p ϕ P E q . Certainly im T Ă F . Observing that T p i β i q “ λ i p i P N q , we see that im T “ F , asrequired. (cid:3) We can now give our characterisation of topological left Noetherianity for A p E q . We noticethat our proof actually implies that for these Banach algebras topological left Noetherianityis equivalent to every closed left ideal being topologically principal. Theorem 4.5.
Let E be a Banach space such that A p E q has a left approximate identity. Thenthe following are equivalent: (a) the Banach algebra A p E q is topologically left Noetherian; (b) every closed left ideal of A p E q is topologically principal; (c) the space E is separable.Proof. It is trivial that (b) implies (a). To see that (c) implies (b), note that, by Theorem4.1, every closed left ideal of A p E q has the form L p F q , for some closed linear subspace F in E . Fixing F P SUB p E q , by Lemma 4.4 there exists T P A p E q such that F “ im T , whichimplies that L p F q “ A p E q T , by Lemma 4.3(i).We show that (a) implies (c) to complete the proof. Suppose that A p E q is topologically leftNoetherian. Then in particular A p E q “ A p E q T ` ¨ ¨ ¨ ` A p E q T n for some T , . . . , T n P A p E q .Observing that L p E q “ A p E q , Lemma 4.2 implies that E “ im T ` ¨ ¨ ¨ ` im T n . Since eachoperator T i is compact, so is each T i , implying that each space im T i is separable. It followsthat E “ im T ` ¨ ¨ ¨ ` im T n is separable. (cid:3) We now give our classification of the closed right ideals of A p E q . Observe that our hypothesison A p E q changes from possessing a left approximate identity to possessing a right approximateidentity. Theorem 4.6.
Let E be a Banach space such that A p E q has a right approximate identity.There is a lattice isomorphism Ξ : p SUB p E q , Ăq Ñ p
CRI p E q , Ăq given by Ξ : F ÞÑ R p F q , with inverse given by p Ξ : I ÞÑ span T P I p im T q p I P CRI p A p E qqq . Proof.
It is clear that Ξ and p Ξ are inclusion preserving. Since a poset isomorphism betweenlattices preserves the lattice structure, once we have shown that Ξ and p Ξ are mutually inverseit will follow that they are lattice isomorphisms.Let F be a closed linear subspace of E and set D “ p Ξ p R p F qq . It is immediate from thedefinitions that D Ă F . Moreover, given x P F , by considering x b λ for some λ P E zt u wesee that x P D . Hence F “ D , and, since F was arbitrary, this shows that p Ξ ˝ Ξ is the identitymap.Let I be a closed right ideal of A p E q , and set F “ p Ξ p I q . It is clear that I Ă R p F q . ByLemma 2.1 the finite-rank operators intersect R p F q densely, so in order to check the reverseinclusion it is sufficient to show that F p E q X R p F q Ă I . Let T P F p E q X R p F q . Then we canwrite T “ ř ni “ x i b λ i , for some n P N , some x , . . . , x n P im T , and some λ , . . . , λ n P E .Fix i P t , . . . , n u . Then x i P F so there exists a sequence p y j q Ă span U P I p im U q such that lim j Ñ8 y j “ x i . Moreover, for each j we can write y j “ S p j q z ` ¨ ¨ ¨ ` S p j q k j z k j , for some k j P N ,some S p j q , . . . , S p j q k j P I, and some z , . . . , z k j P E . For each j , and each p “ , . . . , k j we have ´ S p j q p z j ¯ b λ i “ S p j q p ˝ p z p b λ i q P I . Hence y j b λ i P I for each j , so that, taking the limit as j goes to infinity, x i b λ i P I . As i was arbitrary it follows that T P I . Hence we have shownthat I “ R p F q . As I was arbitrary, we have shown that Ξ ˝ p Ξ is the identity map. (cid:3) Now we set out to characterise when A p E q is topologically right Noetherian, for E a Banachspace as in Theorem 4.6. Lemma 4.7.
Let E and p Ξ be as in Theorem 4.6. Let T , . . . , T n P A p E q and let I “ T A p E q ` ¨ ¨ ¨ ` T n A p E q . Then p Ξ p I q “ im T ` ¨ ¨ ¨ ` im T n . Proof.
