Interpolation sets and the size of quotients of function spaces on a locally compact group

We devise a fairly general method for estimating the size of quotients between algebras of functions on a locally compact group. This method is based on the concept of interpolation sets and unifies the approaches followed by many authors to obtain particular cases. Among the applications we find, we obtain that the quotients WAP(G)/B(G) (G being a locally compact group in the class [IN] or a nilpotent locally compact group) and CB(G)/LUC(G) (G being any non-compact non-discrete locally compact group) contain a linearly isometric copy of \ell_\infty(\kappa(G)) where \kappa(G) is the compact covering number of G, and WAP(G), B(G) and LUC(G) refer, respectively, to the algebra of weakly almost periodic functions, the uniform closure of the Fourier-Stieltjes algebra and the bounded right uniformly continuous functions.


Introduction
The main focus throughout the paper will be on C * -algebras of functions on 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 bounded right uniformly continuous functions, the algebra WAP(G) of weakly almost periodic 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 0 (G) ⊕ C1, where C 0 (G) consists of the functions in CB(G) vanishing at infinity.
The spectra of these algebras A(G) define some of the best-known semigroup compactifications in the sense of [5]. These are compact right (or left) topological semigroups G A having a dense, continuous, homomorphic copy 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 compactification in the sense that any other semigroup compactification is a quotient of G LUC . When G is discrete, G LUC and the Stone-Čech compactification βG are the same. The WAP-compactification G WAP is the spectrum of WAP(G); it is the largest semitopological semigroup compactification. The Bohr or AP-compactifiaction is the spectrum of AP(G) and is the largest topological (semi)group compactification.
The Banach duals of these C * -algebras can also be made into Banach algebras with a convolution type product extending in most cases that of the group algebra L 1 (G). We may recall that L ∞ (G) is the Banach dual of the group algebra L 1 (G) and consists of all scalar-valued functions which are measurable and essentially bounded with respect to the Haar measure; two functions are identified if they coincide on a locally null set, and the norm is given by the essential supremum norm. We may also recall that the product making L ∞ (G) into a Banach algebra is the first (or the second) Arens product on the second dual space L 1 (G) * * of the group algebra, and that LUC(G) * may be seen as a quotient Banach algebra of L 1 (G) * * . For more details, see for instance [17]. These two Banach algebras have been studied extensively in recent years. Particular attention has been given to properties related to Arens regularity of the group algebra L 1 (G) and to the topological centres of G LUC , LUC(G) * and L 1 (G) * * . For the latest, see [8] and the references therein.
The definitions of all these function algebras will be given in the next section. But for the moment the following diagram summarizes already the inclusion relationships known to hold among these algebras. See [13, page 143] 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 the third inclusion; the rest is easy to check. 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 the quotients formed among them has already been taken by many authors. We now give a brief review of what is known in this respect.
In the review below as well as in our study of quotients between the above algebras the compact covering number will appear at several points. We recall that the compact covering number of a topological space X is the smallest cardinal number κ(X) of compact subsets of X required to cover X.
Comparing L ∞ (G) with its subspaces. Already in 1961, Civin and Yood 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 and deduced that the radical of the Banach algebra L ∞ (G) * (with one of the Arens products as a product) is also infinite-dimensional.
This idea was pushed further by Gulick in [34,Lemma 5.2] when G is Abelian, and proved that the quotient L ∞ (G)/CB(G) is even non-separable and so is the radical of L ∞ (G) * . Then Granirer proved in [32] the same results 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-Arens regularity of the group algebra L 1 (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 compact covering of G. A fortiori, the quotient L ∞ (G)/WAP(G) contains the same copy, a fact that was used in [6,Theorem 4.4] to deduce that the group algebra is even extremely non-Arens regular in the sense of Granirer, whenever κ is larger than or equal to the minimal cardinal w(G) of a basis of neighbourhoods at the identity.
It was also proved in [6,Section 4] that the quotient L ∞ (G)/CB(G) always contains a linear isometric copy of ℓ ∞ , yielding extreme non-Arens regularity for the group algebra of compact metrizable groups. Due to a result by Rosenthal 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 on extreme non-Arens regularity of the group algebra was recently settled by the authors of the present paper using a technique inspired by Theorem 2.11. 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] gives that L 1 (G) is extremely non-Arens regular for any infinite locally compact group.
