GROUP REPRESENTATIONS OF BOUNDED COSINE FUNCTIONS

WOJCIECH CHOJNACKI

Dedicated to Professor Jan Kisyński

Abstract. We characterise two classes of locally compact Abelian groups. One of these comprises all groups G such that every bounded continuous cosine function on G with values in a sequentially complete semitopological algebra with identity has a regular group representation. The other comprises all groups G such that every bounded continuous cosine function on G with values in a sequentially complete semitopological algebra with identity has a bounded regular group representation.

1. Introduction

Let G be a locally compact Abelian group with group operation denoted additively, and let A be an algebra with identity over the field ℂ of complex numbers. A homomorphism from G into A is a mapping G: G → A satisfying both Cauchy’s functional equation

G(a + b) = G(a)G(b) (a, b ∈ G)

and the condition

G(0) = e,

where 0 denotes the neutral element of G and e denotes the identity of A. An Avalued cosine function on G is a mapping C: G → A satisfying both d’Alembert’s functional equation

C(a + b) + C(a − b) = 2C(a)C(b)  (a, b ∈ G)  (1.1)

and the condition

C(0) = e.

A cosine function C: G → A has a group representation if there is a homomorphism G: G → A such that, for each a ∈ G,

C(a) = \frac{1}{2}(G(a) + G(−a)).  (1.2)

A semitopological algebra is an algebra equipped with a topology such that the underlying vector space is locally convex and Hausdorff, and the multiplication operation is separately continuous. Henceforth we shall assume that A is a semitopological algebra with identity. If C: G → A is a continuous cosine function such that (1.2) holds for a continuous homomorphism G: G → A, we say that C has a regular group representation. We recall that a mapping with values in a topological vector space is bounded if the image of this mapping is bounded. If C: G → A is a bounded (continuous) cosine function such that (1.2) holds for a bounded (continuous) homomorphism G: G → A, then C will be said to have a bounded (regular) group representation. Given a locally convex vector space E, we denote by L(E) the algebra of all linear continuous operators in E. We shall always consider L(E) as being equipped with the strong operator topology. Under this topology, L(E) is a semitopological algebra. A homomorphism from G into L(E) will often be referred to as a representation of G in E. An L(E)-valued cosine function will at times be referred to as a cosine function in E. Note that if E is either a Banach space or the dual of a Banach space under the ∗-weak topology, and if f is a function from a set S into L(E), then, as recourse to the Banach-Steinhaus theorem shows, f is bounded if and only if it is bounded in norm; that is, sup_{s∈S} ∥f(s)∥ < +∞.

1991 Mathematics Subject Classification. Primary 47D09, 47D03; Secondary 43A65, 22D05, 47B99.

Key words and phrases. Cosine function, group representation, locally compact Abelian group, semitolopogical algebra.

Research supported in part by the Polish Research Council.

This paper deals with the problem of group representability of cosine functions. In the context of operator-valued cosine functions, the problem arose out of a study of the Cauchy problem for second order abstract differential equations. Such a study is greatly facilitated when suitable cosine functions defined on the group ℝ of real numbers and taking on values in L(E), where E is a Banach space, have a regular group representation (cf. [8, Section 2.5], [9, Section III.6], [16, Section III.1.1], [18]). In general, unbounded continuous cosine functions, even as simple as those taking on values in finite-dimensional algebras, may fail to admit a regular group representation; for example, the L(ℂ^{2})-valued cosine function on ℝ given by

t ↦ (\begin{matrix}1 & t^{2}/2 \\ 0 & 1 \end{matrix})

has no regular group representation (cf. [14]). Contrasting with this is a result of [3] stating that every L(H)-valued bounded continuous cosine function on G, H being a Hilbert space and G being a locally compact Abelian group, has a regular group representation. Here the appeal to a Hilbert space structure is essential. In fact, Kisyński [14, 15] and Fattorini [7] (see also [9, Section III.8]) exhibited various L(E)-valued bounded cosine functions on ℝ that have no regular group representation, where E are Banach spaces consisting of odd functions or of odd measures on ℝ equipped either with the norm topology or with the ∗-weak topology.