Since each T i p i “ , . . . , n q belongs to I we have im T ` ¨ ¨ ¨ ` im T n Ă p Ξ p I q . Let x P p Ξ p I q , and let ε ą . Then, by the definition of p Ξ , there exist m P N , S , . . . , S m P I , and y , . . . , y m P E such that } x ´ p S y ` ¨ ¨ ¨ ` S m y m q} ă ε. Since T A p E q ` ¨ ¨ ¨ ` T n A p E q is dense in I , we may in fact suppose that S , . . . , S m P T A p E q ` ¨ ¨ ¨ ` T n A p E q , so that S y ` ¨ ¨ ¨ ` S m y m P im T ` ¨ ¨ ¨ ` im T n . As ε was arbitrary we see that x P im T ` ¨ ¨ ¨ ` im T n . The result now follows. (cid:3) EFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACH ALGEBRAS 13
We omit the proof of the following well known result. In any case, it can be proved in asimilar fashion to Lemma 4.4.
Lemma 4.8.
Let E be a Banach space, and let F be any separable Banach space. Then thereexists an approximable linear map from E to F with dense range. We can now prove the theorem.
Theorem 4.9.
Let E be a Banach space such that A p E q has a right approximate identity.Then the following are equivalent: (a) the Banach algebra A p E q is topologically right Noetherian; (b) every closed right ideal of A p E q is topologically principal; (c) the space E is separable.Proof. It is trivial that (b) implies (a). We show that (a) implies (c). Suppose that A p E q istopologically right Noetherian. Then A p E q “ R p E q is topologically finitely-generated so that,by Lemma 4.7, there exist n P N and T , . . . , T n P A p E q such that E “ im T ` ¨ ¨ ¨ ` im T n . Since each operator T i p i “ , . . . , n q is compact, its image is separable, and hence so is E .Now suppose instead that E is separable, and let I be a closed right ideal of A p E q . Then,by Theorem 4.6, I “ R p F q for some F P SUB p E q . By Lemma 4.8 there exists T P A p E q with im T “ F . By Lemma 4.7 we have p Ξ ´ T A p E q ¯ “ im T “ F , so that, by Theorem 4.6, I “ R p F q “ T A p E q . Since I was arbitrary, this shows that (c) implies (b). (cid:3) We can now prove Theorem 1.5 as a special case of our results above.
Proof of Theorem 1.5.
Of course, under either hypothesis K p E q “ A p E q . By [10, Theorem2.5 (ii)], K p E q has a left approximate identity whenever E has the approximation property.Similarly, by [17, Theorem 3.3] K p E q has a (bounded) two-sided approximate identity when-ever E has the bounded approximation property. Hence the results follow from Theorem 4.5and Theorem 4.9. (cid:3) Remark.
Consider K p ℓ q . Of course, p ℓ q – ℓ , which has BAP by [37, Example 5(a),Chapter II E]. Hence, by Theorem 1.5, K p ℓ q is an example of a Banach algebra which istopologically right Noetherian, but not topologically left Noetherian.We observe that, although it talks about algebraically finitely-generated ideals, the ar-gument given in [11, Corollary 3.2] actually proves that any Banach space E satisfying itshypothesis has the property that B p E q is not topologically left Noetherian. Indeed, the argu-ment there is to demonstrate that there are more (maximal) closed left ideals in B p E q thanthere are finite n -tuples of operators. Hence not every closed left ideal can be topologicallyfinitely-generated. This covers a large class of Banach spaces, including, for example, c , ℓ p for ď p ă 8 , L p r , s for ă p ă 8 , and many other spaces discussed in [11]. In thecase that E is a reflexive Banach space, B p E q is a dual Banach algebra with predual given by E p b E . We shall show in the next section that, for a reflexive Banach space E with the ap-proximation property, B p E q is weak*-topologically left Noetherian whenever K p E q is. Hencein particular, many of the above examples which fail to be } ¨ } -topologically left Noetherianare weak*-topologically left Noetherian.We note however that it is possible for B p E q to be topologically left Noetherian, for aninfinite-dimensional Banach space E . Let E AH be the Banach space constructed by Argyrosand Haydon in [3] with the property that B p E AH q “ C id E AH ` K p E AH q . Since E AH is a predual of ℓ , which has BAP, it satisfies the hypotheses of part (i) and (ii) of Theorem1.5. Since B p E AH q “ K p E AH q , Theorem 4.5 and Lemma 2.2(iii) imply that B p E AH q is } ¨ } -topologically