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 arbitrary topological group, and proved that they are equal if and only if G is pseudocompact (i.e., every continuous scalar-valued function on G is bounded). In 1970, Burckel showed in [9] that CB(G) and WAP(G) are equal if and only if 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 if and only if G is either discrete or compact. This result was extended recently by Filali and Vedenjuoksu in [28,Theorem 4.3] to all topological groups which are not P -groups. The author deduced in [28] that if G is a topological group which is not a P -group, then CB(G) = LUC(G) if and only 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, locally compact 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 linear isometric copy of ℓ ∞ whenever G is a non-precompact topological group which is not a P -group. So this theorem improved actually also Dzinotyiweyi's result for locally compact groups. For non-discrete, P -groups, the quotient CB(G)/LUC(G) was seen to be trivial in the case when for instance G is a Lindelöf P -group (see [28,Theorem 5.1]), but may also contain a linear isometric copy of ℓ ∞ for some other P -groups (see [6,Theorem 3.3]).
In [7,Theorem 3.1], using a technique due to Alas (see [1]), the quotient space 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 precise and 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, if and only if G is a neither compact nor discrete. This leads again to a linear isometric copy of ℓ ∞ (κ) into the quotient L ∞ (G)/WAP(G), and of course may be used to deduce again the extreme non-Arens regularity of of L 1 (G) when κ(G) ≥ w(G) ≥ ω as in [6,Theorem 4.4].
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 means that G LUC or LUC(G) * is a semitopological semigroup if and only if G is compact, i.e., G LUC = G is a compact group and LUC(G) * coincides with the 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 topological centre of G LUC being G, and again by Lau andÜlger in [41,Corollary 3.8] as a corollary of the topological centre of L 1 (G) * * being L 1 (G) [39].
Moreover, Granirer showed in the same paper that if G is non-compact and amenable, then the quotient LUC(G)/WAP(G) contains a linear isometric copy of ℓ ∞ , and so it is not separable. This result was extended by Chou in [10] to E-groups (see below for definition), then by Dzinotyiweyi in [20] to all non-compact locally compact groups, and generalized by Bouziad and Filali in [6, Theorem 2.2] to all non-precompact topological groups. Moreover, as already mentioned above, this result was improved in [6,Theorem 4.2] when G is a non-compact locally compact group, by having a copy of ℓ ∞ (κ) in the quotient LUC(G)/WAP(G).
Comparing WAP(G) with its subspaces. In the "regular" side of the inclusion diagram, when we compare WAP(G) with its subspaces, the situation is not simpler. It is true that the Fourier-Stieltjes algebra B(G) may be dense in WAP(G) (i.e., WAP(G) = B(G)), as in the case of minimally weakly almost periodic groups studied by Veech, Chou and Ruppert, see [53], [12] and [51]. For these groups, WAP(G) = AP(G) ⊕ C 0 (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 0 (G), we may recall first that WAP(G) = AP(G) ⊕ WAP 0 (G). Burckel proved in [9] that C 0 (G) WAP 0 (G) when G is an Abelian, non-compact, locally compact group. In [10], Chou considered E-groups and proved that the quotient WAP 0 (G)/C 0 (G) contains a linear isometric copy of ℓ ∞ . In Section 4, we improve this result by showing that ℓ ∞ may be replaced by an isometric copy of the larger space ℓ ∞ (κ(E)) in each of the quotient space WAP 0 (G)/C 0 (G), where κ(E) is the compact covering of the E-set contained in G. So when G is an SIN -group, these quotients contain a copy of ℓ ∞ (κ(G)). For the same class of groups, we prove also that the quotient WAP(G)/(AP(G) ⊕ C 0 (G)) is non-separable.
