Interpolation sets and the size of quotients of function spaces on a locally compact group
aa r X i v : . [ m a t h . F A ] J u l INTERPOLATION SETS AND THE SIZE OF QUOTIENTSOF FUNCTION SPACES ON A LOCALLY COMPACTGROUP
M. FILALI AND J. GALINDO
Abstract.
We devise a fairly general method for estimating the sizeof quotients between algebras of functions on a locally compact group.This method is based on the concept of interpolation sets and unifiesthe approaches followed by many authors to obtain particular cases.We find in this way that there is a linear isometric copy of ℓ ∞ ( κ ) ineach of the following quotient spaces:– WAP ( G ) /C ( G ) whenever G contains a subset X that is an E -set (see the definition in the paper) and κ = κ ( X ) is the minimalnumber of compact sets required to cover X . In particular, κ = κ ( G ) when G is a SIN -group.–
WAP ( G ) / B ( G ), when G is any locally compact group and κ = κ ( Z ( G )) and Z ( G ) is the centre of G , or when G is either an IN -group or a nilpotent group, and κ = κ ( G ).– WAP ( G ) / B ( G ), when G and κ are as in the foregoing item.– CB ( G ) / LUC ( G ), when G is any locally compact group that is nei-ther compact nor discrete and κ = κ ( G ). Introduction
The main focus throughout the paper will be on C ∗ -algebras of functionson a locally compact group G with identity e . If ℓ ∞ ( G ) denotes the C ∗ -algebra of bounded, scalar-valued functions on G with the supremum norm,our concern will be with the following subalgebras of ℓ ∞ ( G ): the algebra CB ( G ) of continuous bounded functions, the algebra LUC ( G ) of boundedright uniformly continuous functions, the algebra WAP ( G ) of weakly almostperiodic functions, the Fourier-Stieltjes algebra B ( G ), the uniform closure of B ( G ) denoted by B ( G ) and best known as the Eberlein algebra, the algebra AP ( G ) of almost periodic functions, and the algebra C ( G ) ⊕ C
1, where C ( G ) consists of the functions in CB ( G ) vanishing at infinity.The spectra of these algebras A ( G ) define some of the best-known semi-group compactifications in the sense of [5]. These are compact right (or left)topological semigroups G A having a dense, continuous, homomorphic copy Date : June 28, 2018.2010
Mathematics Subject Classification.
Primary 22D15; Secondary 43A46, 43A15,43A60, 54H11.
Key words and phrases. almost periodic functions, Fourier-Stieltjes algebra, weaklyalmost periodic, semigroup compactification, almost periodic compactification, almostperiodic compactification, interpolation sets.Research of the second named author supported by the Spanish Ministry of Science(including FEDER funds), grant MTM2008-04599/MTM and Fundaci´o Caixa Castell´o-Bancaixa, grant P1.1B2008-26. of G contained in their topological centres (i.e, the map x sx ( x xs ) : G A → G A is continuous for each s ∈ G .) For instance, the compactification G LUC is the spectrum of
LUC ( G ), and is usually referred to as the LUC ( G )-or LC ( G )-compactification of G . It is the largest semigroup compactifica-tion in the sense that any other semigroup compactification is a quotientof G LUC . When G is discrete, G LUC and the Stone- ˇCech compactification βG are the same. The WAP -compactification G WAP is the spectrum of
WAP ( G ); it is the largest semitopological semigroup compactification. TheBohr or AP -compactifiaction is the spectrum of AP ( G ) and is the largesttopological (semi)group compactification.The Banach duals of these C ∗ -algebras can also be made into Banachalgebras with a convolution type product extending in most cases that ofthe group algebra L ( G ). We may recall that L ∞ ( G ) is the Banach dualof the group algebra L ( G ) and consists of all scalar-valued functions whichare measurable and essentially bounded with respect to the Haar measure;two functions are identified if they coincide on a locally null set, and thenorm is given by the essential supremum norm. We may also recall that theproduct making L ∞ ( G ) into a Banach algebra is the first (or the second)Arens product on the second dual space L ( G ) ∗∗ of the group algebra, andthat LUC ( G ) ∗ may be seen as a quotient Banach algebra of L ( G ) ∗∗ . Formore details, see for instance [17]. These two Banach algebras have beenstudied extensively in recent years. Particular attention has been given toproperties related to Arens regularity of the group algebra L ( G ) and to thetopological centres of G LUC , LUC ( G ) ∗ and L ( G ) ∗∗ . For the latest, see [8]and the references therein.The definitions of all these function algebras will be given in the nextsection. But for the moment the following diagram summarizes already theinclusion relationships known to hold among these algebras. See [13, page143] for the first inclusion; [13, Lemma 2.1] for the first equality; [42] or [5,Theorem 4.3.13] for the second equality; [5, Corollary 4.4.11] or [9] for thethird inclusion; the rest is easy to check. C ( G ) ⊕ AP ( G ) ⊆ B ( G ) = AP ( G ) ⊕ B ( G ) ⊆ WAP ( G ) = AP ( G ) ⊕ WAP ( G ) ⊆ LUC ( G ) ∩ RUC ( G ) ⊆ LUC ( G ) ⊆ CB ( G ) ⊆ L ∞ ( G ) . When G is finite, the diagram is trivial. When G is infinite and compact,the diagram reduces to CB ( G ) ⊆ L ∞ ( G ).The task of comparing these algebras and estimating the sizes of thequotients formed among them has already been taken by many authors. Wenow give a brief review of what is known in this respect.In the review below as well as in our study of quotients between theabove algebras the compact covering number will appear at several points.We recall that the compact covering number of a topological space X is thesmallest cardinal number κ ( X ) of compact subsets of X required to cover X . Comparing L ∞ ( G ) with its subspaces . Already in 1961, Civin andYood proved in their seminal paper [15] that the quotient space L ∞ ( G ) / CB ( G )is infinite-dimensional for any non-discrete locally compact Abelian group NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 3 and deduced that the radical of the Banach algebra L ∞ ( G ) ∗ (with one ofthe Arens products as a product) is also infinite-dimensional.This idea was pushed further by Gulick in [34, Lemma 5.2] when G isAbelian, and proved that the quotient L ∞ ( G ) / CB ( G ) is even non-separableand so is the radical of L ∞ ( G ) ∗ . Then Granirer proved in [32] the sameresults for any non-discrete locally compact group.A decade later, Young produced, for any infinite locally compact group G , a function in L ∞ ( G ) which is not in WAP ( G ), proving the non-Arensregularity of the group algebra L ( G ) for any such a group, see [54].There was also [6, Theorem 4.2] where the quotient LUC ( G ) / WAP ( G )was seen to contain a linear isometric copy of ℓ ∞ ( κ ) , where κ is the compactcovering of G . A fortiori, the quotient L ∞ ( G ) / WAP ( G ) contains the samecopy, a fact that was used in [6, Theorem 4.4] to deduce that the groupalgebra is even extremely non-Arens regular in the sense of Granirer, when-ever κ is larger than or equal to the minimal cardinal w ( G ) of a basis ofneighbourhoods at the identity.It was also proved in [6, Section 4] that the quotient L ∞ ( G ) / CB ( G ) alwayscontains a linear isometric copy of ℓ ∞ , yielding extreme non-Arens regularityfor the group algebra of compact metrizable groups. Due to a result byRosenthal proved in [48, Proposition 4.7, Theorem 4.8], larger copies of ℓ ∞ cannot be expected in L ∞ ( G ) when G is compact. The question onextreme non-Arens regularity of the group algebra was recently settled bythe authors of the present paper using a technique inspired by Theorem2.11. We actually find in [25] that, for any compact group G , L ∞ ( G ) / CB ( G )contains a copy of L ∞ ( G ). This fact together with [6, Theorem 4.4] givesthat L ( G ) is extremely non-Arens regular for any infinite locally compactgroup. Comparing CB ( G ) with its subspaces . In 1966, Comfort and Ross[16, Theorem 4.1] compared the spaces CB ( G ) and AP ( G ) for an arbitrarytopological group, and proved that they are equal if and only if G is pseudo-compact (i.e., every continuous scalar-valued function on G is bounded). In1970, Burckel showed in [9] that CB ( G ) and WAP ( G ) are equal if and onlyif G is compact. In [3], Baker and Butcher compared CB ( G ) and LUC ( G )for locally compact groups, and proved that these two spaces are equal ifand only if G is either discrete or compact. This result was extended re-cently by Filali and Vedenjuoksu in [28, Theorem 4.3] to all topologicalgroups which are not P -groups. The author deduced in [28] that if G is atopological group which is not a P -group, then CB ( G ) = LUC ( G ) if andonly if G is pseudocompact. In [20], Dzinotyiweyi showed that the quotient CB ( G ) / LUC ( G ) is non-separable if G is a non-compact, non-discrete, locallycompact group. This theorem was generalized in [6, Theorem 3.1] and [7,Theorem 4.1], where CB ( G ) / LUC ( G )) was seen to contain in fact a linearisometric copy of ℓ ∞ whenever G is a non-precompact topological groupwhich is not a P -group. So this theorem improved actually also Dzino-tyiweyi’s result for locally compact groups. For non-discrete, P -groups, thequotient CB ( G ) / LUC ( G ) was seen to be trivial in the case when for instance G is a Lindel¨of P -group (see [28, Theorem 5.1]), but may also contain a lin-ear isometric copy of ℓ ∞ for some other P -groups (see [6, Theorem 3.3]). FILALI AND GALINDO
In [7, Theorem 3.1], using a technique due to Alas (see [1]), the quotientspace CB ( G ) / LUC ( G ) was also seen to contain a linear isometric copy of ℓ ∞ whenever G is a non- SIN topological group.In the locally compact situation, our answer in the present paper is preciseand definite. We prove, in Section 5, that there is a linear isometric copy of ℓ ∞ ( κ ) in CB ( G ) / LUC ( G ), where as before κ is the compact covering G, ifand only if G is a neither compact nor discrete. This leads again to a linearisometric copy of ℓ ∞ ( κ ) into the quotient L ∞ ( G ) / WAP ( G ), and of coursemay be used to deduce again the extreme non-Arens regularity of of L ( G )when κ ( G ) ≥ w ( G ) ≥ ω as in [6, Theorem 4.4]. Comparing
LUC ( G ) with WAP ( G ). In 1972, Granirer showed that LUC ( G ) = WAP ( G ) if and only if G is compact [31].It is not difficult to check that G LUC is a semitopological semigroup (i.e.,the topological centre of G LUC is the whole of G LUC ) if and only if
LUC ( G ) = WAP ( G ). The same observation can be made also for LUC ( G ) ∗ . This meansthat G LUC or LUC ( G ) ∗ is a semitopological semigroup if and only if G iscompact, i.e., G LUC = G is a compact group and LUC ( G ) ∗ coincides withthe measure algebra M ( G ).More recently, Granirer’s result was deduced by Lau and Pym in [40,Proposition 3.6] as a corollary of their main theorem on the topologicalcentre of G LUC being G , and again by Lau and ¨Ulger in [41, Corollary 3.8]as a corollary of the topological centre of L ( G ) ∗∗ being L ( G ) [39].Moreover, Granirer showed in the same paper that if G is non-compactand amenable, then the quotient LUC ( G ) / WAP ( G ) contains a linear iso-metric copy of ℓ ∞ , and so it is not separable. This result was extendedby Chou in [10] to E -groups (see below for definition), then by Dzinotyi-weyi in [20] to all non-compact locally compact groups, and generalized byBouziad and Filali in [6, Theorem 2.2] to all non-precompact topologicalgroups. Moreover, as already mentioned above, this result was improved in[6, Theorem 4.2] when G is a non-compact locally compact group, by havinga copy of ℓ ∞ ( κ ) in the quotient LUC ( G ) / WAP ( G ). Comparing
WAP ( G ) with its subspaces . In the ”regular” side of theinclusion diagram, when we compare WAP ( G ) with its subspaces, the situ-ation is not simpler. It is true that the Fourier-Stieltjes algebra B ( G ) maybe dense in WAP ( G ) (i.e., WAP ( G ) = B ( G )), as in the case of minimallyweakly almost periodic groups studied by Veech, Chou and Ruppert, see[53], [12] and [51]. For these groups, WAP ( G ) = AP ( G ) ⊕ C ( G ). However,if G is a non-compact group, then B ( G ) is far from being dense in WAP ( G )in general as it shall soon be explained.When comparing WAP ( G ) with AP ( G ) and C ( G ), we may recall firstthat WAP ( G ) = AP ( G ) ⊕ WAP ( G ). Burckel proved in [9] that C ( G ) ( WAP ( G ) when G is an Abelian, non-compact, locally compact group. In[10], Chou considered E -groups and proved that the quotient WAP ( G ) /C ( G )contains a linear isometric copy of ℓ ∞ . In Section 4, we improve this resultby showing that ℓ ∞ may be replaced by an isometric copy of the larger space ℓ ∞ ( κ ( E )) in each of the quotient space WAP ( G ) /C ( G ), where κ ( E ) is thecompact covering of the E -set contained in G . So when G is an SIN -group,
NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 5 these quotients contain a copy of ℓ ∞ ( κ ( G )) . For the same class of groups, weprove also that the quotient
WAP ( G ) / ( AP ( G ) ⊕ C ( G )) is non-separable.In Section 5, we deal with the non-compact, IN -groups, and with non-compact nilpotent groups. In this class of groups, the results of the previoussection shall be considerably improved. Rudin proved in [49] that B ( G ) ( WAP ( G ) if G is a locally compact Abelian group and contains a closeddiscrete subgroup which is not of bounded order. This was followed by[47], where Ramirez extended Rudin’s result to any non-compact, locallycompact, Abelian group. Then in [13], Chou extended and strengthened thetheorem to all non-compact IN -groups and nilpotent groups by showingthat the quotient WAP ( G ) / B ( G ) contains a linear isometric copy of ℓ ∞ .We shall strengthen Chou’s result in Section 5 by showing that, in thesecases, there is in fact a linear isometric copy of ℓ ∞ ( κ ) in the quotient spaces WAP ( G ) / B ( G ) , WAP ( G ) / ( AP ( G ) ⊕ C ( G )) and WAP ( G ) / B ( G ) , where κ is as before the compact covering of G. Our method of proof also showsthat
WAP ( G ) / B ( G ) always contains a copy of ℓ ∞ ( κ ( Z ( G ))).It is worthwhile to note that all this confirms an observation made in [5,page 216], and gives indeed an indication on the size and complexity of the WAP -compactification G WAP and the Banach algebra
WAP ( G ) ∗ . Outline.