left and right Noetherian. Further examples come from [28], where the authorsconstruct, for each countably infinite, compact metric space X , a predual of ℓ , say E X , suchthat B p E X q{ K p E X q – C p X q . Since C p X q is topologically Noetherian for any compact metricspace X , a similar argument shows that the Banach algebras B p E X q are topologically left andright Noetherian. In all of these examples the Banach space is hereditarily indecomposable.There is no hereditarily indecomposable Banach space E for which we know that B p E q is nottopologically left/right Noetherian.5. Multiplier Algebras and Dual Banach Algebras
In this section we consider those Banach algebras A whose multiplier algebra is a dual Banachalgebra. We shall focus on the case in which A has a bounded approximate identity. Examplesof such Banach algebras included L p G q for G a locally compact group, and B p E q for E areflexive Banach space with AP. Other examples include the Figà-Talamanca–Herz algebras A p p G q for G a locally compact amenable group, and p P p , , with the predual of M p A p p G qq given by P F p p G q , the algebra of p-pseudo-functions of G . Also L p G q , where G is a locallycompact quantum group in the sense of Kustermans and Vaes, fits into this setting wheneverit has a bounded approximate identity [9].We prove in Proposition 5.3 that, for a Banach algebra A satisfying a fairly mild con-dition, the multiplier algebra M p A q is weak*-topologically left Noetherian whenever A is } ¨ } -topologically left Noetherian. Corollary 1.6 then follows. We go on to prove that for acertain, more restrictive class of Banach algebras there is a bijective correspondence betweenthe closed left ideals of A and the weak*-closed left ideals of M p A q (Theorem 5.10).For background on multiplier algebras see one of [8, 9, 29]. We say that A is faithful if Ax “ t u implies x “ p x P A q , and also xA “ t u implies x “ p x P A q . We recall thatthe canonical map A Ñ M p A q is injective if and only if A is faithful. When A has a boundedapproximate identity, this map is bounded below, so that A is isomorphic to its image inside M p A q . When this is the case we shall identify A with its image inside M p A q .In this section, when we consider a linear functional applied to a vector, we shall often usea subscript to indicate the exact dual pairing. So for example, if A is a Banach algebra, and a P A and f P A , we might write x a, f y p A, A q or x f, a y p A , A q for the value of f applied to a .In [9] Daws considers Banach algebras whose multiplier algebras are also dual Banachalgebras. We shall use the following consequence of Daws’ work, which essentially says thatwhen such a Banach algebra has a bounded approximate identity, the multiplier and dualstructures are compatible in a natural way. Theorem 5.1.
Let A be a Banach algebra with a bounded approximate identity, and supposethat M p A q is a dual Banach algebra, with predual X . Then X may be identified with a closed A -submodule of A ¨ A ¨ A in such a way that (5.1) x f ¨ a, µ y p X, M p A qq “ x f, aµ y p A , A q , (5.2) x a ¨ f, µ y p X, M p A qq “ x f, µa y p A , A q , for all µ P M p A q , and all a P A and f P A with f ¨ a P X / a ¨ f P X respectively. We alsohave (5.3) x x, a y p X, M p A qq “ x x, a y p A , A q , EFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACH ALGEBRAS 15 for all x P X and a P A .Proof. The fact that X may be identified with a closed A -submodule of A ¨ A ¨ A followsimmediately from [9, Theorem 7.9] and the remarks following it, and Equations (5.1) and(5.2) then follow by chasing through the definition of the map θ of that theorem.Equation (5.3) then follows from (5.1): given a P A and x P X , let b P A and f P A satisfy x “ f ¨ b . Then x f ¨ b, a y p X, M p A qq “ x f, ba y p A , A q “ x f ¨ b, a y p A , A q , as required. (cid:3) Remark.
Commutative Banach algebras whose multiplier algebras are dual Banach algebrassatisfying (5.1)/(5.2) were considered by Ülger in [35]. Since Ülger always assumes the exis-tence of a bounded approximate identity, Theorem 5.1 allows the hypothesis of [35, Theorem3.7] to be simplified slightly.We note the following.
Lemma 5.2.