In Section 5, we deal with the non-compact, IN -groups, and with noncompact nilpotent groups. In this class of groups, the results of the previous section shall be considerably improved. Rudin proved in [49] that B(G) WAP(G) if G is a locally compact Abelian group and contains a closed discrete subgroup which is not of bounded order. This was followed by [47], where Ramirez extended Rudin's result to any non-compact, locally compact, Abelian group. Then in [13], Chou extended and strengthened the theorem to all non-compact IN -groups and nilpotent groups by showing that 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 these cases, there is in fact a linear isometric copy of ℓ ∞ (κ) in the quotient spaces WAP(G)/B(G), WAP(G)/(AP(G) ⊕ C 0 (G)) and WAP 0 (G)/B 0 (G), where κ is as before the compact covering of G. Our method of proof also shows that 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 estimate the size of , depends on the existence of sets of interpolation for A 2 (G) that are not sets of interpolation for A 1 (G) (see for instance [6], [7], [10], [13] or [20]). One of the main objectives of the present paper is to make that structure emerge in a clear fashion. A first, but essential, step towards this objective is to work with the right concept of interpolation sets. We will use here the general concept of interpolation set introduced in [24] that extends several related classical ones and show how to apply it in this setting. The resulting interpolation sets are characterized in [24] in term of topological group properties, thereby making them easier to manipulate. We finally illustrate the scope of our approach by studying some concrete cases. We shall in particular study under this light the following quotients: WAP 0 (G) by C 0 (G) and WAP(G) by AP(G)⊕C 0 (G) for E-groups, WAP(G) by B(G), WAP(G) by AP(G) ⊕ C 0 (G) and WAP 0 (G) by B 0 (G) for IN -groups and nilpotent groups, CB(G) by LUC(G) for locally compact groups.
1.1. The function algebras. We start by recalling the definitions of the function algebras we are interested in, for more details the reader is directed for example to [5].
Let G be a topological group. For each function f defined on G, the left translate 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 f ∞ .
A function f ∈ ℓ ∞ (G) is right uniformly continuous when, if for every ǫ > 0, there exists a neighbourhood U of e such that The algebra of right uniformly continuous functions on G is denoted by LUC(G).
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 all continuous positive definite functions on G. Equivalently, B(G) is the space of coefficients of unitary representations of G when G is locally compact. As the Fourier-Stieltjes algebra is not uniformly closed we will work with the Eberlein algebra B(G), which is the uniform closure of B(G), in symbols Let µ be the unique invariant mean on WAP(G) (see [5], or [9]). As stated above, put In [13, page 143], Chou denoted B(G) ∩ WAP 0 (G) by B c (G), and observed that B 0 (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 spectrum (the set of non-zero multiplicative linear functionals) of A(G). Equipped with the topology of pointwise convergence, G A becomes a compact Hausdorff topological space. There is a canonical morphism ǫ A : G → G A given by evaluations 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 its image in G A ) whenever C 0 (G) ⊆ A(G). This is not a necessary condition since it may also happen that ǫ A is injective when C 0 (G) ∩ A(G) = {0} as it 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 when for example T is an A(G)-interpolation set.
The C * -algebra A(G) is left translation invariant when f s ∈ A(G) for every f ∈ A(G) and s ∈ G. When A(G) is left translation invariant, we may define 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), we say that A(G) is admissible.
When A(G) is an admissible C * -subalgebra of CB(G), G A can be equipped with 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 continuous, dense, homomorphic, image of G such that the mappings are continuous for every y ∈ G A and s ∈ G.
The algebras C 0 (G) ⊕ C, AP(G), C 0 (G) ⊕ AP(G), B(G), WAP 0 (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 either discrete or compact, see [3] or [28] for more.
When G is a locally compact group and A is an admissible C * -subalgebra of LUC(G), the semigroup compactification G A has the the joint continuity property, that is, the map A recent account on semigroup compactifications is given in [29].
1.3. A few words on notation. All our groups will be multiplicative and their identity element will be denoted as e. The characteristic function of a set T will be denoted as 1 T . If X is a set and T ⊆ X, given f ∈ ℓ ∞ (X), separates points. If X ⊆ G, we will denote the closure of ǫ A (X) simply as X A , while the closure of X in G will be denoted as X. The reason for this is that in most of our applications the algebra 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 When ǫ A is injective f A can be seen as an extension of f to G A .

Interpolation sets and quotients of function spaces
We begin our work by introducing in precise terms the sets we will be using, and then we prove the impact they have in measuring the size of our quotient spaces A 2 (G)/A 1 (G). This is achieved in Theorem 2.11.
It is worthwhile to note that this theorem may also be applied to obtain most (if not all) of the results concerning the quotient spaces of the various function algebras mentioned in the introduction; it is of course necessary at each 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 definitions and properties shall also be proved for a general Hausdorff topological group whenever this makes sense.
have been a frequent object of study, see [29] and [24] for more details and references. See also [30] for the most recent account on the subject. Approximable interpolation sets appear in the early 70's as a crucial step in Drury's proof of the union theorem of Sidon sets, see [18]. Other wellknown interpolation sets are also approximable as for instance translationfinite 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 discrete groups, see [24] for more on this respect. When It should however be reminded that approximable A(G)-interpolation sets do 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 consequences of the definitions of interpolation and approximable interpolation sets. The identification of T A with the Stone-Čech compactification of T (with the discrete topology) allows us to use the powerful property of extreme disconnectedness of the latter compactification. As the reader will quickly notice this is the key in the arguments leading to the main results in this section. The main results start with a generalization of a theorem proved 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 with the conditions stated in Theorem 2.11 and Corollary 2.12 under which the quotient ) contains a linear isometric copy of ℓ ∞ (κ) for some cardinal κ.