In the development that follows, we characterise—in purely algebraic terms—the class of all locally compact Abelian groups G such that every bounded continuous cosine function on G with values in a sequentially complete semitopological algebra with identity has a regular group representation. We also characterise the class of all locally compact Abelian groups G such that every bounded cosine function on G with values in a sequentially complete semitopological algebra with identity has a bounded regular group representation. It will follow from these characterisations that the group ℤ of integers is in the former class but not in the latter. It will also become apparent that ℝ is in none of the two classes. The latter result (which, of course, can also immediately be deduced from the above-mentioned results of Kisyński and Fattorini) will be a consequence of the existence of a number of L(E)-valued bounded continuous cosine functions on ℝ admitting no regular group representation, where E are Banach spaces consisting of even functions or of even measures on ℝ equipped either with the norm topology or with the ∗-weak topology. Here the appeal to even functions and measures is representative of the approach adopted in handling arbitrary locally compact Abelian groups. It turns out that the use of such functions and measures renders the analysis of certain crucial cosine functions a relatively easy task.

The remainder of the paper is organised as follows. Section 2 introduces the concepts of a c-group and of a bc-group, and reduces the main representability problem to that of characterisation of the classes comprising all c-groups and all bc-groups. Section 3 gives necessary conditions for a given group to be a c-group or to be a bc-group. Section 4 is concerned with compact Abelian groups that are decomposable in a certain sense. The results of that section are used in Section 5 to demonstrate that the sufficient conditions from Section 3 are also necessary. Finally, Section 6 indicates some applications of the results obtained to a study of single operators in Banach spaces.

2. Group representations

In this section, we present our main results concerning group representability of bounded cosine functions. We show that the representability problem can be reduced to a question from harmonic analysis. Being of independent interest, this question will be the subject of a detailed analysis in the subsequent sections. We begin our considerations by introducing some notation and terminology.

Let G be an Abelian group. For subsets A and B of G, and an element a of G, we let

−A = {g ∈ G: − g ∈ A},
A + a = {g ∈ G: g − a ∈ A},
A + B = {g ∈ G: g = a + b for a ∈ A and b ∈ B}.

For n ∈ ℕ with n ≥ 2, we denote by m_{n} the homomorphism from G into itself given by

m_{n}(a) = na (a ∈ G).

We designate by G^{(n)} and G_{(n)} the image and kernel of m_{n}, respectively. If H is an Abelian group, we use G ≃ H to indicate that G and H are isomorphic. If G and H are locally compact, we write G ≅ H to indicate that G and H are topologically isomorphic. If H is a subgroup of G, then G/H denotes the corresponding quotient group. If G is locally compact and H is closed, then G/H will be assumed to have the quotient topology under which this quotient group is locally compact. If {G_{i}}_{i∈I} is an indexed collection of Abelian groups, we write ∏_{i∈I} G_{i} for the direct product of the G_{i}. If I = {1, …, n}, we also write G_{1} ×⋯×G_{n} in place of ∏_{i∈I} G_{i}. The subgroup of all sequences {x_{i}}_{i∈I} in ∏_{i∈I} G_{i} such that x_{i} = 0 for all but finitely many i ∈ I is written ∏_{i∈I}^{∗} G_{i} and is referred to as the weak direct product of the groups G_{i}. If 𝔪 is a cardinal number and if, for some fixed G, G_{i} = G for each i ∈ I, where I is a set of cardinality equal to 𝔪, we write G^{𝔪} for ∏_{i∈I} G_{i} and G^{𝔪∗} for ∏_{i∈I}^{∗} G_{i}. Direct products of locally compact Abelian groups will be assumed to have the product topology. We recall that, for an indexed collection {G_{i}}_{i∈I} of locally compact Abelian groups, ∏_{i∈I} G_{i} is a locally compact Abelian groups if and only if G_{i} is compact for all but finitely many i ∈ I.