Let A be a Banach algebra with a bounded approximate identity, such that M p A q is a dual Banach algebra. Then A is weak*-dense in M p A q .Proof. Let X be the predual of M p A q . Suppose x P A K Ă X . Then, by Theorem 5.1, wemay identify X with a closed subspace of A , and for all a P A we have “ x x, a y p X, M p A qq “x x, a y p A , A q , which implies that x “ . Hence A w ˚ “ p A K q K “ t u K “ M p A q . (cid:3) We now prove a result about weak*-topological left Noetherianity in this setting. Note that,by the previous lemma, the hypothesis is satisfied by any Banach algebra with a boundedapproximate identity whose multiplier algebra is a dual Banach algebra.
Proposition 5.3.
Let A be a Banach algebra such that M p A q admits the structure of adual Banach algebra in such a way that A is weak*-dense in M p A q . Suppose that for everyclosed left ideal I in A there exists n P N and there exist µ , . . . , µ n P M p A q such that I “ A µ ` ¨ ¨ ¨ ` A µ n . Then M p A q is weak*-topologically left Noetherian. In particular, M p A q is weak*-topologically left Noetherian whenever A is } ¨ } -topologically left Noetherian.Proof. Let I be a weak*-closed left ideal of M p A q . Since A is weak*-dense in M p A q , whichis unital, Lemma 2.1 implies that A X I is weak*-dense in I . On the other hand, A X I isa closed left ideal in A , so there exists n P N , and there exist µ , . . . , µ n P M p A q such that A X I “ A µ ` ¨ ¨ ¨ ` A µ n . It follows that I “ A µ ` ¨ ¨ ¨ ` A µ nw ˚ “ M p A q µ ` ¨ ¨ ¨ ` M p A q µ nw ˚ . As I was arbitrary the result follows. (cid:3) We are now able to prove Corollary 1.6 concerning the weak*-topological left/right Noethe-rianity for algebras of the form M p G q and B p E q . Note that, by Proposition 3.5 and thediscussion at the end of Section 4, these algebras are often not } ¨ } -topologically left/rightNoetherian Proof of Corollary 1.6.
This follows from Proposition 5.3, Theorem 1.4, and Theorem 1.5. (cid:3)
Definition 5.4.
Let A be a Banach algebra. We say that A is a compliant Banach algebra if A is faithful and M p A q is a dual Banach algebra in such a way that, for each a P A , the maps M p A q Ñ A given by µ ÞÑ µa and µ ÞÑ aµ are weak*-weakly continuous. In this article we shall consider the ideal structure of compliant Banach algebras, but wenote that they appear to have interesting properties more broadly and are worthy of furtherstudy. In the papers [18] and [19] Hayati and Amini consider Connes amenability of certainmultiplier algebras which are also dual Banach algebras. In our terminology, [19, Theorem3.3] says that if A is a compliant Banach algebra with a bounded approximate identity, then A is amenable if and only if M p A q is Connes amenable.We have the following family of examples of compliant Banach algebras. Lemma 5.5.
Let A be a Banach algebra with a bounded approximate identity which is Arensregular and an ideal in its bidual. Then A is a compliant Banach algebra.Proof. By [24, Theorem 3.9] A as an algebra with Arens multiplication may be identified with M p A q . Arens regularity implies that A is a dual Banach algebra with predual A . Given a P A , the maps µ ÞÑ µa and µ ÞÑ aµ are weak*-continuous as maps from A to itself, andhence they are weak*-weakly continuous when considered as maps from A to A . (cid:3) It follows from Lemma 5.5 that c p N q is an example of a compliant Banach algebra. Afamily of examples that will be important to us is the following: Corollary 5.6.
Let E be a reflexive Banach space with the approximation property. Then K p E q is a compliant Banach algebra.Proof. By [38, Theorem 3] K p E q is Arens regular. Moreover K p E q “ B p E q , so that we seethat K p E q is an ideal in its bidual. Hence the result follows from the previous lemma. (cid:3) The following lemma is also useful for finding examples.
Lemma 5.7.