The underlying structure in many of the proofs that estimatethe size of A ( G ) / A ( G ) for C ∗ -subalgebras A ( G ) ⊆ A ( G ) of ℓ ∞ ( G ), de-pends on the existence of sets of interpolation for A ( G ) that are not sets ofinterpolation for A ( G ) (see for instance [6], [7], [10], [13] or [20]). One ofthe main objectives of the present paper is to make that structure emergein a clear fashion. A first, but essential, step towards this objective is towork with the right concept of interpolation sets. We will use here thegeneral concept of interpolation set introduced in [24] that extends severalrelated classical ones and show how to apply it in this setting. The result-ing interpolation sets are characterized in [24] in term of topological groupproperties, thereby making them easier to manipulate. We finally illustratethe scope of our approach by studying some concrete cases. We shall in par-ticular study under this light the following quotients: WAP ( G ) by C ( G )and WAP ( G ) by AP ( G ) ⊕ C ( G ) for E -groups, WAP ( G ) by B ( G ), WAP ( G )by AP ( G ) ⊕ C ( G ) and WAP ( G ) by B ( G ) for IN -groups and nilpotentgroups, CB ( G ) by LUC ( G ) for locally compact groups.1.1. The function algebras.
We start by recalling the definitions of thefunction algebras we are interested in, for more details the reader is directedfor example to [5].Let G be a topological group. For each function f defined on G , the lefttranslate f s of f by s ∈ G is defined on G by f s ( t ) = f ( st ). For each s ∈ G ,the left translation operator L s : ℓ ∞ ( G ) → ℓ ∞ ( G ) is defined as L s ( f ) = f s .The supremum norm of an element f ∈ ℓ ∞ ( G ) will be denoted as k f k ∞ .A function f ∈ ℓ ∞ ( G ) is right uniformly continuous when, if for every ǫ >
0, there exists a neighbourhood U of e such that | f ( s ) − f ( t ) | < ǫ whenever st − ∈ U. The algebra of right uniformly continuous functions on G is denoted by LUC ( G ) . FILALI AND GALINDO
A function f ∈ CB ( G ) is almost periodic when the set of all its left(equivalently, right) translates is a relatively norm compact subset in CB ( G ) . The algebra of almost periodic functions on G is denoted by AP ( G ) . A function f ∈ CB ( G ) is weakly almost periodic when the set of all its left(equivalently, right) translates makes a relatively weakly compact subset in CB ( G ) . The algebra of weakly almost periodic functions on G is denoted by WAP ( G ) . The
Fourier-Stieltjes algebra B ( G ) is the linear span of the set of allcontinuous positive definite functions on G . Equivalently, B ( G ) is the spaceof coefficients of unitary representations of G when G is locally compact.As the Fourier-Stieltjes algebra is not uniformly closed we will work withthe Eberlein algebra B ( G ), which is the uniform closure of B ( G ) , in symbols B ( G ) = B ( G ) k·k ∞ .Let µ be the unique invariant mean on WAP ( G ) (see [5], or [9]). As statedabove, put WAP ( G ) = { f ∈ WAP ( G ) : µ ( | f | ) = 0 } , B ( G ) = { f ∈ B ( G ) : µ ( | f | ) = 0 } . In [13, page 143], Chou denoted B ( G ) ∩ WAP ( G ) by B c ( G ), and observedthat B ( G ) = B c ( G ) when G is locally compact.1.2. The spectrum as a compactification.
Let G be a topological group, A ( G ) ⊆ ℓ ∞ ( G ) be a unital C ∗ -subalgebra and denote by G A the the spec-trum (the set of non-zero multiplicative linear functionals) of A ( G ). Equippedwith the topology of pointwise convergence, G A becomes a compact Haus-dorff topological space. There is a canonical morphism ǫ A : G → G A givenby evaluations ǫ A ( s )( f ) = f ( s ) , for every f ∈ A ( G ) and s ∈ G. This map is continuous if and only if A ( G ) ⊆ CB ( G ), and injective on G if and only if A ( G ) separates the points of G . We may recall, for example,that the map ǫ A is injective on G (and in fact a homeomorphism onto itsimage in G A ) whenever C ( G ) ⊆ A ( G ) . This is not a necessary conditionsince it may also happen that ǫ A is injective when C ( G ) ∩ A ( G ) = { } asit is the case when G is a locally compact, maximally almost periodic and A ( G ) = AP ( G ). It may also happen that ǫ A is injective on a given subset T of G. We will then identify T as a subset of G A . This situation occurs whenfor example T is an A ( G )-interpolation set.The C ∗ -algebra A ( G ) is left translation invariant when f s ∈ A ( G ) forevery f ∈ A ( G ) and s ∈ G. When A ( G ) is left translation invariant, we maydefine for every x ∈ G A and f ∈ A ( G ) , the function on G by xf ( s ) = x ( f s ) . When 1 , f s and xf are in A ( G ) for every s ∈ G, x ∈ G A and f ∈ A ( G ) , wesay that A ( G ) is admissible .When A ( G ) is an admissible C ∗ -subalgebra of CB ( G ), G A can be equippedwith the product G A given by xy ( f ) = x ( yf ) for every x, y ∈ G A and f ∈ A ( G ) .G A then becomes a semigroup compactification of the topological group G in the sense of [5]. This means that G A is a compact semigroup having a NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 7 continuous, dense, homomorphic, image of G such that the mappings x xy : G A → G A and x ǫ A ( s ) x : G A → G A are continuous for every y ∈ G A and s ∈ G .The algebras C ( G ) ⊕ C , AP ( G ), C ( G ) ⊕ AP ( G ), B ( G ), WAP ( G ) ⊕ C , WAP ( G ) and LUC ( G ) are all known to be admissible, see for example [5].But when G is locally compact, CB ( G ) is not admissible unless G is eitherdiscrete or compact, see [3] or [28] for more.When G is a locally compact group and A is an admissible C ∗ -subalgebraof LUC ( G ), the semigroup compactification G A has the the joint continuityproperty , that is, the map( s, x ) ǫ A ( s ) x : G × G A → G A is continuous.A recent account on semigroup compactifications is given in [29].1.3. A few words on notation.
All our groups will be multiplicative andtheir identity element will be denoted as e . The characteristic function of aset T will be denoted as 1 T . If X is a set and T ⊆ X , given f ∈ ℓ ∞ ( X ),we define k f k T = sup {| f ( x ) | : x ∈ T } so that k f k ∞ = k f k X . The morphism ǫ A maps G into G A faithfully if A ( G ) separates points. If X ⊆ G , we willdenote the closure of ǫ A ( X ) simply as X A , while the closure of X in G willbe denoted as X. The reason for this is that in most of our applications thealgebra A ( G ) separates points of G and therefore ǫ A may be used to identify G with a subset of G A .A standard application of Gelfand duality identifies A ( G ) with CB ( G A ).Under this identification, to every f ∈ A ( G ) there corresponds f A ∈ CB ( G A )in such a way that the following diagram commutes(1) G ǫ A / / f ❇❇❇❇❇❇❇❇ G A f A (cid:15) (cid:15) C When ǫ A is injective f A can be seen as an extension of f to G A .2. Interpolation sets and quotients of function spaces
We begin our work by introducing in precise terms the sets we will beusing, and then we prove the impact they have in measuring the size of ourquotient spaces A ( G ) / A ( G ). This is achieved in Theorem 2.11.It is worthwhile to note that this theorem may also be applied to obtainmost (if not all) of the results concerning the quotient spaces of the variousfunction algebras mentioned in the introduction; it is of course necessary ateach time to construct the required interpolation sets.Our final main results in this section and in the rest of the paper concern C ∗ -algebras of bounded functions on a locally compact group, but definitionsand properties shall also be proved for a general Hausdorff topological groupwhenever this makes sense. FILALI AND GALINDO
Definition 2.1.