Hereafter we shall assume that G is a locally compact Abelian group. We shall denote by 𝔅(G) the σ-algebra of all Borel subsets of G. If a ∈ G, then δ_{a} will stand for the Dirac measure on G concentrated at a. We denote by M(G) the space of all complex bounded regular Borel measures on G. With the norm ∥μ∥ = |μ|(G), where |μ| denotes the total variation of the measure μ ∈ M(G), M(G) is a Banach space. M(G) is also a Banach algebra with identity under the convolution multiplication

(μ ∗ ν)(E) = ∫_{G} μ(E − a) dν(a) (μ, ν ∈ M(G), E ∈ 𝔅(G))

and with δ_{0} as identity. We designate by M_{a}(G) the Banach algebra of all atomic measures in M(G). A measure μ ∈ M(G) will be termed even if

μ(E) = μ(−E) (E ∈ 𝔅(G)).

The even measures belonging to M(G) form a Banach algebra that will be denoted by M_{e}(G). M_{ae}(G) will signify the Banach algebra of all atomic measures in M_{e}(G).

There exists a natural cosine function C on G taking on values in M_{ae}(G) defined by

C(a) = \frac{1}{2}(δ_{a} + δ_{−a}) (a ∈ G).

C will be referred to as the basic cosine function on G.

A mapping G ∋ a ↦ ν_{a} ∈ M(G) will be called a homomorphism if

ν_{a} ∗ ν_{b} = ν_{a+b}  (2.1)

for any a, b ∈ G. A homomorphism G ∋ a ↦ ν_{a} ∈ M(G) will be said to be bounded if sup_{a∈G} ∥ν_{a}∥ < +∞. A homomorphism G ∋ a ↦ ν_{a} ∈ M_{ae}(G) will be termed a c-homomorphism (the “c” is for “cosine”) if

ν_{a} + ν_{−a} = δ_{a} + δ_{−a}  (2.2)

for each a ∈ G. A locally compact Abelian group that admits a c-homomorphism will be called a c-group. A locally compact Abelian group admitting a bounded c-homomorphism will be called a bc-group.

In the forthcoming discussion of group representability of bounded cosine functions, c-groups and bc-groups will play a crucial role. Prior to presenting details, we state two fundamental theorems about the shape of c-groups and bc-groups. Much of the paper will be devoted to proving these results.

Theorem 2.1. A locally compact Abelian group G is a c-group if and only if G^{(2)} is either a countable torsion group or a group isomorphic with ℤ × F, where F is a finite Abelian group.

Theorem 2.2. A locally compact Abelian group G is a bc-group if and only if G^{(2)} is finite.

Note that, by virtue of Theorems 2.1 and 2.2, the class of c-groups is essentially larger than that of bc-groups. In fact, the group ℤ is in the former class but not in the latter.

It is clear from the definition of a c-homomorphism that a given locally compact Abelian group G is a c-group if and only if the basic cosine function on G has a group representation. Similarly, a locally compact Abelian group G is a bc-group if and only if the basic cosine function on G has a bounded group representation. As we shall see shortly, these statements are particular cases of much stronger results (cf. Theorems 2.3, 2.4, and 2.5).

We start with a technicality.

Proposition 2.1. Let G be a locally compact Abelian group such that G^{(2)} is countable, and let A be a semitopological algebra with identity. Then every group representation of an A-valued continuous cosine function on G is regular.