Let A be as in Theorem 5.1, and suppose that the identification of that theoremyields X “ A ¨ A “ A ¨ A . Then A is a compliant Banach algebra.Proof. Let p µ α q be a net in M p A q which converges to some µ P M p A q in the weak*-topology.Fix a P A , and let f P A be arbitrary. Then lim α x f, aµ α y p A , A q “ lim α x f ¨ a, µ α y p X, M p A qq “ x f ¨ a, µ y p X, M p A qq “ x f, aµ y p A , A q . This shows that the map µ ÞÑ aµ is weak*-weakly continuous, and by an analogous argumentso is the map µ ÞÑ µa . As a was arbitrary this proves the lemma. (cid:3) Compliance is a fairly restrictive condition, as the next Proposition illustrates.
Proposition 5.8. (i)
Let A be a compliant Banach algebra, and let l denote the firstArens product on A . Then A is an ideal in p A , l q . (ii) Let K be a locally compact space. The Banach algebra C p K q is compliant if and onlyif K is discrete. (iii) Let G be a locally compact group. The Banach algebra L p G q is compliant if and onlyif G is compact.Proof. (i) If A is compliant then, for every a P A , the maps given by L a : b ÞÑ ab and R a : b ÞÑ ba are weakly compact. Hence L a p A q , R a p A q Ă A by [26, Theorem 3.5.8]. Given a P A we have L a : Ψ ÞÑ a l Ψ p Ψ P A q , and we see that A is a right ideal in A . Similarly it is a left ideal.(ii) Whenever K is discrete C p K q is Arens regular with bidual given by ℓ p K q . Hence thealgebra is compliant by Lemma 5.7. EFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACH ALGEBRAS 17
Now suppose instead that C p K q is compliant. The dual space of C p K q may be identifiedwith M p K q , and, as such, for each x P K , we may define an element ε x P C p K q by ε x : µ ÞÑ ż t x u µ p µ P M p K qq . By part (i) we know that, for any x P K and any f P C p K q , we have ε x l f P C p K q . Wethen calculate that, for µ P M p K q , we have x f l ε x , µ y “ x ε x , µ ¨ f y “ ż t x u f d µ “ µ pt x uq f p x q so that ε x l f is equal to f p x q at x , and everywhere else. Therefore, given any x P K , bychoosing any f P C p K q not vanishing at x , we see that the point mass at x is continuous.Hence K is discrete.(iii) If L p G q is compliant, then, by part (i) and [15], G is compact. Suppose instead that G is compact. By examining the maps in [9, Theorem 7.9] we see that the identificationin Theorem 5.1 is the usual inclusion C p G q Ñ L p G q . Then C p G q “ L p G q ¨ L p G q “ L p G q ¨ L p G q by [14, Proposition 2.39(d)]. Hence, by Lemma 5.7, L p G q is compliant. (cid:3) For compliant Banach algebras there is a bijective correspondence between the closed leftideals of A and the weak*-closed left ideals of M p A q as we describe below in Theorem 5.10.The next section will be devoted to applications of this result. Lemma 5.9.
Let I be a closed left ideal of a compliant Banach algebra A , and let µ P I w ˚ Ă M p A q . Then Aµ Ă I .Proof. Let p µ α q be a net in I converging to µ in the weak*-topology and let a P A . For eachindex α we have aµ α P I . Since A is compliant, the net aµ α converges weakly to aµ in A .Hence aµ P I w “ I . As a was arbitrary, the result follows. (cid:3) Theorem 5.10.
Let A be a compliant Banach algebra with a bounded approximate identity.The map I ÞÑ I w ˚ , defines a bijective correspondence between closed left ideals in A and weak*-closed left idealsin M p A q . The inverse is given by J ÞÑ A X J, for J a weak*-closed left ideal in M p A q .Proof. First we take an arbitrary closed left ideal I in A and show that A X I w ˚ “ I . Certainly I Ă A X I w ˚ . Let a P A X I w ˚ . Then by Lemma 5.9 we have Aa Ă I . Since A has a boundedapproximate identity, this implies that a P I . As a was arbitrary, we must have I “ A X I w ˚ .It remains to show that, given a weak*-closed left ideal J of M p A q , we have A X J w ˚ “ J ,and this follows from Lemma 2.1 and Lemma 5.2. (cid:3) Finally we show that for compliant Banach algebras the converse of Proposition 5.3 holds,so that weak*-topological left Noetherianity of M p A q can be characterised in terms of a } ¨ } -topological condition on A . Proposition 5.11.