Let G be a topological group and A ( G ) ⊆ ℓ ∞ ( G ). A subset T ⊆ G is said to be(i) an A ( G ) -interpolation set if every bounded function f : T → C canbe extended to a function f : G → C such that f ∈ A ( G ).(ii) an approximable A ( G ) -interpolation set if it is an A ( G )-interpolationset and for every neighbourhood U of e , there are open neighbour-hoods V , V of e with V ⊆ V ⊆ U such that, for each T ⊆ T there is h ∈ A ( G ) with h ( V T ) = { } and h ( G \ ( V T )) = { } . Remark 2.2. A ( G )-interpolation sets for some concrete algebras A ( G ) ⊆ ℓ ∞ ( G ) have been a frequent object of study, see [29] and [24] for more detailsand references. See also [30] for the most recent account on the subject.Approximable interpolation sets appear in the early 70’s as a crucial stepin Drury’s proof of the union theorem of Sidon sets, see [18]. Other well-known interpolation sets are also approximable as for instance translation-finite sets considered by Ruppert in [50] (and called R W -sets by Chou in[14]) that turn to be the approximable WAP ( G )-interpolation sets of discretegroups, see [24] for more on this respect.When G is discrete, the definition of approximable A ( G )-interpolation setis much simpler. In that case T ⊂ G is an approximable A ( G )-interpolationset if and only if T is an A ( G )-interpolation such that 1 T ∈ A ( G ), or equiv-alently, if every function supported on T is in A ( G ).It should however be reminded that approximable A ( G )-interpolation setsdo not make sense for every C ∗ -subalgebra A ( G ) of ℓ ∞ ( G ). For example,no subset in a non-compact locally compact group can be an approximable AP ( G )-interpolation set, see [24, Section 3 and Corollary 4.24].2.1. The quotients.
The following lemma contains some elementary con-sequences of the definitions of interpolation and approximable interpolationsets. The identification of T A with the Stone- ˇCech compactification of T (with the discrete topology) allows us to use the powerful property of ex-treme disconnectedness of the latter compactification. As the reader willquickly notice this is the key in the arguments leading to the main resultsin this section. The main results start with a generalization of a theoremproved by Chou [13] for B ( G ) (Lemma 2.5) to arbitrary C ∗ -subalgebras of ℓ ∞ ( G ) . Along with some rather technical lemmas, this provides us withthe conditions stated in Theorem 2.11 and Corollary 2.12 under which thequotient A ( G ) / A ( G ) ( A ( G ) ⊂ A ( G ) being admissible C ∗ -subalgebras of CB ( G )) contains a linear isometric copy of ℓ ∞ ( κ ) for some cardinal κ. Lemma 2.3.
Let G be a topological group. Let A ( G ) be a C ∗ -subalgebra of ℓ ∞ ( G ) with ∈ A ( G ) and T ⊆ G . (i) T is an A ( G ) -interpolation set if and only if ǫ A is injective on T and there is a homeomorphism between T A and βT d , the Stone-ˇCech-compactification of T equipped with the discrete topology, thatleaves the points of T fixed. (ii) T is an A ( G ) -interpolation set if and only if for every pair of subsets T , T ⊂ T , T ∩ T = ∅ implies T A ∩ T A = ∅ . NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 9 (iii) If T is an A ( G ) -interpolation set and f : T → C is a bounded func-tion, then f has an extension f ∈ A ( G ) with k f k ∞ = k f k T . (iv) If T is an approximable A ( G ) -interpolation set, then for every boundedfunction h : T → C and every neighbourhood U of the identity, thereis f ∈ A ( G ) such that f (cid:12)(cid:12) T = h, f ( G \ U T ) = { } and k f k ∞ = k h k T . Proof.
First observe that ǫ A is injective on every A ( G )-interpolation set T :if t = t ∈ T , there is f ∈ ℓ ∞ ( T ) with f ( t ) = f ( t ). Take ¯ f ∈ A ( G )extending f . By (1) ¯ f = ¯ f A ◦ ǫ A , hence ǫ A ( t ) = ǫ A ( t ).Assertion (i) follows then from the universal property defining the Stone-ˇCech compactification of a discrete space. In fact, the restriction of theevaluation map ǫ A to T gives a homeomorphism of the discrete set T d ontoits image in G A . So T A is a (topological) compactification of T d , and we mayapply [21, Corollary 3.6.3].Assertion (ii) follows also directly from a well-known characterization ofthe Stone- ˇCech compactification of a discrete space, see for instance [21,Corollary 3.6.2].To prove (iii), let f : T → C with k f k T = M be given. If B M is the closeddisc of radius M centered at 0 (in C ), we can use (i) and the universalproperty of βT d to find a continuous function f β : T A → B M with f β (cid:12)(cid:12) T = f .Then, by Tietze’s extension theorem, f β can be extended to a continuousfunction f A : G A → B M , the restriction f A (cid:12)(cid:12) G is then the desired extension.To prove (iv), let T be an approximable A ( G )-interpolation set. First, wefind, using (iii), f ∈ A with f (cid:12)(cid:12) T = h and k f k ∞ = k h k T . The definition ofapproximable A ( G )-interpolation sets provides two neighbourhoods V , V with V ⊆ V ⊆ U and f ∈ A such that f ( V T ) = { } and f ( G \ V T ) = { } . Using [21, 3.2.20], we can assume (taking the minimum of f and the functionthat is constant and equal to 1) that k f k ∞ = 1. The product f · f thencoincides with h on T and vanishes off V T . (cid:3) Remark 2.4.
Note that if in the lemma above A ( G ) ⊆ CB ( G ), then T isnecessarily discrete since every bounded function on T must be continuous.Observe as well that the sole existence of an infinite A ( G )-interpolationset T in G , implies that G A contains a copy of βT d , where T d is the discreteset T . The compactification G A is therefore large and topologically involved.The following theorem, due to Chou [13], has its roots in a result ofRamirez (see Theorem 2.3 of [19]) in the Abelian setting. This theorem isused by Chou, loc. cit., to find an isometric copy of ℓ ∞ inside WAP ( G ) / B ( G )for a discrete group G. This was originally the departing point of our paper.
Theorem 2.5. (Chou, [13, Lemma 3.11] ) Let G be a discrete group. Asubset T ⊆ G fails to be a B ( G ) -interpolation set if and only if there is abounded function f ∈ ℓ ∞ ( G ) , with k f k ∞ = 1 such that f ( G \ T ) = { } and k φ − f k T ≥ for all φ ∈ B ( G ) . Remark 2.6.
It is an immediate consequence of the previous theorem that B ( G )-interpolation sets are also B ( G )-interpolation sets (i.e., Sidon sets).We do not know whether Theorem 2.5 remains valid for all locally compactgroups.The result in Theorem 2.5 is more natural when the function algebra isa C ∗ -subalgebra. It is not surprising therefore that it holds for any C ∗ -subalgebra. Next lemma proves even more. Lemma 2.7.
Let G be a topological group, A ( G ) ⊆ A ( G ) ⊆ ℓ ∞ ( G ) betwo C ∗ -subalgebras with ∈ A ( G ) , and let ( T η ) η<κ be a family of disjointsubsets of G such that (i) each T η fails to be an A ( G ) -interpolation set, (ii) T = S η<κ T η is an approximable A ( G ) -interpolation set.Then for each open neighbourhood U of e , there is a function f ∈ A ( G ) with k f k ∞ = 1 such that f ( G \ U T ) = { } and k f − φ k T η ≥ for every η < κ and every φ ∈ A ( G ) . Proof.
Let T = S η<κ T η be an approximable A ( G )-interpolation set asstated in the lemma. Let U be an open neighbourhood of e .To avoid cumbersomeness, we abuse our notation and use the same lettersto denote subsets of T and their images in G A .Then, by Statement (ii) of Lemma 2.3, each T η must contain two disjointsubsets T ,η , T ,η such that T ,η A ∩ T ,η A = ∅ . Define for each η < κ, afunction h η : G → [ − ,
1] supported on T η with h η ( T ,η ) = { } and h η ( T ,η ) = {− } . Then consider the function h : G → [ − ,
1] supported on T and given by h ( t ) = h η ( t ) if t ∈ T η for some η < κ. By Statement (iv) of Lemma 2.3, there is a a function f ∈ A ( G ) such that f ( G \ U T ) = 0 , f (cid:12)(cid:12) T = h and k f k ∞ = k h k T = 1 . Let now φ be any function in A ( G ) , and take ε >
0. Given η < κ , we aregoing to prove that k f − φ k T η ≥ − ε .Take p η ∈ T ,η A ∩ T ,η A and pick t ,η ∈ T ,η and t ,η ∈ T ,η with | φ ( t ,η ) − φ A ( p η ) | < ε and | φ ( t ,η ) − φ A ( p η ) | < ε, where φ A denotes the extension of φ to G A . Then2 = | h η ( t ,η ) − h η ( t ,η ) | = | h ( t ,η ) − h ( t ,η ) | = | f ( t ,η ) − f ( t ,η ) | (2) ≤ | f ( t ,η ) − φ ( t ,η ) | + | φ ( t ,η ) − φ A ( p η ) | + | φ A ( p η ) − φ ( t ,η ) | + | φ ( t ,η ) − f ( t ,η ) | . It follows that either | f ( t ,η ) − φ ( t ,η ) | ≥ − ε or | f ( t ,η ) − φ ( t ,η ) | ≥ − ε .Since ε > k f − φ k T η ≥
1. Since k f k ∞ = 1 and f ( G \ U T ) = { } , we see that f is the required function. (cid:3) For the main theorem in this section, we need to recall the followingdefinitions. These sets are also essential for the rest of the paper.
NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 11
Definition 2.8.
Let G be a topological group, T be a subset of G and U be a neighbourhood of e . We say that T is right U -uniformly discrete if U s ∩ U s ′ = ∅ for every s = s ′ ∈ T. The set T being left U -uniformly discrete is defined analogously. We say that T is right uniformly discrete (resp. left uniformly discrete ) when it is right U -uniformly discrete (resp. left U -uniformly discrete) for some neighbourhood U of e. If T is both left and right uniformly discrete, we say that T isuniformly discrete. Lemma 2.9.
Let G be a locally compact group, A ( G ) be a C ∗ -subalgebra of CB ( G ) , U be a compact neighbourhood of e , and T ⊆ G be an approximable A ( G ) -interpolation set that can be partitioned as T = S η<κ T η with U T η ∩ U T η ′ = ∅ whenever η = η ′ . Then there is a compact neighbourhood V ofthe identity with V ⊆ U such that given any two functions f, g ∈ ℓ ∞ ( G ) supported in V T and a function c = ( c η ) η<κ ∈ ℓ ∞ ( κ ) such that (3) f (cid:12)(cid:12) V T η = c ( η ) g (cid:12)(cid:12) V T η for each η < κ, one has that: (i) If g is continuous, then so is f . (ii) If T is right U -uniformly discrete, A ( G ) is an admissible C ∗ -subalgebraof LUC ( G ) , then g ∈ A ( G ) , implies f ∈ A ( G ) .Proof. First, consider the two neighbourhoods V and V provided by thedefinition of approximable A ( G )-interpolation sets for the neighbourhood U . We take V as V , and we can obviously assume that V ⊆ U .That f is well defined follows from the relation U T η ∩ U T η ′ = ∅ .Let g ∈ CB ( G ) and f ∈ ℓ ∞ ( G ) be functions with f ( G \ V T ) = g ( G \ V T ) = { } , related as in (3). We prove that f is continuous considering separatelycontinuity at interior points of V T and points that do not belong to theinterior of
V T .Let s ∈ G that is not an interior point of V T . Since s can be approachedfrom G \ V T , we see by continuity that g ( s ) = 0. By checking the caseswhen s ∈ V T and when s / ∈ V T , we deduce that f ( s ) = 0 as well. Now let( x α ) be a net in G converging to s . We can assume that either ( x α ) ⊂ V T or ( x α ) ⊂ G \ V T . In the former case, we use the fact that | f ( x α ) | ≤k c k ∞ · | g ( x α ) | and conclude that lim α f ( x α ) = 0 = f ( s ). In the other casewhen (( x α ) ⊂ G \ V T ), it is clear that lim α f ( x α ) = 0 = f ( s ). The continuityof f at s follows.Suppose now that s is an interior point of V T . Pick η < κ such that s = vt with v ∈ V and t ∈ T η , and let W be a neighbourhood of the identitywith W s ⊆ V T with | g ( s ) − g ( s ′ ) | < ǫ for every s ′ ∈ W s.