Proof. Since G/G_{(2)} ≃ G^{(2)}, it follows that G/G_{(2)} is countable, and so, by Baire’s theorem for locally compact regular spaces (cf. [13, §5.28]), G/G_{(2)} is discrete. Thus G_{(2)} is an open subgroup of G. Let C: G → A be a continuous cosine function, and let G: G → A be a homomorphism satisfying (1.2). If a ∈ G_{(2)}, then a = −a, and so G(a) = C(a). Hence, since C is continuous, G is continuous on G_{(2)}. Now that G_{(2)} is an open neighbourhood of 0 in G and the multiplication operation of A is separately continuous, it is easy to see that G is continuous on the whole of G. □

We have the following fundamental result:

Theorem 2.3. Let A be a sequentially complete semitopological algebra with identity. If a locally compact Abelian group G is a c-group, then every A-valued bounded continuous cosine function on G has a regular group representation. Similarly, if a locally compact Abelian group G is a bc-group, then every A-valued bounded continuous cosine function on G has a bounded regular group representation.

Proof. Let G be a locally compact Abelian group admitting a c-homomorphism G ∋ a ↦ ν_{a} ∈ M_{ae}(G), and let C: G → A be a bounded continuous cosine function. For each a ∈ G, define an element G(a) of A by setting

G(a) = ∫_{G} C(g) dν_{a}(g).

Here, of course, the integral reduces to a series with (at most) countably many Avalued summands, and the series is convergent in view of the sequential completeness of A and the boundedness of C (we draw upon the standing assumption that the topology of A is determined by a family of pseudonorms and is Hausdorff). In view of (1.1) and the boundedness of C, the function G×G ∋ (g, h) ↦ C(g)C(h) ∈ A is bounded. Hence, since the multiplication operation of A is separately continuous, it follows that

G(a)G(b) = ∫_{G×G} C(g)C(h) d(ν_{a} ⊗ ν_{b})(g, h)

for any a, b ∈ G. Now, by (2.1) and the evenness of the ν_{a} (a ∈ G),

∫_{G×G} C(g)C(h) d(ν_{a} ⊗ ν_{b})(g, h) = \frac{1}{2} ∫_{G×G}(C(g + h) + C(g − h)) d(ν_{a} ⊗ ν_{b})(g, h)

= ∫_{G×G} C(g + h) d(ν_{a} ⊗ ν_{b})(g, h) = ∫_{G} C(g) d(ν_{a} ∗ ν_{b})(g) = G(a + b).

Thus G: a ↦ G(a) is a homomorphism from G into A. It is clear that G is bounded if the c-homomorphism a ↦ ν_{a} is bounded. In view of (2.2) and the evenness of C, G satisfies (1.2). Finally, by Theorems 2.1 and 2.2 (which will be established later without recourse to the theorem being proved), G^{(2)} is at most countable, and so, by Proposition 2.1, G is continuous. □

Given a locally compact Abelian group G, let C_{0}(G) be the space of all complex continuous functions on G vanishing at infinity, and let C_{u}(G) be the space of all complex bounded uniformly continuous functions on G. Let λ_{G} denote the Haar measure of G, let L^{∞}(G) be the space of all classes of complex essentially bounded λ_{G}-measurable functions on G, and let L^{1}(G) be the space of all classes of complex functions which are λ_{G}-integrable on G. Let C_{0e}(G), C_{ue}(G), L_{e}^{∞}(G), and L_{e}^{1}(G) be the spaces of all even elements of C_{0}(G), C_{u}(G), L^{∞}(G), and L^{1}(G), respectively. For a ∈ G, let T_{a} denote the operator of translation by a defined for functions by (T_{a}f)(b) = f(a + b) and for measures by T_{a}μ = μ ∗ δ_{−a}. It is obvious that each T_{a} defines a linear isometry of C_{0}(G), C_{u}(G), L^{∞}(G), L^{1}(G), and M(G). For a ∈ G, let

C(a) = \frac{1}{2}(T_{a} + T_{−a}).  (2.3)

Clearly, each C(a) defines a linear operator of unit norm on C_{0e}(G), C_{ue}(G), L_{e}^{∞}(G), L_{e}^{1}(G), and M_{e}(G), and moreover C: a ↦ C(a) is a cosine function in each of the spaces C_{0e}(G), C_{ue}(G), L_{e}^{∞}(G), L_{e}^{1}(G), and M_{e}(G). C is continuous in C_{0e}(G), C_{ue}(G), and L_{e}^{1}(G) if each of these spaces is equipped with the usual norm topology, and is also continuous in L_{e}^{∞}(G) and M_{e}(G) if each of the latter spaces is given the standard ∗-weak topology.