Let A be a compliant Banach algebra with a bounded approximate identity.Then M p A q is weak*-topologically left Noetherian if and only if every closed left ideal I in A has the form I “ Aµ ` ¨ ¨ ¨ ` Aµ n , for some n P N , and some µ , . . . , µ n P M p A q .Proof. The “if” direction follows from Proposition 5.3 and Lemma 5.2. Conversely, supposethat M p A q is weak*-topologically left Noetherian, and let I be a closed left ideal of A . Thenthere exist n P N and µ , . . . , µ n P M p A q such that I w ˚ “ M p A q µ ` ¨ ¨ ¨ ` M p A q µ nw ˚ “ Aµ ` ¨ ¨ ¨ ` Aµ nw ˚ , where we have used Lemma 5.2 to get the second equality. Hence, by applying Theorem 5.10twice, we obtain I “ I w ˚ X A “ Aµ ` ¨ ¨ ¨ ` Aµ nw ˚ X A “ Aµ ` ¨ ¨ ¨ ` Aµ n . The result follows. (cid:3) Some Classification Results
In this section we use Theorem 5.10 to give classifications of the weak*-closed left ideals of M p G q , for G a compact group, and of the weak*-closed left ideals of B p E q , for E a reflexiveBanach space with the approximation property. We then observe how this gives us someclassification results for the closed right submodules of the preduals.Let G be a compact group and suppose the for each π P p G we have chosen a linear subspace E π ď H π . Then we define J rp E π q π P p G s : “ ! µ P M p G q : π p µ qp E π q “ , π P p G ) . We shall show that these are exactly the weak*-closed left ideals of M p G q . Lemma 6.1.
Let G be a compact group and let E π ď H π p π P p G q . Then (6.1) span ! ξ ˚ π η : π P p G, ξ P E π , η P H π ) K “ J rp E π q π P p G s . Proof.
Routine calculation. (cid:3)
Theorem 6.2.
Let G be a compact group. Then the weak*-closed left ideals of M p G q are givenby J rp E π q π P p G s , as p E π q π P p G runs over the possible choices of linear subspaces E π ď H π p π P p G q .Proof. By Lemma 6.1 each space J rp E π q π P p G s is weak*-closed, and it is easily checked thatit is a left ideal. Moreover, by Theorem 3.2 each closed left ideal of L p G q has the form L p G q X J rp E π q π P p G s , for some choice of subspaces E π ď H π p π P p G q . By Proposition 5.8, L p G q is a compliant Banach algebra, so we may apply Proposition 5.10 to see that this mustbe the full set of weak*-closed left ideals. (cid:3) Recall the definition of I B p E q p F q given in Equation (4.3). Theorem 6.3.
Let E be a reflexive Banach space with the approximation property. Then theweak*-closed left ideals of B p E q are exactly given by I B p E q p F q , as F runs through SUB p E q .The weak*-closed right ideals are given by R B p E q p F q , as F runs through SUB p E q . EFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACH ALGEBRAS 19
Proof.
For any closed linear subspace F Ă E the left ideal I B p E q p F q is weak*-closed since wehave I B p E q p F q “ t x b λ : x P F, λ P E u K , where x b λ denotes an element of the predual E p b E . Similarly we have R B p E q p F q “ t x b λ : x P E, λ P F K u K , so that these right ideals are weak*-closed. Observe that for F P SUB p E q , we have L B p E q p F K q “ t T P B p E q : im T Ă F K u “ t T P B p E q : im T Ă F K u (6.2) “ t T P B p E q : ` im T ˘ K Ą F u “ t T P B p E q : ker T Ą F u “ I B p E q p F q , so that, by Theorem 4.1, every closed left ideal of K p E q has the form I B p E q p F q X K p E q forsome F P SUB p E q . By Corollary 5.6 K p E q is compliant, so we may apply Proposition 5.10to see that every weak*-closed left ideal has the required form. A similar argument applies tothe weak*-closed right ideals. (cid:3) Finally we show how, using the following proposition, we can describe the closed left/rightsubmodules of E p b E , for E a reflexive Banach space with AP. We also obtain a descriptionof the left-translation-invariant closed subspaces of C p G q , for G a compact group (comparewith [1, Theorem 2]). Proposition 6.4.