Let w ∈ W besuch that s ′ = wvt and notice that s ′ = ws = wvt = v t for some v ∈ V and t ∈ T implies that t ∈ T η . Therefore, | f ( s ) − f ( s ′ ) | = | c ( η ) g ( s ) − c ( η ) g ( s ′ ) | for every s ′ ∈ W s, and so the continuity of f at interior points follows as well.We now assume A ⊆ LUC ( G ). Define a function ϕ on T by ϕ ( t ) = c ( η )for every t ∈ T η . Since T is an A ( G )-interpolation set, we may extend ϕ toa function ϕ ∈ A ( G ) . By Lemma 2.3 (iv), we can assume that ϕ ( G \ V T ) = { } . If g A and ϕ A denote the respective extensions of g and ϕ to G A , wedefine f ∗ : G A → C by f ∗ ( vp ) = ϕ A ( p ) · g A ( vp ) if v ∈ V and p ∈ T A (4) f ∗ ( x ) = 0 if x / ∈ V T A .We check that f ∗ is a well-defined, continuous extension of f to G A . (1) f ∗ is well defined. It might happen that some vp ∈ V T A admits twodifferent decompositions. We check that the definition of f ∗ does not dependof the choice of the decomposition. Suppose therefore that v p = v p with v , v ∈ V and p , p ∈ T A .If p = p , we may choose T , T ⊂ T such that T A ∩ T A = ∅ , p ∈ T A and p ∈ T A (this is possible by (ii) of Lemma 2.3). Since T is approximable, we maypick h ∈ A ( G ) such that h ( V T ) = { } and h ( G \ V T ) = { } . Recalling that multiplication by elements of G is continuous on G A , itis clear that h A ( v p ) = 1. By the same reason, if ( t α ) is a net in T converging to p , we have that v p = lim α v t α . But since T is right U -uniformly discrete, no element v t α can be in V T , hence h A ( v p ) must bezero. This contradiction shows that(5) v p = v p implies p = p . This shows already that f ∗ is well defined, since the equalities v p = v p and p = p give us f ∗ ( v p ) = ϕ A ( p ) g A ( v p ) = f ∗ ( v p ) . (In fact, v and v must be also equal by the same argument, but this isenough for our purposes.) (2) f ∗ coincides with f on G . Since
V T A ∩ G = V ( T A ∩ G ) = V T and f ( G \ V T ) = { } , we readily see that f and f ∗ coincide on G \ V T . Let onthe other hand s = vt with v ∈ V and t ∈ T η . Then f ∗ ( s ) = ϕ ( t ) g ( s ) = c ( η ) g ( s ) = f ( s ) . (3) f ∗ is continuous. Using the joint continuity property, we see that
V T A is closed in G A . So the continuity of f ∗ at the points outside of V T A is clear.We divide the case x = vp ∈ V T A into two subcases. Suppose first that x is an interior point V T A , and let ( q α ) be a net in G LUC converging to x .Then ( q α ) is eventually of the form ( v α p α ) with ( v α ) in V and ( p α ) in T A . By taking subnets if necessary, we may assume that lim α v α = v in V andlim α p α = p in T A . Accordingly, x = vp = v p , and applyig 5, we see that p = p . Therefore, f ∗ ( q α ) = ϕ A ( p α ) g A ( q α ) −→ ϕ A ( p ) g A ( x ) = ϕ A ( p ) g A ( x ) = f ∗ ( x ) , NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 13 as required. The second subcase is when x is outside the interior of V T A . Here, we may assume that the net ( q α ) given to converge to x is also outside V T A , and so g A ( q α ) = 0 for every α . Since g A is continuous, we deduce that f ∗ ( x ) = ϕ A ( p ) g A ( x ) = 0 , as required.From (1), (2), (3) we conclude that f ∈ A . (cid:3) Remarks 2.10. (i) A known theorem due to Veech asserts that the leftaction of a locally compact group G on G LUC is free, i.e., gx = x for every x ∈ G LUC and g ∈ G, g = e , see [52], or [46] for a shorter proof. The proofof the previous Lemma reveals that Veech’s property in fact holds in G A atany point in the closure of the approximable A ( G )-interpolation sets with A ⊂ LUC ( G ). That is, if T is any such a set, x ∈ T A and g = e in G , then gx = x and xg = x in G A . This property was proved in G WAP in [4] and [23]using t -sets. t -Sets are by [24] approximable WAP ( G )-interpolation sets.We will return to these matters in a forthcoming work.(ii) It could also be worth to mention that for metrizable locally compactgroups the condition on T in (ii) is redundant. Indeed, by [24, Theorem 4.9],every LUC ( G )-interpolation subset of a metrizable group is right uniformlydiscrete. Theorem 2.11.
Let G be a locally compact group and let A ( G ) ⊂ A ( G ) ⊆ LUC ( G ) be two unital C ∗ -subalgebras of ℓ ∞ ( G ) with A ( G ) admissible. Let,in addition, U be a compact neighbourhood of the identity such that T is right U -uniformly discrete. Suppose that G contains a family of sets { T η : η < κ } such that (i) T η ∩ T η ′ = ∅ for every η = η ′ < κ , (ii) T η fails to be an A ( G ) -interpolation set for every η < κ , and (iii) T = S η<κ T η is an approximable A ( G ) -interpolation set.Then there is a linear isometry Ψ : ℓ ∞ ( κ ) → A ( G ) / A ( G ) .Proof. Let V be the neighbourhood of the identity provided by Lemma 2.9.Since T = S η<κ T η is an approximable A ( G )-interpolation set and each T η fails to be an A ( G )-interpolation set, we take from Lemma 2.7 a function f ∈ A ( G ) with k f k ∞ = 1 such that f ( G \ V T ) = { } and k f − φ k T η ≥ φ ∈ A ( G ) and η < κ. For each c = ( c η ) η<κ ∈ ℓ ∞ ( κ ), we define the function f c : G → C supportedin V T by f c ( vt ) = c η f ( vt ) if t ∈ T η and η < κ, i.e., with the notation of Lemma 2.9, f c (cid:12)(cid:12) V T η = c ( η ) f (cid:12)(cid:12) V T η .Then f c ∈ A ( G ) by (ii) of Lemma 2.9. Obviously, the map Ψ : ℓ ∞ ( κ ) → A ( G ) / A ( G ) given byΨ( c ) = f c + A ( G ) for every c ∈ ℓ ∞ ( κ )is linear. We next check that it is isometric. The same argument of [13, Theorem 3.12] shows now that, for every η < κ, k Ψ (( c η ) η<κ ) k A ( G ) / A ( G ) = inf {k f c − φ k ∞ : φ ∈ A ( G ) }≥ inf {k f c − φ k T η : φ ∈ A ( G ) } = inf {k c η f − φ k T η : φ ∈ A ( G ) } = | c η | inf {k f − φ k T η : φ ∈ A ( G ) }≥ | c η | , where the last inequality follows from the choice of f . Since, obviously, k Ψ( c ) k A ( G ) / A ( G ) ≤ k f c k ∞ = k c k , for every c = ( c η ) η<κ ∈ ℓ ∞ ( κ ) , we see that Ψ is the required isometry. (cid:3) Corollary 2.12.
If in the above theorem A ( G ) = CB ( G ) and T is notassumed to be right U -uniformly discrete but still U T η ∩ U T η ′ = ∅ , then thequotient CB ( G ) / A ( G ) contains a linearly isometric copy of ℓ ∞ ( κ ) .Proof. The proof of Theorem 2.11 remains valid in this case applying (i) ofLemma 2.9 instead of (ii). (cid:3)
Remark 2.13.
Two C ∗ -subalgebras of ℓ ∞ ( G ) may be different, and yet pro-duce a small quotient (i.e., separable), for example if G is a minimally weaklyalmost periodic group ([14], [49], [53]) then WAP ( G ) / AP ( G ) = C ( G ). If G = SL (2 , R ) , then WAP ( G ) = C ( G ) ⊕ C , and so WAP ( G ) /C ( G ) = C . In the theorem and corollary above, we have just met conditions under whichthis is not so.
Corollary 2.14.
Under the hypotheses of Theorem 2.11 or Corollary 2.12,the quotient space A ( G ) / A ( G ) is non-separable. Interpolation sets
The definitions in this section gather the topological group-theoretic prop-erties that will correspond to the interpolation sets needed in the three sec-tions that follow. Once these interpolation sets are at hand, an applicationof Theorem 2.11 and Corollary 2.12 will lead immediately to the desiredconclusion on the quotients.In addition to the uniformly discrete sets defined in the previous sectionswe shall also need the following sets.
Definition 3.1.
Let G be a non-compact topological group. We say that asubset S of G is(i) right translation-finite if every infinite subset L ⊆ G contains afinite subset F such that T { b − T : b ∈ F } is finite; left translation-finite if every infinite subset L ⊆ G contains a finite subset F suchthat T { T b − : b ∈ F } is finite; and translation-finite when it is bothright and left translation-finite.(ii) right translation-compact if every non-relatively compact subset L ⊆ G contains a finite subset F such that T { b − S : b ∈ F } is relativelycompact; left translation-compact if every non-relatively compactsubset L ⊆ G contains a finite subset F such that T { Sb − : b ∈ F } NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 15 is relatively compact; and translation-compact when it is both leftand right translation-compact.(iii) a right t -set ( left t -set ) if there exists a compact subset K of G containing e such that gS ∩ S (respectively, Sg ∩ S ) is relativelycompact for every g / ∈ K ; and a t -set when it is both a right and aleft t -set.We also need to establish the range of locally compact groups to whichour methods apply in the next two sections, these are those locally com-pact groups for which the existence of a good supply of WAP -functions isguaranteed.Recall that a locally compact group G is an IN − group if it has an invari-ant neighbourhood of the identity. We recall also from [10], that a locallycompact group G is an E-group if it contains a non-relatively compact set X such that for each neighbourhood U of e, the set \ { x − U x : x ∈ X ∪ X − } is again a neighbourhood of e. The set X is called an E-set . This is a largeclass of locally compact groups. This includes of course all non-compact
SIN − groups, the groups with a non-compact centre such as the matrixgroup GL ( n, R ), and the direct product of any E -group with any locallycompact group.A detailed study of approximable LUC - and
WAP ( G )-interpolation sets,with some precise characterizations, is carried out in the recent paper [24].We summarize in Lemma 3.2 the results that will be needed in the presentpaper. Lemma 3.2. ( [24, Lemma 4.8 and Proposition 3.3 (iii)] ) Let G be a topo-logical group and let T ⊂ G . (i) If the underlying topological space of G is normal, then all discreteclosed subsets of G are approximable CB ( G ) -interpolation sets. (ii) If T is right (resp. left) uniformly discrete , then T is an approx-imable LUC ( G ) -interpolation set (resp. RUC ( G ) -interpolation set). (iii) If G is assumed to be metrizable, then every LUC ( G ) -interpolationset (resp. RUC ( G ) -interpolation set) is right (left) uniformly dis-crete. (iv) If G is an E -group and T is an E -set in G which is right (or left)uniformly discrete with respect to U for some neighbourhood U of the identity such that U T is translation-compact, then T is anapproximable WAP ( G ) -interpolation set. (v) If G is a metrizable E -group, T ⊂ G is a WAP ( G ) -interpolation setif and only if U T is translation-compact for some compact neigh-bourhood U of the identity such that T is right (or left) uniformlydiscrete with respect to U . The following Lemma will be needed later on in Section 5.