We are now in a position to state the following converse to Theorem 2.3:

Theorem 2.4. Let G be a locally compact Abelian group, and let C be the cosine function on G given by (2.3). Then G is a c-group (bc-group) if any of the following statements is valid:

(i) C: G → L(M_{e}(G)) has a (bounded) group representation; (ii) C: G → L(L_{e}^{1}(G)) has a (bounded) group representation; (iii) C: G → L(L_{e}^{∞}(G)) has a (bounded) group representation; (iv) C: G → L(C_{ue}(G)) has a (bounded) group representation;

(v) C: G → L(C_{0e}(G)) has a (bounded) group representation.

Proof. If we add to the above list the statement:

(vi) G is a c-group (bc-group),

then, to prove the theorem, it will suffice to establish the validity of the following chain of implications:

(i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) ⇒ (vi).

Below we establish successively each of the implications forming the above chain.

(i) ⇒ (ii). Let G be a representation of G in M_{e}(G) such that (1.2) holds for the L(M_{e}(G))-valued cosine function C. To prove that (ii) follows from (i), it suffices to show that L_{e}^{1}(G), regarded as a subspace of M_{e}(G), is an invariant subspace for all the G(a) (a ∈ G). Fix a ∈ G arbitrarily. Let f ∈ L_{e}^{1}(G), and let V be an open symmetric neighbourhood of 0 in G. If b ∈ V, then

∥C(b)G(a)f − G(a)f∥ = ∥G(a)(C(b)f − f)∥ ≤ ∥G(a)∥ ∥C(b)f − f∥. (2.4)

Hereafter, adhering to the standard convention, we shall identify any given element k of L^{1}(G) with the measure that is absolutely continuous with respect to λ_{G} and has k for density. Now, for each g ∈ L^{1}(G) and each h ∈ L_{e}^{1}(G), we have

g ∗ h = ∫_{G} h(b)C(b)g dλ_{G}(b),

where the right-hand side is to be interpreted as the Bochner integral of the function G ∋ b ↦ h(b)C(b)g ∈ L^{1}(G). In view of (2.4), for any non-negative λ_{G}-measurable even function φ on G with support in V such that ∫_{G} φ dλ_{G} = 1, we have

\begin{aligned}∥(G(a)f) ∗ φ − G(a)f∥ & = ∥∫_{G} φ(b)(C(b)G(a)f − G(a)f) dλ_{G}(b)∥ \\ & ≤ ∥G(a)∥ sup_{b∈V} ∥C(b)f − f∥. \end{aligned}  (2.5)

As L_{e}^{1}(G) is an ideal in M_{e}(G), (G(a)f) ∗ φ is an element of L_{e}^{1}(G). Clearly, sup_{b∈V} ∥C(b)f − f∥ tends to zero as V runs over a fundamental system of open symmetric neighbourhoods of 0 in G. Hence, in view of (2.5), G(a)f is the limit of a net in L_{e}^{1}(G) (with values of the form (G(a)f)∗φ). Since L_{e}^{1}(G) is closed in M_{e}(G), it follows that G(a)f belongs to L_{e}^{1}(G). Thus L_{e}^{1}(G) is an invariant subspace for all the G(a) (a ∈ G).

(ii) ⇒ (iii). Let G be a representation of G in L_{e}^{1}(G) such that (1.2) holds for the L(L_{e}^{1}(G))-valued cosine function C. For a bounded linear operator T on L_{e}^{1}(G), let T′ denote the dual of T acting on L_{e}^{∞}(G). Clearly, C′: a ↦ C(a)′ is a cosine function in L_{e}^{∞}(G), G′: a ↦ G(a)′ is a representation of G in L_{e}^{∞}(G) such that (1.2) holds, and G′ is bounded if and only G is bounded. Since C′ coincides with the L(L_{e}^{∞}(G))-valued cosine function C, we see that the latter cosine function has a group representation which may be assumed bounded if the L(L_{e}^{1}(G))-valued cosine function C has a bounded group representation.

