Approximable WAP- and LUC-interpolation sets

If we require the extended functions to be almost periodic, then interpolation sets are usually known as I 0 -sets and were introduced by Hartman and Ryll-Nardzewsky [26]. For further details and recent results on I 0 -sets, see for example the papers by Galindo and Hernández [19,20] Graham and Hare [21,23,24], Graham, Hare and Körner [22] or Hernández [27].

Interpolation sets for the functions in the Fourier-Stieltjes algebra B(G) are usually known as Sidon sets when the group G is discrete and Abelian and weak Sidon sets in general, see for instance the works by Lopez and Ross [31], and Picardello [32]. Sidon sets are in fact uniformly approximable as proved by Drury in [8]. This means that in addition of being interpolation sets for the Fourier-Stieltjes algebra, the characteristic function of the set can be uniformly approximated by members of the algebra; in other words, the characteristic function of the set belongs to the Eberlein algebra B(G) = B(G)

• ∞ . This fact has important consequences as for instance Drury’s union theorem: the union of two Sidon subsets of a discrete Abelian group remains Sidon.

Ruppert [38] and Chou [6] considered interpolation sets for the algebra of weakly almost periodic functions on discrete groups and semigroups, again with the extra condition that the characteristic function of the set is weakly almost periodic. This is equivalent to the property that all bounded functions vanishing off the set being weakly almost periodic. These interpolation sets were called translation-finite sets (after their combinatorial characterization) by Ruppert and R W -sets (after their interpolation properties) by Chou. Let ℓ ∞ (G) be the C*-algebra of bounded, scalar-valued functions on G with the supremum norm and let A(G) ⊆ ℓ ∞ (G). In the present paper we introduce the notion of approximable A(G)-interpolation sets in such a way that it is suitable for functions defined on any topological group G and reduces to that of uniformly approximable Sidon sets when G is discrete and A(G) = B(G), and to that of R Wsets or translation-finite sets when G is discrete and A = WAP(G), the algebra of weakly almost periodic functions on G. Since we shall be dealing with closed subalgebras of ℓ ∞ (G), save some brief digressions around B(G), we shall omit the adverb “uniformly” in our definition (cf. Definition 3.1).

As the reader might expect, approximable A(G)-interpolation sets can be found in abundance if the algebra is large while they might be hard to find if the algebra is too small. As extreme cases, we could mention that all subsets of G have the property if A(G) = ℓ ∞ (G), while no metrizable locally compact group can have infinite approximable AP(G)-interpolation sets, where AP(G) is the algebra of almost periodic functions on G (see below).

Our principal concern shall be with the algebras LUC(G), of bounded functions which are uniformly continuous with respect to the right uniformity of G, and WAP(G), of weakly almost periodic functions on G. A combinatorial characterization of approximable LUC(G)-and WAP(G)-interpolation sets will be presented in Section 4.

But beforehand, we deal in Section 3 with the more straightforward cases of the algebra CB(G) of bounded, continuous, scalar-valued functions and the algebra C 0 (G) of continuous functions vanishing at infinity on G. It turns out that for the algebras C 0 (G), CB(G) and LUC(G), interpolation sets and approximable interpolation sets are the same for any topological group. This is also true for the algebra B(G) when G is an Abelian discrete group, a fact that follows from Drury’s theorem since, as we shall see in Proposition 3.5, uniformly approximable Sidon sets (in the sense of Dunkl-Ramirez [9]) are the same as our approximable B(G)-interpolation sets when G is discrete.

We do not know if this stays true for any locally compact group, but we give a partial result towards its affirmation for locally compact metrizable groups. It should be remarked that approximability cannot be expected within the algebra B(G). In fact, Dunkl and Ramirez noted in [9, Remark 5.5, page 59] that if T is a Sidon set in a discrete Abelian group G, and 1 T denotes the characteristic function of T , then 1 T ∈ B(G) if and only if T is finite. They observed also that for G = Z, this holds for any subset of G.

It is quick to see that the closed discrete sets are CB(G)-interpolation sets when the topological space underlying G is normal, and the finite sets are C 0 (G)interpolation sets (Proposition 3.3). In Section 4, we see that the right (left) uniformly discrete sets are approximable interpolation sets for the algebra LUC(G) (RUC(G)) for any topological grou