Lemma 3.3.
Let G be a locally compact group, let H be a closed subgroupof G and let T ⊂ H . (i) If U T is a right t -set in H for some compact neighbourhood U ofthe identity e in H , then there is a compact neighbourhood V of e in G such that V T is a right t -set in G . (ii) If in addition T is central, then the left analogue of Statment (i)holds also.Proof. Let U be a compact neighbourhood of e in H , and suppose that U T is a right t-set in H . By definition there is a compact subset K ⊆ H suchthat gU T ∩ U T is relatively compact whenever g / ∈ K . Let V be a compactsymmetric neighbourhood of the identity in G such that V ∩ H = U and let K ′ = V KV .Let g ∈ G but g / ∈ K ′ , and consider a net ( g α ) in gV T ∩ V T with noconvergent subnet. Then, for each α , there are v α , w α ∈ V and t α , s α ∈ T such that g α = v α t α = gw α s α , and so t α s − α = v − α gw α for every α. Note that neither of the nets ( s α ) and ( t α ) has a convergent subnet since V is compact. We can assume that ( v α ) and ( w α ) converge to v, w ∈ V ,respectively. Therefore ( t α s − α ) is a net in H which converges to h = v − gw .Since H is closed, h ∈ H . Therefore, the net ( t α s − α ) is eventually in ( hV ) ∩ H = h ( V ∩ H ) = hU . This means that the net ( t α ) may be seen in hU T ∩ U T ,and therefore hU T ∩ U T is not relatively compact. But that would implythat h ∈ K , and so g ∈ K ′ , whence a contradiction. Thus, V T is a right t -set in G. To prove the analogous statement for left t -sets, we suppose in additionthat T is central and put again K ′ = V KV.
Using the fact that T is central,the same argument leads to s − α t α = w α gv − α for every α. By taking subnets, we see that ( t α ) is eventually in T U h ∩ T U = U T h ∩ U T with h ∈ V gV . Thus, h must in K, and so g is in K ′ . (cid:3) To state a well-known necessary condition for a subset T ⊆ G to be a B ( G )-interpolation set we need the concept of large squares that we recallfrom [13, Definition 3.3]:A finite subset F of G is an n -square if F = AB where | A | = | B | = n and | F | = n . A subset of a group is then said to contain large squares if itcontains an n -square for every n ∈ N .Large squares are incompatible with Sidon sets, as proved in [13, Propo-sition 3.4]. We restate here this theorem, stressing on B ( G ). Theorem 3.4. (Chou, [13] ) Let G be a topological group. A B ( G ) -interpolationset cannot contain large squares.Proof. Let G d be the group G with the discrete topology. If T is a B ( G )-interpolation set, then T is also a B ( G d )-interpolation set. By Remark 2.6, T is then a B ( G d )-interpolation set and, by [13, Proposition 3.4], T cannotcontain large squares. (cid:3) NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 17 The quotients of
WAP ( G ) by C ( G ) and WAP ( G ) by AP ( G ) ⊕ C ( G )In [10], Chou considered E -groups and proved that the quotient space WAP ( G ) /C ( G ) contains a linear isometric copy of ℓ ∞ . In this section,we strengthen this result and prove that if G is an E -group, then there isa linear isometric copy of ℓ ∞ ( κ ) in the quotient WAP ( G ) /C ( G ) where κ is the compact covering number of an E -set contained in G. In particular, κ = κ ( G ) when G is an SIN -group.Our method applies further to show that the quotient
WAP ( G ) / AP ( G ) ⊕ C ( G ) is non-separable. Theorem 4.1.
Let G be a non-compact locally compact E -group havingan E -set X with a compact covering number κ . Then the quotient space WAP ( G ) /C ( G ) contains a linear isometric copy of ℓ ∞ ( κ ) .Proof. Let V be a fixed compact symmetric neighborhood of e . Then weconsider a set T ⊂ X as that constructed in Section 2 of [23]. This set hasthe following properties:(i) κ ( T ) = | T | = κ ( X ).(ii) T is right V -uniformly discrete.(iii) V T is a t -set.For completeness, we recall from [23] the construction of the set T since thisshall be needed in the proof. We may assume that e ∈ X and start with x = e. . Suppose that the elements x β have been selected for all β < α with α < κ . Set X α = [ β ,β ,β <α V x ǫ β x ǫ β V x ǫ β V, where each ǫ i = ± . Since κ ( X α ) < κ, we pick x α in X \ X α for our set T .In this way, we form a set T = { x α : α < κ } .We obtain from Lemma 3.2 that T is an approximable WAP ( G )-interpolationset. Since every infinite subset of T is uniformly discrete and C ( G )-interpolationsets must be relatively compact, and so finite (see the proof of Proposition3.3 of [24]) any decomposition T = S η<κ T η as a disjoint union of κ -manyinfinite subsets leaves us in position to apply Theorem 2.11 and finish theproof. (cid:3) We deduce first that the quotient
WAP ( G ) / ( AP ( G ) ⊕ C ( G )) containsan isomorphic copy of ℓ ∞ ( κ ). Corollary 4.2.
Let G be a non-compact locally compact E -group havingan E -set X with a compact covering number κ . Then the quotient space WAP ( G ) / ( AP ( G ) ⊕ C ( G )) contains an isomorphic copy of ℓ ∞ ( κ ) .Proof. We only have to recall that
WAP ( G ) = AP ( G ) ⊕ WAP ( G ), [42].Therefore WAP ( G ) / ( AP ( G ) ⊕ C ( G )) is isomorphic to WAP ( G ) /C ( G ),and apply then Theorem 4.1. (cid:3) If we want to use our Theorem 2.11 to obtain a linear isometric copyof ℓ ∞ ( κ ) in the quotient WAP ( G ) / AP ( G ) we need first an approximable WAP ( G )-interpolation set which is not an AP ( G )-interpolation set. If G , forinstance, is discrete this means we need a translation-finite set that is not an I -set. Such sets can be easily found in Z , the additive group of integers: T = { n + n : n ∈ N } ∪ { n : n ∈ N } is such an example, see [30, Example 1.5.2]for a (simple) proof. For arbitrary discrete groups, an example as simple asthat has escaped to us. A considerably more complicated construction canbe used to obtain an approximable WAP ( G )-interpolation set that is not a B ( G )-interpolation set, a fortiori not an AP ( G )-interpolation set when G is an IN-group, a nilpotent group or a group with large enough centre, seeSection 5 (note that the set T above is a Sidon set, i.e., a B ( G )-interpolationset).We present however, on the lines of Theorem 2.11, an ad-hoc constructionof a linear isomorphism of ℓ ∞ ( κ ) into WAP ( G ) / ( AP ( G ) ⊕ C ( G )) with normat most one whose inverse has norm at most 2. A detailed look at theproof reveals that it is actually based in finding an approximable WAP ( G )-interpolation set that is not an approximable AP ( G )-interpolation set. Whatmakes this construction different from our general approach is that this setcould even be an AP ( G )-interpolation set. Recall that in a non-compactlocally compact group no AP ( G )-interpolation set is approximable. Theorem 4.3.
Let G be a non-compact, locally compact E-group havingan E-set X with a compact covering number κ . Then the Banach space WAP ( G ) contains a linear isometric copy L of ℓ ∞ ( κ ) such that k f k ≤ k f + AP ( G ) ⊕ C ( G ) k q ≤ k f k for every f ∈ L. In particular, the quotient space
WAP ( G ) / ( AP ( G ) ⊕ C ( G )) is non-separable.Proof. Let V be a fixed compact symmetric neighbourhood of e in G , T be the approximable WAP ( G )-interpolation set used in Theorem 4.1 and { T η : η < κ } be any partition of T into κ -many infinite subsets.We begin with a function f ∈ WAP ( G ) with f ( G \ V T ) = { } and f ( T ) = { } . Consider the function f c ∈ WAP ( G ), supported in V T , definedin Theorem 2.11. Clearly, the map c f c : ℓ ∞ ( κ ) → WAP ( G )is a linear isometry, so we only need to check that the quotient mapΨ : ℓ ∞ ( κ ) → WAP ( G ) / ( AP ( G ) ⊕ C ( G )satisfies k f c + AP ( G ) ⊕ C ( G ) k q ≥ k c k for every c ∈ ℓ ∞ ( κ ). Without lossof generality, we may assume that k c k = 1 . We claim that k f c + g + h k ≥ g + h ∈ AP ( G ) ⊕ C ( G ) . Suppose, otherwise, that k f c + g + h k < g + h ∈ AP ( G ) ⊕ C ( G ) , pick ǫ > k f c + g + h k < − ǫ. Then, in particular, | f c ( x ) + g ( x ) + h ( x ) | < − ǫ for every x ∈ T. Fix η < κ such that 1 − ǫ/ < | c η | . Since h ∈ C ( G ) , we may fix as well x ∈ T η such that | h ( x ) | < ǫ/ . Let( x n ) n ≥ be any sequence in T η with x ǫ = x n for every n ∈ N , ǫ = ± xx − n x m = vx β ∈ V T for some β < κ and v ∈ V. By the definitionof T , this is possible for at most two m ′ s : x β = x ǫ and so x m = x n x − vx ǫ , or x β = x ǫn and so x m = x n x − vx ǫn . In other words, for every fixed n ∈ N , thereexists at most two m for which xx − n x m ∈ V T.