(iii) ⇒ (iv). Let G be a representation of G in L_{e}^{∞}(G) such that (1.2) holds for the L(L_{e}^{∞}(G))-valued cosine function C. The implication (iii) ⇒ (iv) will follow once we show that C_{ue}(G), regarded as a subspace of L_{e}^{∞}(G), is an invariant subspace for all the G(a) (a ∈ G). To this end, we employ an argument analogous to the one used in establishing the implication (i) ⇒ (ii). We suitably change spaces and norms, and choose φ to be continuous so as to ensure that, for a ∈ G and f ∈ C_{ue}(G), (G(a)f) ∗ φ is in C_{ue}(G).

(iv) ⇒ (v). Let G be a representation of G in C_{ue}(G) such that (1.2) holds for the L(C_{ue}(G))-valued cosine function C. To prove that (iv) follows from (v), it suffices to show that C_{0e}(G), regarded as a subspace of C_{ue}(G), is an invariant subspace for all the G(a) (a ∈ G). For each a ∈ G, C_{0e}(G) ∋ f ↦ (G(a)f)(0) ∈ ℂ is a linear bounded functional on C_{0e}(G). By the Riesz representation theorem, there exists a unique ν_{a} ∈ M_{e}(G) such that

(G(a)f)(0) = ∫_{G} f dν_{a} (f ∈ C_{0e}(G)).

Clearly ∥ν_{a}∥ ≤ ∥G(a)∥. Recall that for f ∈ C_{0}(G) and μ ∈ M(G) the convolution f ∗ μ is the element of C_{0}(G) defined by

(f ∗ μ)(a) = ∫_{G} f(a − b) dμ(b) (a ∈ G).

Now, if f ∈ C_{0e}(G) and a, b ∈ G, then, since G(a)f and ν_{a} are even, we have

(G(a)f)(b) = (C(b)G(a)f)(0) = (G(a)C(b)f)(0) = (f ∗ ν_{a})(b).

Thus, for each a ∈ G, the mapping f ↦ f ∗ ν_{a} transforms C_{0e}(G) into itself, and so C_{0e}(G) is an invariant subspace for all the G(a) (a ∈ G).

(v) ⇒ (vi). Let G be a representation of G in C_{0e}(G) such that (1.2) holds for the L(C_{0e}(G))-valued cosine function C. By the argument from the previous paragraph, for each a ∈ G, there exists a unique ν_{a} ∈ M_{e}(G) such that

G(a)f = f ∗ ν_{a} (f ∈ C_{0e}(G))

and ∥ν_{a}∥ = ∥G(a)∥. Hence, if a, b ∈ G and f ∈ C_{0e}(G), then

f ∗ ν_{a+b} = G(a + b)f = G(a)G(b)f = f ∗ ν_{b} ∗ ν_{a}.

We thus see that (2.1) holds. Analogously, (1.2) gets translated into (2.2). Let π_{a} denote the mapping from M(G) onto M_{a}(G) that takes measures into their corresponding atomic parts. As is known, π_{a} is a contractive projection, is a homomorphism of convolution algebras, and maps M_{e}(G) onto M_{ae}(G). A moment’s reflection shows that G ∋ a ↦ π_{a}(ν_{a}) ∈ M_{ae}(G) is a c-homomorphism such that ∥π_{a}(ν_{a})∥ ≤ ∥G(a)∥ for each a ∈ G. Consequently, G is a c-group, and when G is bounded, G is in fact a bc-group. □

As an immediate consequence of Theorems 2.3 and 2.4, we obtain the following:

Theorem 2.5. Let G be a locally compact Abelian group. Consider the following conditions:

(i) for every sequentially complete semitopological algebra A with identity, any A-valued bounded continuous cosine function on G has a regular group representation;

(i’) for every Banach space E, any L(E)-valued bounded (strongly) continuous

cosine function on G has a regular group representation; (ii) for every sequentially complete semitopological algebra A with identity, any A-valued bounded continuous cosine function on G has a bounded regular group representation;

(ii’) for every Banach space E, any L(E)-valued bounded (strongly) continuous

cosine function on G has a bounded regular group representation. Then conditions (i) and (i’) are equivalent and are satisfied if and only if G is a c-group; similarly, conditions (ii) and (ii’) are equivalent and are satisfied if and only if G is a bc-group.

3. Characterising c-groups and bc-groups: sufficient conditions

The purpose of this section is to prove the sufficiency parts of Theorems 2.1 and 2.2.

Let G be a locally compact Abelian group. We denote by Ĝ the dual group of G. The pairing between elements of G and Ĝ will be indicated by (·, ·). Given a measure μ ∈ M(G), we denote by μ̂ the Fourier-Stieltjes transform of μ, that is,

μ̂(χ) = ∫_{G}(a, −χ) dμ(a) (χ ∈ Ĝ).

Let Φ(Ĝ) be the set of all odd functions φ on Ĝ having the following properties:

(i) φ(Ĝ \ (Ĝ)_{(2)}) ⊂ {−1, 1} and φ((Ĝ)_{(2)}) = {0}; (ii) for each a ∈ G, there exists μ_{a} ∈ M_{a}(G) such that

μ̂_{a}(χ) = φ(χ)(1 − (2a, −χ))  (χ ∈ Ĝ).  (3.1)

Note that if φ ∈ Φ(Ĝ) and Ĝ ≠ (Ĝ)_{(2)}, then φ(Ĝ \ (Ĝ)_{(2)}) = {−1, 1}.

Proposition 3.1. Let G be a locally compact Abelian group such that Φ(Ĝ) is nonvoid, and let φ be a function in Φ(Ĝ). Retaining the notation from the definition above, for each a ∈ G, put

ν_{a} = \frac{1}{2}(δ_{a} + δ_{−a} + μ_{a} ∗ δ_{−a}).  (3.2)

Then the mapping G ∋ a ↦ ν_{a} ∈ M_{ae}(G) is a c-homomorphism. If there exists μ ∈ M(G) such that μ̂ = φ, then that c-homomorphism is bounded.

Proof. By (3.1) and (3.2), for each a ∈ G and each χ ∈ Ĝ,

ν̂_{a}(χ) = \frac{1 + φ(χ)}{2}(a, χ) + \frac{1 − φ(χ)}{2}(a, −χ).

Taking into account that φ is an odd function mapping Ĝ \ (Ĝ)_{(2)} into {−1, 1} and (Ĝ)_{(2)} onto {0}, we see that, for each a ∈ G and each χ ∈ Ĝ,

ν̂_{a}(χ) = (a, ε(χ)χ),

where

ε(χ) = \begin{cases}φ(χ) & if χ ∈ Ĝ \ (Ĝ)_{(2)}, \\ 1 & if χ ∈ (Ĝ)_{(2)}. \end{cases}

From this it follows that, for each a ∈ G, ν_{a} is even and the mapping a ↦ ν_{a} is a c-homomorphism of G.

If φ = μ̂ for some μ ∈ M_{a}(G), then, by (3.1), for each a ∈ G we have μ_{a} = μ − μ ∗ δ_{2a}, whence ∥μ_{a}∥ ≤ 2∥μ∥. Thus, on account of (3.2), the c-homomorphism a ↦ ν_{a} is bounded. The result follows. □

Proposition 3.2. The set Φ(ℤ̂) is non-empty.