(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-
Cech-compactification of T equipped with the discrete topology, that leaves the points of T fixed.

(ii) T is an A(G)-interpolation set if and only if for every pair of subsets
. Assertion (i) follows then from the universal property defining the Stone-Cech compactification of a discrete space. In fact, the restriction of the evaluation map ǫ A to T gives a homeomorphism of the discrete set T d onto To prove (iii), let f : T → C with f T = M be given. If B M is the closed disc of radius M centered at 0 (in C), we can use (i) and the universal property of βT d to find a continuous function f β : G is then the desired extension. To prove (iv), let T be an approximable A(G)-interpolation set. First, we find, using (iii), Using [21, 3.2.20], we can assume (taking the minimum of f 2 and the function that is constant and equal to 1) that f 2 ∞ = 1. The product f 1 · f 2 then coincides with h on T and vanishes off V 2 T . Remark 2.4. Note that if in the lemma above A(G) ⊆ CB(G), then T is necessarily discrete since every bounded function on T must be continuous.
Observe as well that the sole existence of an infinite A(G)-interpolation set T in G, implies that G A contains a copy of βT d , where T d is the discrete set T . The compactification G A is therefore large and topologically involved.
The following theorem, due to Chou [13], has its roots in a result of Ramirez (see Theorem 2.3 of [19]) in the Abelian setting. This theorem is used 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.
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 compact groups.
The result in Theorem 2.5 is more natural when the function algebra is a C * -subalgebra. It is not surprising therefore that it holds for any C *subalgebra. Next lemma proves even more.
Proof. Let T = η<κ T η be an approximable A 2 (G)-interpolation set as stated in the lemma. Let U be an open neighbourhood of e.
To avoid cumbersomeness, we abuse our notation and use the same letters to denote subsets of T and their images in Then consider the function h : G → [−1, 1] supported on T and given by Let now φ be any function in A 1 (G), and take ε > 0. Given η < κ, we are going to prove that f − φ Tη ≥ 1 − ε.
For the main theorem in this section, we need to recall the following definitions. These sets are also essential for the rest of the paper. 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 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 Uuniformly 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 is uniformly 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 Proof. First, consider the two neighbourhoods V 1 and V 2 provided by the definition of approximable A(G)-interpolation sets for the neighbourhood U . We take V 1 as V , and we can obviously assume that V 2 ⊆ 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 ) = {0}, related as in (3). We prove that f is continuous considering separately continuity at interior points of V T and points that do not belong to the interior of V T .
Let s ∈ G that is not an interior point of V T . Since s can be approached from G \ V T , we see by continuity that g(s) = 0. By checking the cases when 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 ( The continuity of 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 identity with W s ⊆ V T with |g(s) − g(s ′ )| < ǫ for every s ′ ∈ W s. Let w ∈ W be such that s ′ = wvt and notice that s ′ = ws = wvt = v 0 t 0 for some v 0 ∈ V and t 0 ∈ T implies that t 0 ∈ T η . Therefore, 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 to a function ϕ ∈ A(G). By Lemma 2.3 (iv), we can assume that ϕ(G \ V T ) = {0}. If g A and ϕ A denote the respective extensions of g and ϕ to G A , we define f * : 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 two different decompositions. We check that the definition of f * does not depend of the choice of the decomposition. Suppose therefore that v 1

Recalling that multiplication by elements of G is continuous on G
This shows already that f * is well defined, since the equalities v 1 p 1 = v 2 p 2 and p 1 = p 2 give us (In fact, v 1 and v must be also equal by the same argument, but this is enough for our purposes.) we readily see that f and f * coincide on G \ V T . Let on the other hand s = vt with v ∈ V and t ∈ T η . Then (3) f * is continuous. Using the joint continuity property, we see that 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.
By taking subnets if necessary, we may assume that lim α v α = v 0 in V and lim α p α = p 0 in T A . Accordingly, x = vp = v 0 p 0 , and applyig 5, we see that 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 From (1), (2), (3) we conclude that f ∈ A.