Therefore, for every fixed n , NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 19 f η ( xx − n x m ) = 0 for every m except maybe for these two m ′ s . Moreover,since for each n , the set { xx − n x m : m ∈ N } is not relatively compact, wemay choose m such that | h ( xx − n x m ) | < ǫ/ g ∈ AP ( G ), by taking subsquences if necessary, we may fix n ∈ N such that that k r x n g − r x m g k < ǫ/ n, m ≥ n ;and so, in particular, | g ( x ) − g ( xx − n x m ) | = | g ( xx − n x n ) − g ( xx − n x m ) | = | r x n g ( xx − n ) − r x m g ( xx − n ) |≤ k r x n g − r x m g k < ǫ/ . Therefore, for every fixed n = n and m ≥ n chosen suitably, we have2 − ǫ < | c η | = | c η f η ( x ) − c η f η ( xx − n x m ) | == | f c ( x ) − f c ( xx − n x m ) + g ( x ) − g ( xx − n x m ) + h ( x ) − h ( xx − n x m ) − g ( x ) + g ( xx − n x m ) − h ( x ) + h ( xx − n x m ) |≤ | f c ( x ) + g ( x ) + h ( x ) | + | f c ( xx − n x m ) + g ( xx − n x m ) + h ( xx − n x m ) | + | g ( x ) − g ( xx − n x m ) | + | h ( x ) − h ( xx − n x m ) |≤ k f c + g + h k + k f c + g + h k + k r x n g − r x m g k + | h ( x ) | + | h ( xx − n x m ) |≤ (1 − ǫ ) + (1 − ǫ ) + ǫ/ ǫ/ − ǫ. This is clearly absurd, so we must have k f c + g + h k ≥ (cid:3) Both theorems in this section will be considerably improved in the nextsection when G is an IN -group or a nilpotent group.5. The quotient of
WAP ( G ) by B ( G )The situation is much more delicate with WAP ( G ) / B ( G ). Already inthe cases dealt with by Rudin in [49] and by Ramirez in [47] proving that B ( G ) ( WAP ( G ) the arguments were quite involved. Elaborating on thework by Rudin and Ramirez, Chou proved in [13] that the quotient space WAP ( G ) / B ( G ) contains a linear isometric copy of ℓ ∞ whenever G is a non-compact, locally compact, IN -group or a nilpotent group. In all these papersthe key argument consists in constructing a t -set that contains large squares.We follow here that thread and find copies of ℓ ∞ ( κ ) for κ as large as possiblein WAP ( G ) / B ( G ) by applying Theorem 2.11.More precisely, we shall strengthen Chou’s theorems by showing thatthere is a copy of ℓ ∞ ( κ ) in the quotient WAP ( G ) / B ( G ) when G is either anIN-group or a nilpotent group and κ = κ ( G ), and that, in general, for everylocally compact group a copy of ℓ ∞ ( κ ( Z ( G ))) can be found in the quotient WAP ( G ) / B ( G ) . The following technical lemma establishes that a group cannot be coveredby β -cosets of finitely many different subgroups of index larger than β . Thisis similar to a theorem, known at least from the times of [45], in which onlyfinitely many cosets are allowed. Lemma 5.1.
Let G be any group with | G | = κ . Suppose that there is afinite collection { H , . . . , H n } of subgroups of G such that G can be coveredby β < κ right-cosets of them, i.e., such that (6) G = n [ j =1 [ i ∈ I j H j x i,j , with | I | + · · · + | I n | = β < κ. Then some of the subgroups H j has index at most β .Proof. This is proved by induction on n . The theorem is obvious if n = 1.Assume the theorem has been proved for unions of cosets of n − ∗ ) G = n [ j =1 [ i ∈ I j H j x i,j , with | I | + · · · + | I n | = β < κ. If | G : H n | > β , there is x ∈ G such that x / ∈ S i ∈ I n H n x i,n . Since H n x ∩ H n x i,n = ∅ for every i ∈ I n , we obtain H n x ⊆ n − [ j =1 [ i ∈ I j H j x i,j , and so( ∗∗ ) H n = n − [ j =1 [ i ∈ I j H j x i,j x − ∩ H n = n − [ j =1 [ i ∈ I j ( H j ∩ H n ) y i,j , where the y i,j ’s have been suitably chosen in H n (if h j x i,j x − = y i,j is in H j x i,j x − ∩ H n , then x i,j x − = h − j y i,j ). Applying our inductive hypothesis,we deduce that there is j with | H n : ( H n ∩ H j ) | ≤ β . (One may alsoproceed directly and replace H n from ( ∗∗ ) in ( ∗ ), then apply the inductivehypothesis).There is therefore a family { z s : s ∈ S } ⊂ H n with | S | ≤ β such that H n = S s ∈ S ( H n ∩ H j ) z s and we may replace (6) by G = n − [ j =1 [ i ∈ I j H j x i,j [ [ i ∈ I n [ s ∈ S H j z s x i,n ! . Since this is a cover of G by cosets of at most n − G we deduce from our inductive hypothesis that some of the subgroups H j ,1 ≤ j ≤ n −
1, has index at most β . (cid:3) Lemma 5.2.
Let G be a locally compact group containing a normal subgroup H ⊂ G . If | G : H | = κ ≥ ω , then G contains a family { T η : η < κ } of subsetssuch that, putting T = S η<κ T η , (i) Ht ∩ Ht ′ = ∅ for every t = t ′ ∈ T . (ii) T η contains large squares for every η < κ . (iii) If U ⊂ H is compact, then U T g ∩ U T and gU T ∩ U T are relativelycompact relatively compact for every g / ∈ t − U t with t ∈ T . NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 21
Proof.
For each η < κ , let in this proof C η ( H ) denote the set C η ( H ) = { g ∈ G : | Cl ( gH ) | ≤ η } , where Cl ( gH ) denotes the conjugacy class of gH in G/H . If A ⊆ G , Cl ( A )will stand for the set { g − ag : g ∈ G, a ∈ A } . For n < ω , we define h A i n tobe the set of all products of at most n elements in A ∪ A − .We define for each η < κ and n < ω two collections of finite sets C η,n = { x η,n,k : 1 ≤ k ≤ n } and D η,n = { y η,n,k : 1 ≤ k ≤ n } . These sets are defined recursively. First we order the set κ × ω in thecanonical way ([37, 3.12]): For η, η ′ < κ and n, n ′ < ω , we define ( η, n ) < ( η ′ , n ′ ) if either max( η, n ) < max( η ′ , n ′ ), or max( η, n ) = max( η ′ , n ′ ) and η < η ′ , or max( η, n ) = max( η ′ , n ′ ), η = η ′ and n < n ′ . This way, the cardinalof the set { ( η, n ) : ( η, n ) < ( η , n ) } is less than κ for every ( η , n ) ∈ κ × ω .We now define x , , = y , , = e and set C , = { x , , } , D , = { y , , } .Assume that the sets C γ,j and D γ,j and the elements x η,n,l have beendefined for ( γ, j ) < ( η, n ) and 1 ≤ l < k . We describe how to define x η,n,k .Define first the following subsets of G/H by R η,n,k = [ ( γ,j ) < ( η,n ) ( C γ,j H ∪ D γ,j H ) [ k − [ l =1 x η,n,l H ! and S η,n,k = [ ( γ,j ) ≤ ( η,n ) C γ,j H [ [ ( γ,j ) < ( η,n ) D β,i H [ k − [ l =1 y η,n,l H ! . Consider a cardinal number f ( η, n, k ) such that κ > f ( η, n, k ) > n |h R η,n,k i | + |h S η,n,k i | . Note that, in particular, f ( η, n, k ) is finite when κ = ω .We now choose x η,n,k such that(7) x η,n,k H / ∈ D R η,n,k , Cl (cid:0) R η,n,k ∩ C f ( η,n,k ) (cid:1) E . Finding this element is possible because |h R η,n,k i H/H | ≤ f ( η, n, k ) < κ = | G : H | and conjugacy classes of elements of C f ( η,n,k ) do not have, by defini-tion, more than f ( η, n, k )-elements.Once the elements in C η,n are defined in this way, we define the elementsin D η,n . If y η,n, , . . . , y η,n,k − have already been defined, we use the followingClaim to define y η,n,k . Claim 1:
There is y η,n,k ∈ G such that (8) y η,n,k H / ∈ h S η,n,k i and (9) C η,n C − η,n \ y η,n,k R η,n,k y − η,n,k H = { e } . We first enumerate ( C η,n C − η,n \ { e } ) T Cl ( R η,n,k ) H as { a , . . . , a l } . Letthen R j = { r ∈ R η,n,k : rH ∈ Cl ( a j H ) } and choose for each j , 1 ≤ j ≤ l , and each r ∈ R j an element y j,r ∈ G andan element h j,r ∈ H with r = y − j,r a j h j,r y j,r . Suppose now that no y ∈ G can be found so that conditions (8) and (9)are satisfied. In that case some R j must be non-empty and, indeed, G = h S η,n,k i [ l [ j =1 [ r ∈ R j [ h ∈ H L j,r,h , where L j,r,h = (cid:8) g ∈ G : r = g − a j hg (cid:9) . Observe now that that L j,r,h H ⊆ C G/H ( a j H ) y j,r H , where C G/H ( a j H ) = { gH ∈ G/H : ga j H = a j gH } is the centralizer of a j H . Therefore,(10) G/H = h S η,n,k i [ l [ j =1 [ r ∈ R j C G/H ( a j H ) y j,r H . If the elements of the set h S η,n,k i are viewed as cosets of the trivial subgroup { e G/H } , we find G/H as a union of less than n | R η,n,k | + |h S η,n,k i | cosets.Since the latter number is less than f ( η, n, k ) some of them must correspondto a subgroup of index at most f ( η, n, k ), by Lemma 5.1. Thus, there is j ,1 ≤ j ≤ n such that | G/H : C G/H ( a j H ) | = | Cl ( a j H ) | < f ( η, n, k ). Weconclude that a j ∈ C f ( η,n,k ) .Since a j H ∩ Cl ( R η,n,k ) = ∅ , we find that a j ∈ Cl ( R n,η,k ∩ C f ( η,n,k ) ) H . If a j = x − η,n,k x η,n,k and k > k , this goes against condition (7) in the choiceof x η,n,k and finishes the proof of the claim.We have therefore constructed two families C η,n = { x η,n,k : 1 ≤ k ≤ n } and D η,n = { y η,n,k : 1 ≤ k ≤ n } , ( η, n ) ∈ κ × ω, with properties (7), (8) and (9). We now check that the sets T η = [ n<ω ( D η,n C η,n ) , η < κ satisfy the desired properties.First of all we see that | D η,n C η,n | = n . If this were not the case, therewould be 1 ≤ k , k , k , k ≤ n with k = k and k < k such that y η,n,k x η,n,k = y η,n,k x η,n,k . But then y η,n,k ∈ h C η,n , y η,n,k i which goesagainst our choice of the elements in D η,n . Therefore, T η contains n -squaresfor every n .In order to prove the last statement, we take U a compact subset of H and g / ∈ U . Choose t = y η ,n ,k x η ,n ,k ′ and t = y η ,n ,k x η ,n ,k ′ , u , u ∈ U ,with gu t = u t ∈ gU T ∩ U T.