Let p A, X q be a dual Banach algebra. Then there is a bijective correspon-dence between the closed right A -submodules of X and the weak*-closed left ideals of A givenby Y ÞÑ Y K , for Y a closed right A -submodule of X .Proof. It is quickly checked that Y K is a left ideal, whenever Y is a right submodule, and that I K is a right submodule whenever I is a left ideal. Equation (2.1) now tells us that the givencorrespondence is bijective, with inverse given by I ÞÑ I K , for I a weak*-closed left ideal. (cid:3) In the following corollary, given a Banach space E , and a closed subspaces F Ă E and D Ă E , we identify F p b E with the closure of the algebraic tensor product F b E inside E p b E , and similarly for E p b D . Corollary 6.5.
Let E be a reflexive Banach space with the approximation property. Then theclosed right B p E q -submodules of E p b E are given by F p b E p F P SUB p E qq . The closed left B p E q -submodules are given by E p b D p D P SUB p E qq . Proof.
Given F P SUB p E q it is easily seen that F p b E is a closed right submodule. Hence I : “ p F p b E q K is a weak*-closed left ideal by Proposition 6.4. It is easily checked that E ˝ I “ F K , and hence I “ L B p E q p F K q “ I B p E q p F q by (6.2). Since the correspondence given inProposition 6.4 is bijective, and since by Theorem 6.3 every weak*-closed ideal has the form I B p E q p F q for some F , it must be that every closed right submodule has the form F p b E forsome F P SUB p E q .The result about closed left submodules is proved analogously. (cid:3) Corollary 6.6.
Let G be a compact group, and let X Ă C p G q be a closed linear subspace,which is invariant under left translation. Then there exists a choice of linear subspaces E π ď H π p π P p G q such that X “ span ! ξ ˚ π η : π P p G, ξ P E π , η P H π ) . Proof.
In fact, by the weak*-density of the discrete measures in M p G q , the closed right sub-modules of C p G q coincide with the closed linear subspaces invariant under left translation(compare with [36, Lemma 3.3]). By Proposition 6.4 X has the form I K , for some weak*-closed left ideal I of M p G q . It now follows from Theorem 6.2 and Lemma 6.1 that X has thegiven form. (cid:3) Acknowledgements.
This work was supported by the French “Investissements d’Avenir”program, project ISITE-BFC (contract ANR-15-IDEX-03). The article is based on part ofthe author’s PhD thesis, and as such he would like to thank his doctoral supervisors GarthDales and Niels Laustsen, as well as his examiners Gordon Blower and Tom Körner, for theircareful reading of earlier versions of this material and their helpful comments. We would alsolike to thank Yemon Choi for some helpful email exchanges. Finally, we would like to thank theanonymous referee for his/her many insightful comments, and in particular for conjecturingProposition 5.8(ii).
References [1] C. A. Akemann, Invariant subspaces of C p G q , Pacific J. Math. (1968), 421–424.[2] A. Atzmon, Nonfinitely generated closed ideals in group algebras, J. Funct. Anal. (1972), 231–249.[3] S. Argyros and R. Haydon, A hereditarily indecomposable L –space that solves the scalar-plus-compactproblem, Acta Math. (2011), 1–54.[4] D. Blecher and T. Kania, Finite generation in C*-algebras and Hilbert C*-modules,
Studia Math. (2014), 143–151.[5] P. G. Casazza, Approximation properties, in
Handbook of the geometry of Banach spaces pages 271–316,North Holland, Amsterdam, 2001.[6] K. Casto, Are convolution algebras ever “topologically Noetherian”?,
Mathoverflow ,https://mathoverflow.net/questions/185741/are-convolution-algebras-ever-topologically-noetherian.[7] R. Choukri, A concept of finiteness in topological algebras, in
Topological algebras and applications pages131–137, Contemp. Math., , Amer. Math. Soc., Providence, RI, 2007.[8] H. G. Dales,
Banach algebras and automatic continuity , Volume of London Mathematical SocietyMonographs, New Series , Oxford Science Publications. The Clarendon Press, Oxford University Press,New York, 2000.[9] M. Daws,