Remarks 2.10. (i) A known theorem due to Veech asserts that the left action 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 proof of the previous Lemma reveals that Veech's property in fact holds in G A at any 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 compact groups 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 uniformly discrete.

Theorem 2.11. Let G be a locally compact group and let
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 1 (G)-interpolation set for every η < κ, and (iii) T = η<κ T η is an approximable A 2 (G)-interpolation set.
Proof. Let V be the neighbourhood of the identity provided by Lemma 2.9.
Since T = η<κ T η is an approximable A 2 (G)-interpolation set and each T η fails to be an A 1 (G)-interpolation set, we take from Lemma 2.7 a function f ∈ A 2 (G) with f ∞ = 1 such that i.e., with the notation of Lemma 2.9, f c V Tη = c(η)f V Tη . Then f c ∈ A 2 (G) by (ii) of Lemma 2.9. Obviously, the map Ψ : ℓ ∞ (κ) → A 2 (G)/A 1 (G) given by is linear. We next check that it is isometric.
The same argument of [13,Theorem 3.12] shows now that, for every η 0 < κ, where the last inequality follows from the choice of f . Since, obviously, we see that Ψ is the required isometry.
Corollary 2.12. If in the above theorem A 2 (G) = CB(G) and T is not assumed to be right U -uniformly discrete but still U T η ∩ U T η ′ = ∅, then the quotient CB(G)/A 1 (G) contains a linearly isometric copy of ℓ ∞ (κ).
Proof. The proof of Theorem 2.11 remains valid in this case applying (i) of Lemma 2.9 instead of (ii).
Corollary 2.14. Under the hypotheses of Theorem 2.11 or Corollary 2.12, the quotient space A 2 (G)/A 1 (G) is non-separable.

Interpolation sets
The definitions in this section gather the topological group-theoretic properties that will correspond to the interpolation sets needed in the three sections that follow. Once these interpolation sets are at hand, an application of Theorem 2.11 and Corollary 2.12 will lead immediately to the desired conclusion on the quotients.
In addition to the uniformly discrete sets defined in the previous sections we shall also need the following sets. containing e such that gS ∩ S (respectively, Sg ∩ S) is relatively compact for every g / ∈ K; and a t-set when it is both a right and a left t-set.
We also need to establish the range of locally compact groups to which our methods apply in the next two sections, these are those locally compact groups for which the existence of a good supply of WAP-functions is guaranteed.
Recall that a locally compact group G is an IN −group if it has an invariant neighbourhood of the identity. We recall also from [10], that a locally compact 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 is again a neighbourhood of e. The set X is called an E-set. This is a large class of locally compact groups. This includes of course all non-compact SIN −groups, the groups with a non-compact centre such as the matrix group GL(n, R), and the direct product of any E-group with any locally compact 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 present paper.

ii) If T is right (resp. left) uniformly discrete , then T is an approximable LUC(G)-interpolation set (resp. RUC(G)-interpolation set). (iii) If G is assumed to be metrizable, then every LUC(G)-interpolation set (resp. RUC(G)-interpolation set) is right (left) uniformly discrete. (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 2 for some neighbourhood U of the identity such that U T is translation-compact, then T is an if and only if U T is translation-compact for some compact neighbourhood U of the identity such that T is right (or left) uniformly discrete with respect to U 2 .
The following Lemma will be needed later on in Section 5.  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 such that gU T ∩ U T is relatively compact whenever g / ∈ K. Let V be a compact symmetric 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 no convergent 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 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 −1 α ) is a net in H which converges to h = v −1 gw. Since H is closed, h ∈ H. Therefore, the net (t α s −1 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 imply that 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 addition that T is central and put again K ′ = V KV. Using the fact that T is central, the same argument leads to s −1 α t α = w α gv −1 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 ′ .
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 recall from [13, Definition 3.3]: A finite subset F of G is an n-square if F = AB where |A| = |B| = n and |F | = n 2 . A subset of a group is then said to contain large squares if it contains an n-square for every n ∈ N.
Large squares are incompatible with Sidon sets, as proved in [13, Proposition 3.4]. We restate here this theorem, stressing on B(G). 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 cannot contain large squares. 4. The quotients of WAP 0 (G) by C 0 (G) and WAP(G) by AP(G) ⊕ C 0 (G) In [10], Chou considered E-groups and proved that the quotient space WAP 0 (G)/C 0 (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 is a linear isometric copy of ℓ ∞ (κ) in the quotient WAP 0 (G)/C 0 (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 0 (G) is non-separable.