We order the 3-tuples ( η i , n i , k i ) lexicographically with respect to the lastentry, that is ( η i , n i , k i ) > ( η ′ i , n ′ i , k ′ i ) if either ( η i , n i ) > ( η ′ i , n ′ i ) or ( η i , n i ) =( η ′ i , n ′ i ) and k i > k ′ i . NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 23
Assume that ( η , n ) ≥ ( η , n ).Let now gu t = u t with t = y η ,n ,k x η ,n ,k ′ and t = y η ,n ,k x η ,n ,k ′ , u , u ∈ U be any other element of gU T ∩ U T .Then g = u y η ,n ,k x η ,n ,k ′ x − η ,n ,k ′ y − η ,n ,k u − = u y η ,n ,k x η ,n ,k ′ x − η ,n ,k ′ y − η ,n ,k u − . Let ( η i , n i , k i ) = max { ( η i , n i , k i ) : 1 ≤ i ≤ } . There must then be i ,1 ≤ i ≤ i = i , with η i = η i , n i = n i and y η i ,n i ,k i = y η i ,n i ,k i , forotherwise y η i ,n i ,k i ∈ S η i ,n i ,k i . Claim 2 : It is not possible that ( η , n ) = ( η , n ) > ( η , n ). Should thisbe the case, then y η ,n ,k = y η ,n ,k and x η ,n ,k ′ x − η ,n ,k ′ ∈ y − η ,n ,k (cid:16) y η ,n ,k x η ,n ,k ′ x − η ,n ,k ′ y − η ,n ,k (cid:17) y η ,n ,k H. It follows from our condition (9) in the choice of y η ,n ,k (Claim 1, page 21)that x η ,n ,k ′ = x − η ,n ,k ′ but this is only possible if g ∈ U (take into accounthat gu t = u t and that y η ,n ,k = y η ,n ,k ), and the claim is proved.The same argument shows that it is not possible that ( η , n ) = ( η , n ) > ( η i , n i ), with i = 3 , η , n ) = ( η , n ) or ( η , n ) = ( η , n ). But, sincethe element gu t = u t was chosen arbitrarily in gU T ∩ U T , it follows that gU T ∩ U T ⊆ ( gU D η ,n C η ,n ) ∪ ( U D η ,n C η ,n ) , and this is a relatively compact set.We now prove check that U T g ∩ U T is compact. Choose t = y η ,n ,k x η ,n ,k ′ and t = y η ,n ,k x η ,n ,k ′ , u , u ∈ U with u t g = u t ∈ U T g ∩ U T.
Let u y η ,n ,k x η ,n ,k ′ g = u y η ,n ,k x η ,n ,k ′ , u , u ∈ U be any other element of U T g ∩ U T with ( η , n ) ≥ ( η , n ). We have that g = t − u − u t = x − η ,n ,k ′ y − η ,n ,k u − u y η ,n ,k x η ,n ,k ′ ∈ x − η ,n ,k ′ y − η ,n ,k y η ,n ,k x η ,n ,k ′ H. As in the preceding case we assume that ( η , n ) ≥ ( η , n ) and it isenough to see that neither ( η , n ) = ( η , n ) > ( η , n ), nor ( η , n ) =( η , n ) > ( η , n ).If ( η , n ) = ( η , n ) > ( η , n ), then necessarily ( η , n , k ) = ( η , n , k )and t − u − u t ∈ x − η ,n ,k ′ x η ,n ,k ′ H . If k ′ > k ′ , then x η ,n ,k ′ ∈ R η ,n ,k ′ ,against the election of x η ,n ,k ′ . But k ′ = k ′ , implies that t = t (recallthat ( η , n , k ) = ( η , n , k )) and g ∈ t − U t . We rule out analogouslythe possibility ( η , n ) = ( η , n ) and argue as above to prove that U T g ∩ U T ⊆ ( U D η ,n C η ,n ) g ∪ ( U D η ,n C η ,n ) , and conclude that U T g ∩ U T is relatively compact. (cid:3)
Corollary 5.3.
Let G be a locally compact group, H a subgroup of G and N G ( H ) = { g ∈ G : gH = Hg } be the normalizer of H in G . If | N G ( H ) : H | = κ ≥ ω , then N G ( H ) contains a family { T η : η < κ } of subsetssuch that (i) if T = S η<κ T η , then Ht ∩ Ht ′ = ∅ for every t = t ′ ∈ T ; (ii) T η contains large squares for every η < κ ; (iii) if T = S η<κ T η and U ⊂ H is compact, then U T g ∩ U T and gU T ∩ U T are relatively compact relatively compact for every g / ∈ t − U t with t ∈ T .Proof. Since H is a normal subgroup of N G ( H ), we can apply Lemma 5.2 to N G ( H ). Statements (i) and (ii) remain the same if N G ( H ) is replaced by G .As for Statement (iii), one notices that gU T ∩ U T = ∅ and U T g ∩ U T = ∅ both imply that g ∈ N G ( H ), and so this statement follows also from Lemma5.2. (cid:3) We obtain a first consequence for groups with large center.
Theorem 5.4.
Let G be a locally compact group, Z ( G ) be the algebraiccenter of G and put κ = κ ( Z ( G )) . If κ = 1 , then there is always a linearisometry Ψ : ℓ ∞ ( κ ) → WAP ( G ) / B ( G ) .Proof. The center Z ( G ), as every locally compact Abelian group, alwayscontains an open subgroup G topologically isomorphic to R n × K , with K compact. Therefore, G must contain two subgroups H ⊂ H ⊆ Z ( G ) with H open in H and | H : H | = κ . Indeed, if | Z ( G ) : G | ≥ ω , then we take H = Z ( G ) and H = G . If G has finite index and Z ( G ) is not compact,then κ = ω and so we may take H = Z and H = { e } .Let { T η : η < κ } be the family of subsets of H provided by Lemma 5.2.By Theorem 3.4, none of them is a B ( G )-interpolation set. If T = S η<κ T η and U is a compact neighbourhood of the identity in H with U ⊂ H , then,since H is commutative, U T is a t-set in H . By Lemma 3.3, we can finda compact neighbourhood V of the identity in G , such that V T is a t-setin G . If V is chosen so that V ∩ H ⊂ H (remember H is open in H ),then T is V -uniformly discrete, and by Lemma 3.2, T is an approximable WAP ( G )-interpolation set.It suffices now to apply Theorem 2.11. (cid:3) Theorem 5.2 can be readily applied to discrete groups. To further expandits applicability we follow the usual path applying well-known structuretheorems. The following Lemma for instance is the analog of Lemma 4.4 of[13].
Lemma 5.5.
Let G be a locally compact group and let N be a closed subgroupof G . (i) If N is normal, the quotient map π : G → G/N induces linearisometries
Π :
WAP ( G/N ) / B ( G/N ) → WAP ( G ) / B ( G ) and Π : WAP ( G/N ) / B ( G/N ) → WAP ( G ) / B ( G ) . NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 25 (ii) If N is open, there are linear isometries Ψ :
WAP ( N ) / B ( N ) → WAP ( G ) / B ( G ) and Ψ : WAP ( N ) / B ( N ) → WAP ( G ) / B ( G ) . Proof.
We first prove (i). The map φ φ ◦ π clearly defines a linear isometry˜ π : WAP ( G/N ) → WAP ( G ). By [11, Theorem], we have˜ π ( B ( G/N )) = ˜ π ( CB ( G/N )) ∩ B ( G ) . By [9], we have ˜ π ( WAP ( G/N )) = ˜ π ( CB ( G/N )) ∩ WAP ( G ) . Since B ( G ) ⊆ WAP ( G ), we see that˜ π ( B ( G/N )) = ˜ π ( CB ( G/N )) ∩ WAP ( G ) ∩ B ( G ) = ˜ π ( WAP ( G/N )) ∩ B ( G )(11)so that ˜ π induces a linear isomorphismΠ : WAP ( G/N ) / B ( G/N ) → WAP ( G ) / B ( G ) , given by Π( φ + B ( G/N )) = ˜ π ( φ ) + B ( G ) . We check that Π is an isometry. If φ ∈ WAP ( G/N ), k Π( φ + B ( G/N )) k = k ˜ π ( φ ) + B ( G ) k = inf {k ˜ π ( φ ) + ψ k : ψ ∈ B ( G ) }≤ inf {k ˜ π ( φ + ψ ) k : ψ ∈ B ( G/N ) } = inf {k φ + ψ k : ψ ∈ B ( G/N ) } = k φ + B ( G/N ) k . For the reverse inequality, we follow the path of Lemma 2.3 of [12] andconsider the invariant mean µ N on WAP ( N ). For φ ∈ WAP ( G ), we definethe function φ N : G → C by φ N ( g ) = µ N ( φ g ) , where φ g ( h ) = φ ( gh ) . By invariance of µ N , the function φ N is constant on the cosets of N andtherefore induces a continuous function on G/N . Clearly, k φ N k G/N ≤ k φ k G .Now Lemma 2.3 of [12] proves in fact that φ N ∈ WAP ( G/N ). Moreover,by first considering positive-definite functions, it is also easily checked that ψ N ∈ B ( G/N ) for every ψ ∈ B ( G ).Note as well that for φ ∈ WAP ( G/N ), we have ˜ π ( φ ) N = φ .Now if φ ∈ WAP ( G/N ) and ψ ∈ B ( G ), k ˜ π ( φ ) + ψ k ≥ k (˜ π ( φ ) + ψ ) N k = k ˜ π ( φ ) N + ψ N k = k φ + ψ N k≥ k φ + B ( G/N ) k . And the remaining inequality k Π( φ + B ( G/N )) k = k ˜ π ( φ ) + B ( G ) k ≥ k φ + B ( G/N ) k follows. We prove now the analogue statements for
WAP ( G ) and B ( G ) . Wecheck first that ˜ π maps WAP ( G/N ) into
WAP ( G ) and B ( G/N ) into B ( G ). Consider the adjoint of ˜ π, this is the map given by˜ π ∗ : WAP ( G ) ∗ → WAP ( G/N ) ∗ , ˜ π ∗ ( ν ) = ν ◦ ˜ π. Note that if µ ∈ WAP ( G ) ∗ is invariant then ˜ π ∗ ( µ ) ∈ WAP ( G/N ) ∗ is invari-ant. To see this, let ¯ s = π ( s ) ∈ G/N and f ∈ WAP ( G/N ) and note that˜ π ( f ¯ s ) = (˜ π ( f )) s , and so˜ π ∗ ( µ )( f ¯ s ) = µ (˜ π ( f ¯ s ) = µ ((˜ π ( f )) s ) = µ (˜ π ( f )) = ˜ π ∗ ( µ )( f ) . Thus, ˜ π ∗ ( µ ) is the invaraint on WAP ( G/N ) . Let now f ∈ WAP ( G/N ) and µ be the invariant mean on WAP ( G ).Then ˜ π ( f ) ∈ WAP ( G ) and µ ( | ˜ π ( f ) | ) = µ ( | f ◦ π | ) = µ ( | f | ◦ π ) = ˜ π ∗ ( µ )( | f | ) = 0 . Thus, ˜ π ( f ) ∈ WAP ( G ) . To see that ˜ π ( f ) ∈ B ( G ) when f ∈ B ( G/N ), weargue in a similar way using the fact that ˜ π ( f ) ∈ B ( G ) by [12, Theorem].Accordingly, ˜ π ( B ( G/N )) ⊆ ˜ π ( WAP ( G/N )) ∩ B ( G ) . The reverse inclusion is checked as follows. If f ∈ ˜ π ( WAP ( G/N )) ∩ B ( G ) , then by (11) f is clearly in ˜ π ( B ( G/N )) . So let g ∈ B ( G/N ) with f = ˜ π ( g ) . We only need to make sure that ˜ π ∗ ( µ )( | g | ) = 0 . But this is alsoclear from the following identity.˜ π ∗ ( µ )( | g | = µ (˜ π ( | g | )) = µ ( | g | ◦ π ) = µ ( | g ◦ π | ) = µ ( | ˜ π ( g ) | ) = µ ( | f | ) . Thus, we obtain the analogue of (11)˜ π ( B ( G/N )) = ˜ π ( WAP ( G/N )) ∩ B ( G )(12)so that ˜ π induces a linear isomorphismΠ : WAP ( G/N ) / B ( G/N ) → WAP ( G ) / B ( G ) , given by Π ( φ + B ( G/N )) = ˜ π ( φ ) + B ( G ) . To check that Π is an isometry, we proceed precisely as for Π.For the proof of (ii), we associate to each φ ∈ WAP ( N ) the function(13) φ N ( g ) = φ ( g ) if g ∈ N and φ N ( g ) = 0 if g / ∈ N. Then φ N is in WAP ( G ) by [43, Lemma 5.4], [9, Theorem 3.14] or [10, Lemma2.4]. If φ happens to be in B ( N ) then φ N ∈ B ( G ) by [35, page 280] or [13,Lemma 4.1] and this obviously extends to B ( N ) and B ( G ).Define then Ψ : WAP ( N ) / B ( N ) → WAP ( G ) / B ( G ) byΨ( φ + B ( N )) = φ N + B ( G ) . It is easy to check that Ψ is a linear isometry.To prove the second statement of (ii), note that the extension of φ N defined in (13) is clearly in WAP ( G ) if φ ∈ WAP ( N ), and in B ( G ) if φ ∈ B ( N ). It is again straightforward to verify thatΨ : WAP ( N ) / B ( N ) → WAP ( G ) / B ( G ) , Ψ ( φ + B ( N )) = φ N + B ( G )is the required linear isometry. (cid:3) NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 27
We reach finally our main results.