Multipliers, self-induced and dual Banach algebras , Dissertationes Math., Volume , 2010.[10] P. Dixon, left approximate identities in algebras of compact operators on Banach spaces,
Proc. Roy. Soc.Edinburgh Sect. A (1986), 169–175.[11] H. G. Dales, T. Kania, T. Kochanek, P. Koszmider and N. J. Laustsen, Maximal left ideals in the Banachalgebra of operators on a Banach space,
Studia Math. (2013), 245–286.[12] H. G. Dales and W. Żelazko, Generators of maximal left ideals in Banach algebras,
Studia Math. (2012), 173–193.[13] P. Eymard, L’algèbre de Fourier d’un group localement compact,
Bull. Soc. Math. France (1964),181–236.[14] G. Folland, A course in abstract harmonic analysis , Studies in Advanced Mathematics, CRC Press, BocaRaton, FL, 1995.[15] M. Grosser, L p G q as an ideal in its second dual space, Proc. Amer. Math. Soc. (1979), 363–364.[16] N. Grønbæk, Morita equivalence for Banach algebras, J. Pure Appl. Algebra (1995), 183–219.[17] N. Grønbæk and G. Willis, Approximate identities in Banach algebras of compact operators. Canad.Math. Bull. (1993), 45–53. EFT IDEALS OF BANACH ALGEBRAS AND DUAL BANACH ALGEBRAS 21 [18] B. Hayati and M. Amini, Connes–amenability of multiplier Banach algebras,
Kyoto J. Math. (2010),41–50.[19] B. Hayati and M. Amini, Dual multiplier Banach algebras and Connes–amenability, Publ. Math Debrecen (2015), 169–182.[20] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integrationtheory, group representations , Die Grundlehren der mathematischen Wissenschaften, Bd. 115. AcademicPress, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963.[21] E. Hewitt and K. A. Ross,
Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups.Analysis on locally compact Abelian groups , Die Grundlehren der mathematischen Wissenschaften, Band152. Springer-Verlag, New York-Berlin, 1970.[22] A. Joyal and R. Street, An introduction to Tannaka duality and quantum groups. In
Category theory(Como, 1990) , 413–492, Lecture Notes in Math. , Springer, Berlin, 1991.[23] E. Kanuith and A. T.–M. Lau, Spectral synthesis for A p G q and subspaces of V N p G q , Proc. Amer. Math.Soc. (2001), 3253–3263.[24] H. C. Lai, Multipliers of a Banach algebra in the second conjugate algebra as an idealizer,
Tôhoku Math.J. (1974), 431–452.[25] N. J. Laustsen and J. T. White, Subspaces that can and cannot be the kernel of a bounded operator ona Banach space, to appear in Proceedings of the 23rd International Conference on Banach Algebras andApplications , arXiv:1811.02393.[26] R. Megginson,
An introduction to Banach space theory , Graduate Texts in Mathematics , Springer-Verlag, New York, 1998.[27] C. Meniri, Linearly compact rings and selfcogenerators,
Rendiconti del Seminario Matematico della Uni-versita di Padova , (1984), 99–116.[28] P. Motakis, D. Puglisi, D. Zisimopoulou, A hierarchy of Banach spaces with C p K q Calkin algebras,
IndianaUniv. Math. J. (2016), 39–67.[29] T. W. Palmer, Banach algebras and the general theory of *-algebras. Vol. I. Algebras and Banach algebras ,Volume of Encyclopedia of Mathematics and its Applications , Cambridge University Press, Cambridge,1994.[30] R. Prosser,
On the ideal structure of operator algebras , Mem. Amer. Math. Soc. Volume (1963).[31] V. Runde, Amenability for dual Banach algebras, Studia Math (2001), 47–66.[32] A. M. Sinclair and A. W. Tullo, Noetherian Banach algebras are finite dimensional,
Math. Ann. (1974), 151–153.[33] A. Szankowski, B p H q does not have the approximation property, Acta Math. , (1981), 89–108.[34] R. Szwarc, Groups acting on trees and approximation properties of the Fourier algebra, J. Funct. Anal. (1991), 320–343.[35] A. Ülger, A characterization of the closed unital ideals of the Fourier–Stieltjes algebra B p G q of a locallycompact amenable group G , J. Funct. Anal. (2003), 90–106.[36] J. T. White, Finitely-generated left ideals in Banach algebras in groups and semigroups,
Studia Math. (2017), 67–99.[37] P. Wojtaszczyk,
Banach spaces for analysts , Cambridge Studies in Advanced Mathematics , CambridgeUniversity Press, Cambridge, 1991.[38] N. Young, Periodicity of functionals and representations of normed algebras on reflexive spaces, Proc.Edinburgh Math. Soc.20