Theorem 4.1. Let G be a non-compact locally compact E-group having an E-set X with a compact covering number κ. Then the quotient space WAP 0 (G)/C 0 (G) contains a linear isometric copy of ℓ ∞ (κ).
Proof. Let V be a fixed compact symmetric neighborhood of e. Then we consider a set T ⊂ X as that constructed in Section 2 of [23]. This set has the following properties: For completeness, we recall from [23] the construction of the set T since this shall be needed in the proof. We may assume that e ∈ X and start with x 0 = e.. Suppose that the elements x β have been selected for all β < α with α < κ. Set where each ǫ i = ±1. 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 0 (G)-interpolation set. Since every infinite subset of T is uniformly discrete and C 0 (G)-interpolation sets must be relatively compact, and so finite (see the proof of Proposition 3.3 of [24]) any decomposition T = η<κ T η as a disjoint union of κ-many infinite subsets leaves us in position to apply Theorem 2.11 and finish the proof.

Corollary 4.2. Let G be a non-compact locally compact E-group
having an E-set X with a compact covering number κ. Then the quotient space WAP(G)/ (AP(G) ⊕ C 0 (G)) contains an isomorphic copy of ℓ ∞ (κ).
If we want to use our Theorem 2.11 to obtain a linear isometric copy of 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, for instance, is discrete this means we need a translation-finite set that is not an I 0 -set. Such sets can be easily found in Z, the additive group of integers: T = {3 n + n : n ∈ N} ∪ {3 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 as that has escaped to us. A considerably more complicated construction can be 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, see Section 5 (note that the set T above is a Sidon set, i.e., a B(G)-interpolation set).
We present however, on the lines of Theorem 2.11, an ad-hoc construction of a linear isomorphism of ℓ ∞ (κ) into WAP(G)/(AP(G)⊕C 0 (G)) with norm at most one whose inverse has norm at most 2. A detailed look at the proof reveals that it is actually based in finding an approximable WAP(G)interpolation set that is not an approximable AP(G)-interpolation set. What makes this construction different from our general approach is that this set could even be an AP(G)-interpolation set. Recall that in a non-compact locally compact group no AP(G)-interpolation set is approximable.

Theorem 4.3. Let G be a non-compact, locally compact E-group having an E-set X with a compact covering number κ. Then the Banach space WAP(G) contains a linear isometric copy
In particular, the quotient space WAP(G)/(AP(G)⊕C 0 (G)) is non-separable.
Since h ∈ C 0 (G), we may fix as well x ∈ T η such that |h(x)| < ǫ/4. Let (x n ) n≥1 be any sequence in T η with x ǫ = x n for every n ∈ N, ǫ = ±1. Suppose xx −1 n x m = vx β ∈ V T for some β < κ and v ∈ V. By the definition of T , this is possible for at most two m ′ s: x β = x ǫ and so x m = x n x −1 vx ǫ , or x β = x ǫ n and so x m = x n x −1 vx ǫ n . In other words, for every fixed n ∈ N, there exists at most two m for which xx −1 n x m ∈ V T. Therefore, for every fixed n, f η (xx −1 n x m ) = 0 for every m except maybe for these two m ′ s. Moreover, since for each n, the set {xx −1 n x m : m ∈ N} is not relatively compact, we may choose m such that |h(xx −1 n x m )| < ǫ/4. Now since g ∈ AP(G), by taking subsquences if necessary, we may fix n 0 ∈ N such that that r xn g − r xm g < ǫ/2 for every n, m ≥ n 0 ; and so, in particular, Therefore, for every fixed n = n 0 and m ≥ n 0 chosen suitably, we have This is clearly absurd, so we must have 2f c + g + h ≥ 1 as required.
Both theorems in this section will be considerably improved in the next section when G is an IN -group or a nilpotent group.

The quotient of WAP(G) by B(G)
The situation is much more delicate with WAP(G)/B(G). Already in the 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 the work 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 noncompact, locally compact, IN -group or a nilpotent group. In all these papers the 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 possible in WAP(G)/B(G) by applying Theorem 2.11.
More precisely, we shall strengthen Chou's theorems by showing that there is a copy of ℓ ∞ (κ) in the quotient WAP(G)/B(G) when G is either an IN-group or a nilpotent group and κ = κ(G), and that, in general, for every locally 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 covered by β-cosets of finitely many different subgroups of index larger than β. This is similar to a theorem, known at least from the times of [45], in which only finitely many cosets are allowed.