Theorem 5.6.
Let G be a a non-compact, locally compact, IN -group andput κ = κ ( G ) . Then there is a linear isometry Ψ : ℓ ∞ ( κ ) → WAP ( G ) / B ( G ) .Proof. Let G denote the connected component of G . By Theorem 2.13 of[33], there is an open normal subgroup N of G that contains a compactnormal subgroup K with N/K
Abelian.Suppose first that κ ( N ) = κ . Then κ = κ ( N/K ). By Theorem 5.4, thereis a linear isometric copy of ℓ ∞ ( κ ) in WAP ( N/K ) / B ( N/K ). We applythen (i) and (ii) of Lemma 5.5 to obtain a linear isometric copy of ℓ ∞ ( κ ) in WAP ( G ) / B ( G ).If κ ( N ) < κ , it follows that κ = | G : N | . We apply Lemma 5.2 to thediscrete group G/N . Let { T η : η < κ } be the collection of subsets obtainedin that Lemma (in this case the subgroup H of that Lemma is trivial) and let T = S η<κ T η . By (iii) in that Lemma, the set T is a t -set (note that U = { e } in this case, hence T g ∩ T and gT ∩ T are finite if g = e ) while each of the sets T η contains large squares. Therefore, T is a WAP ( G/N )( G )-interpolationset by Lemma 3.2, while none of the sets T η is a B ( G/N )( G )-interpolationset by Theorem 3.4.By Theorem 2.11 there is an isometric embedding ℓ ∞ ( κ ) → WAP ( G/N ) / B ( G/N ) . Lemma 5.5 then provides the desired copy of ℓ ∞ ( κ ) in WAP ( G ) / B ( G ). (cid:3) Theorem 5.6 leads naturally also to an improvement of [13, Theorem 4.6].
Theorem 5.7.
Let G be a a non-compact, locally compact, nilpotent groupand put κ = κ ( G ) . Then the quotient WAP ( G ) / B ( G ) contains a linearisometric copy of ℓ ∞ ( κ ) . Proof.
The case κ ( Z ( G )) = κ ( G ) is already proved in Theorem 5.4. So wemay assume that κ ( Z ( G )) < κ ( G ). We argue by induction on the length n of the upper central series of G (the nilpotency length of G ) { e } = G ⊂ G ⊂ . . . ⊂ G n − ⊂ G n = G with Z i +1 ( G ) /Z i ( G ) = Z ( G/Z i ( G )) . If n = 1, then G is Abelian and so Theorem 5.4 or Theorem 5.6 applies.Assume as inductive hypothesis that the claim holds for groups of nilpo-tency length at most n − G has nilpotency length n . Since κ ( G ) = κ ( Z ( G )) + κ ( G/Z ( G )) and the case κ ( Z ( G )) = κ ( G ) has alreadybeen ruled out, we can assume that κ ( G/Z ( G )) = κ ( G ). Our inductivehypothesis ( κ ( G/Z ( G )) has nilpotency length n −
1) and Lemma 5.5 thenprovide the desired isometry. (cid:3)
When G is an IN -group or a nilpotent group, we recover and improvefurther the results obtained in Section 4. Corollary 5.8.
Let G be a non-compact IN -group or a nilpotent groupand let κ be the compact covering of G . Then each of the quotient spaces WAP ( G ) / ( AP ( G ) ⊕ C ( G )) and WAP ( G ) / B ( G ) contains a linear isomet-ric copy of ℓ ∞ ( κ ) . Proof.
That the first quotient contains a copy of ℓ ∞ ( κ ) follows directly fromTheorem 5.6 and Theorem 5.7 if we recall the inclusion AP ( G ) ⊕ C ( G ) ⊆ B ( G ) (see [13, page 143]).For the second quotient, we argue as in Theorem 5.6. None of the sets T η , η < κ, constructed in all cases needed in the proof of Theorem 5.6, isa B ( G )-interpolation set. On the other hand, proceeding precisely as inTheorems 5.6 and 5.7 (and using the right statements of Lemma 5.5), wesee that T = ∪ η<κ T η is an approximable WAP ( G )-interpolation set. (cid:3) On the quotient of CB ( G ) by LUC ( G )When G is non-compact, non-discrete, locally compact group, Dzinotyi-weyi showed in [20] that the quotient CB ( G ) / LUC ( G ) is non-separable.When G is a non-precompact, topological group which is not a P-group,this theorem was generalized and improved in [6, Theorem 3.1] and [7, The-orem 4.1], where a linear isometric copy of ℓ ∞ was proved to be containedin CB ( G ) / LUC ( G ). This section is concerned again with locally compactgroups. Our theorem is then more precise and definite. We prove, there is alinear isometric copy of ℓ ∞ ( κ ) in CB ( G ) / LUC ( G ), where as before κ is thecompact covering G, if and only if G is neither compact nor discrete. Lemma 6.1.
Every non-discrete locally compact group contains a faithfullyindexed sequence { x n : n ∈ N } that converges to the identity.Proof. A locally compact group always contains a compact subgroup K such that G/K is a metrizable topological space (see [2, Theorem 4.3.29],for instance). Infinite compact groups on the other hand always containnon-trivial convergent sequences ([2, Theorem 4.1.7 and Exercise 4.1.f]). If K is infinite we are done. If K is finite, G is non-discrete and metrizable, ittherefore contains non-trivial convergent sequences. (cid:3) Theorem 6.2.
Let G be a locally compact group. Then CB ( G ) / LUC ( G ) contains a linear isometric copy of ℓ ∞ ( κ ( G )) if and only if G is neithercompact nor discrete.Proof. The necessity is clear since CB ( G ) = LUC ( G ) if G is either compactor discrete.If G is not compact we can find a compact neighbourhood of the identity U and a U -right uniformly discrete subset X = { x α : α < κ } ⊆ G with κ = κ ( G ). This is clear if G is σ -compact. If κ > ω , we consider H = h U i , the subgroup generated by U . Then κ = | G : H | and any system ofrepresentatives of right cosets of H constitutes an H -right uniformly discreteset of cardinality κ .Partition X in κ -many countable subsets X = S α<κ X α . Enumerate, foreach α < κ, X α = { x α,n : n < ω } . Since G is not discrete, U contains (byLemma 6.1) a faithfully indexed sequence S = { s n : n < ω } converging tothe identity. With these ingredients, we define T α,n = { s j x α,n : 1 ≤ j ≤ n } , T α = [ n T α,n and T = [ α T α . Obviously,
U T α ∩ U T α ′ = ∅ for every α = α ′ < κ . NTERPOLATION SETS AND QUOTIENTS OF FUNCTION SPACES 29
Each set T α fails to be an LUC ( G )-interpolation set. Indeed, the function f : T α → C such that f ( s j x α,n ) = 1 for every j, n ∈ N with 1 ≤ j ≤ n and f ( s j +1 x α,n ) = − j, n ∈ N with 1 ≤ j + 1 ≤ n cannot coincide on T α with any φ ∈ LUC ( G ), since given ε >
0, we canchoose j large enough and n ≥ j + 1 so that | φ ( s j x α,n ) − φ ( s j +1 x α,n ) | < ε while f ( s j x α,n ) − f ( s j +1 x α,n ) = 2 . We now prove that T is an approximable CB ( G )-interpolation set. Sincethe sequence ( s j ) is taken in U and X is right U -uniformly discrete, we seethat the open set U x α,n of G contains no point from T other than s j x α,n , for 1 ≤ j ≤ n . Thus, T is discrete.Next we check that T is closed. Let x / ∈ T . If for some α < κ, n < ω, and 1 ≤ j ≤ n , we have s j x α,n ∈ U x, then x α,n ∈ s − j U x ⊆ U x . Note alsothat s j x α,n may be in U x for at most one α since U T α ∩ U T α ′ = ∅ for every α = α ′ < κ . Thus, U x ∩ T ⊆ { s j x α,n ∈ T : x α,n ∈ U x, α < κ n < ω, ≤ j ≤ n } . Since X is right uniformly discrete and U x is relatively compact, the set { x α,n ∈ U x : α < κ n < ω, ≤ j ≤ n } = X ∩ U x must be finite. Therefore, there is k such that U x ∩ T ⊆ { s j x α,n i : 1 ≤ j ≤ n i , i = 1 , . . . , k } . We conclude that
U x ∩ T is finite, and so T is closed. Since the topologicalspace underlying G is normal, T is an approximable CB ( G )-interpolationset by Lemma 5.2.Corollary 2.12 now implies that CB ( G ) / LUC ( G ) contains a linear isomet-ric copy of ℓ ∞ ( κ ) with κ = | X | = κ ( G ). (cid:3) The equivalence of the first two statements of the following Corollary wereproved by Baker and Butcher in [3], see also [28] for a different proof.
Corollary 6.3.
Let G be a locally compact group with a compact coveringnumber κ . Then the following statements are equivalent. (1) G is neither compact nor discrete. (2) CB ( G ) = LUC ( G ) . (3) CB ( G ) / LUC ( G ) contains a linear isometric copy of ℓ ∞ ( κ ( G )) .Proof. (1) = ⇒ (3) is proved in the theorem above. (3) = ⇒ (2) is obviousand (2) = ⇒ (1) is clear. (cid:3) Remark 6.4.
Theorem 6.2 implies a fortiori that the space L ∞ ( G ) / LUC ( G )as well as L ∞ ( G ) / WAP ( G ) contains a linear isometric copy of ℓ ∞ ( κ ) . Thearguments used in [6, Section 4], may be applied again to deduce that thegroup algebra L ( G ) is extremely non-Arens regular whenever κ is greateror equal to the local weight w ( G ) of G (this is the least cardinality of anopen base at the identity of G .) To obtain the full result, however, harderwork is necessary. This is achieved in our recent article [25]. Acknowledgement.
This paper was written when the first author wasvisiting University of Jaume I in Castell´on in December 2010-January 2011.He would like to thank Jorge Galindo for his hospitality and all the folks atthe department of mathematics in Castell´on. The work was partially sup-ported by Grant INV-2010-20 of the 2010 Program for Visiting Researchersof University Jaume I. This support is also gratefully acknowledged.
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Mahmoud Filali, Department of Mathematical Sciences, University of Oulu,Oulu, Finland.E-mail: [email protected]