Lemma 5.1. Let G be any group with |G| = κ. Suppose that there is a finite collection {H 1 , . . . , H n } of subgroups of G such that G can be covered by β < κ right-cosets of them, i.e., such that 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 − 1 different subgroups and suppose with |I 1 | + · · · + |I n | = β < κ.
we obtain and so where the y i,j 's have been suitably chosen in . Applying our inductive hypothesis, we deduce that there is j 0 with |H n : (H n ∩ H j 0 )| ≤ β. (One may also proceed directly and replace H n from ( * * ) in ( * ), then apply the inductive hypothesis).
There is therefore a family {z s : s ∈ S} ⊂ H n with |S| ≤ β such that H n = s∈S (H n ∩ H j 0 )z s and we may replace (6) by Since this is a cover of G by cosets of at most n − 1 different subgroups of G we deduce from our inductive hypothesis that some of the subgroups H j , 1 ≤ j ≤ n − 1, has index at most β.
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 subsets such that, putting T = η<κ T η , (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 relatively compact relatively compact for every g / ∈ t −1 U 2 t with t ∈ T .
Proof. For each η < κ, let in this proof C η (H) denote the set where Cl(gH) denotes the conjugacy class of gH in G/H. If A ⊆ G, Cl(A) will stand for the set {g −1 ag : g ∈ G, a ∈ A}. For n < ω, we define A n to be the set of all products of at most n elements in A ∪ A −1 .
Finding this element is possible because | R η,n,k 8 H/H| ≤ f (η, n, k) < κ = |G : H| and conjugacy classes of elements of C f (η,n,k) do not have, by definition, more than f (η, n, k)-elements.
Once the elements in C η,n are defined in this way, we define the elements in D η,n . If y η,n,1 , . . . , y η,n,k−1 have already been defined, we use the following Claim to define y η,n,k .
We first enumerate (C η,n C −1 η,n \ {e}) Cl (R η,n,k ) H as {a 1 , . . . , a l }. Let then 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 and an element h j,r ∈ H with r = y −1 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, where L j,r,h = g ∈ G : r = g −1 a j hg . Observe now that that L j,r,h H ⊆ C G/H (a j H)y j,r H, where is the centralizer of a j H. Therefore, If the elements of the set S η,n,k 8 are viewed as cosets of the trivial subgroup {e G/H }, we find G/H as a union of less than n 2 |R η,n,k | + | S η,n,k 8 | cosets.
Since the latter number is less than f (η, n, k) some of them must correspond to 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). We conclude 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 −1 η,n,k 1 x η,n,k 2 and k 2 > k 1 , this goes against condition (7) in the choice of x η,n,k 2 and finishes the proof of the claim.
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 Lemma 5.2.
We obtain a first consequence for groups with large center. Let {T η : η < κ} be the family of subsets of H 2 provided by Lemma 5.2. By Theorem 3.4, none of them is a B(G)-interpolation set. If T = η<κ T η and U is a compact neighbourhood of the identity in H 2 with U ⊂ H 1 , then, since H 2 is commutative, U T is a t-set in H 2 . By Lemma 3.3, we can find a compact neighbourhood V of the identity in G, such that V T is a t-set in G. If V is chosen so that V 4 ∩ H 2 ⊂ H 1 (remember H 1 is open in H 2 ), then T is V 2 -uniformly discrete, and by Lemma 3.2, T is an approximable WAP(G)-interpolation set.
It suffices now to apply Theorem 2.11.
Theorem 5.2 can be readily applied to discrete groups. To further expand its applicability we follow the usual path applying well-known structure theorems. The following Lemma for instance is the analog of Lemma 4.4 of [13]. Proof. We first prove (i). The map φ → φ•π clearly defines a linear isometrỹ π : WAP(G/N ) → WAP(G). By [11,Theorem], we havẽ π(B(G/N )) =π(CB(G/N )) ∩ B(G).
Since B(G) ⊆ WAP(G), we see that so thatπ induces a linear isomorphism given by Π(φ + B(G/N )) =π(φ) + B(G). We check that Π is an isometry. If φ ∈ WAP(G/N ), For the reverse inequality, we follow the path of Lemma 2.3 of [12] and consider the invariant mean µ N on WAP(N ). For φ ∈ WAP(G), we define the function φ N : G → C by By invariance of µ N , the function φ N is constant on the cosets of N and therefore induces a continuous function on G/N . Clearly, φ N G/N ≤ φ 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).
For the proof of (ii), we associate to each φ ∈ WAP(N ) the function It is easy to check that Ψ is a linear isometry.
We reach finally our main results.
Proof. Let G 0 denote the connected component of G. By Theorem 2.13 of [33], there is an open normal subgroup N of G that contains a compact normal subgroup K with N/K Abelian.
If κ(N ) < κ, it follows that κ = |G : N |. We apply Lemma 5.2 to the discrete group G/N . Let {T η : η < κ} be the collection of subsets obtained in that Lemma (in this case the subgroup H of that Lemma is trivial) and let T = 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)-interpolation set by Lemma 3. Proof. The case κ(Z(G)) = κ(G) is already proved in Theorem 5.4. So we may 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 0 ⊂ G 1 ⊂ . . . ⊂ G n−1 ⊂ 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 nilpotency length at most n − 1 and suppose G has nilpotency length n. Since κ(G) = κ(Z(G)) + κ(G/Z(G)) and the case κ(Z(G)) = κ(G) has already been ruled out, we can assume that κ(G/Z(G)) = κ(G). Our inductive hypothesis (κ(G/Z(G)) has nilpotency length n − 1) and Lemma 5.5 then provide the desired isometry.
When G is an IN -group or a nilpotent group, we recover and improve further the results obtained in Section 4.
Corollary 5.8. Let G be a non-compact IN -group or a nilpotent group and let κ be the compact covering of G. Then each of the quotient spaces WAP(G)/(AP(G) ⊕ C 0 (G)) and WAP 0 (G)/B 0 (G) contains a linear isometric copy of ℓ ∞ (κ).
Proof. That the first quotient contains a copy of ℓ ∞ (κ) follows directly from Theorem 5.6 and Theorem 5.7 if we recall the inclusion AP(G) ⊕ C 0 (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, is a B 0 (G)-interpolation set. On the other hand, proceeding precisely as in Theorems 5.6 and 5.7 (and using the right statements of Lemma 5.5), we see that T = ∪ η<κ T η is an approximable WAP 0 (G)-interpolation set.

On the quotient of CB(G) by LUC(G)
When G is non-compact, non-discrete, locally compact group, Dzinotyiweyi 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, Theorem 4.1], where a linear isometric copy of ℓ ∞ was proved to be contained in CB(G)/LUC(G). This section is concerned again with locally compact groups. Our theorem is then more precise and definite. We prove, there is a linear isometric copy of ℓ ∞ (κ) in CB(G)/LUC(G), where as before κ is the compact covering G, if and only if G is neither compact nor discrete. Lemma 6.1. Every non-discrete locally compact group contains a faithfully indexed 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 contain non-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, it therefore contains non-trivial convergent sequences. 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 neither compact nor discrete.
Proof. The necessity is clear since CB(G) = LUC(G) if G is either compact or discrete.
If G is not compact we can find a compact neighbourhood of the identity U and a U 2 -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 = U , the subgroup generated by U . Then κ = |G : H| and any system of representatives of right cosets of H constitutes an H-right uniformly discrete set of cardinality κ.
We now prove that T is an approximable CB(G)-interpolation set. Since the sequence (s j ) is taken in U and X is right U 2 -uniformly discrete, we see that 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 −1 j U x ⊆ U 2 x. Note also that 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 2 x, α < κ n < ω, 1 ≤ j ≤ n}.
Since X is right uniformly discrete and U 2 x is relatively compact, the set {x α,n ∈ U 2 x : α < κ n < ω, 1 ≤ j ≤ n} = X ∩ U 2 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 topological space underlying G is normal, T is an approximable CB(G)-interpolation set by Lemma 5.2. Corollary 2.12 now implies that CB(G)/LUC(G) contains a linear isometric copy of ℓ ∞ (κ) with κ = |X| = κ(G).
The equivalence of the first two statements of the following Corollary were proved 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 covering number κ. Then the following statements are equivalent.
(1) G is neither compact nor discrete.
(2) CB(G) = LUC(G). 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). The arguments used in [6,Section 4], may be applied again to deduce that the group algebra L 1 (G) is extremely non-Arens regular whenever κ is greater or equal to the local weight w(G) of G (this is the least cardinality of an open base at the identity of G.) To obtain the full result, however, harder work is necessary. This is achieved in our recent article [25].