Proof. Denoting by 𝕋 the multiplicative group of complex numbers with unit modulus, let φ be the function on 𝕋 given by

φ(e^{iθ}) = \begin{cases}1 & if 0 < θ < π, \\ 0 & if either θ = 0 or θ = π, \\ −1 & if π < θ < 2π. \end{cases}

It is readily seen that φ has the Fourier expansion of the form

φ(e^{iθ}) ∼ \frac{4}{π} ∑_{k=0}^{∞} \frac{sin(2k + 1)θ}{2k + 1} = \frac{2}{πi} ∑_{k∈ℤ} \frac{1}{2k + 1}e^{i(2k+1)θ}.

Hence, if a ∈ ℤ, then

φ(e^{iθ})(1 − e^{−2iaθ}) ∼ \frac{2}{iπ} (∑_{k∈ℤ} \frac{1}{2k + 1}e^{i(2k+1)θ} − ∑_{k∈ℤ} \frac{1}{2k + 1}e^{i(2(k−a)+1)θ})

= \frac{2}{iπ} ∑_{k∈ℤ} (\frac{1}{2k + 1} − \frac{1}{2(k + a) + 1}) e^{i(2k+1)θ}

= \frac{4a}{iπ} ∑_{k∈ℤ} \frac{1}{(2k + 1)(2(k + a) + 1)}e^{i(2k+1)θ}.

For each a ∈ ℤ, let μ_{a} be the element of the space l^{1}(ℤ) of all complex summable sequences on ℤ defined by

μ_{a}(n) = \begin{cases}0 & if n ≡ 0 (mod 2), \\ \frac{4a}{iπn(n + 2a)} & if n ≡ 1 (mod 2). \end{cases}  (3.3)

Of course, l^{1}(ℤ) can be identified with M_{a}(ℤ), and so each μ_{a} (a ∈ ℤ) can be viewed as an atomic measure on ℤ. Identifying in a standard way ℤ̂ with 𝕋 and applying the Fourier inversion theorem, we obtain, for each a ∈ ℤ and each 0 ≤ θ < 2π,

μ̂_{a}(e^{iθ}) = φ(e^{iθ})(1 − e^{−2iaθ}).

Hence φ ∈ Φ(ℤ̂). □

If G is a locally compact Abelian group, then by G_{d} we mean G with the discrete topology. By bG we denote the Bohr compactification of G.

Proposition 3.3. Let G be a locally compact Abelian group. If Φ((G_{d}) ) is nonempty, then Φ(Ĝ) is also non-empty.

Proof. Let φ ∈ Φ((G_{d}) ). Since (G_{d}) ≅ b(Ĝ) and since Ĝ can canonically be embedded into b(Ĝ) (as a Borel subgroup; cf. [11]), we may meaningfully speak about the restriction φ_{|Ĝ} of φ to Ĝ. Clearly, φ_{|Ĝ} is an odd function mapping Ĝ \ (Ĝ)_{(2)} into {−1, 1} and (Ĝ)_{(2)} onto {0}. Since M_{a}(G_{d}) can be identified with M_{a}(G), we see that, for each a ∈ G, there is μ_{a} ∈ M_{a}(G) such that (3.1) holds with φ replaced by φ_{|Ĝ}. Hence φ_{|Ĝ} ∈ Φ(Ĝ). □

For a subgroup H of a locally compact Abelian group G, we denote by H^{⊥} the annihilator of H in Ĝ, that is, the closed subgroup of Ĝ defined as

H^{⊥} = {χ ∈ Ĝ: (a, χ) = 1 for all a ∈ H}.

Recall that if H is closed, then (G/H) ≅ H^{⊥} and Ĥ ≅ Ĝ/H^{⊥}. Note also that if H_{1} and H_{2} are closed subgroups of G such that H_{1} ⊂ H_{2}, then (H_{2}/H_{1}) ≅ H_{1}^{⊥}/H_{2}^{⊥}. Indeed, G/H_{2} ≅ (G/H_{1})/(H_{2}/H_{1}), and so H_{2}^{⊥} is topologically isomorphic with the annihilator of H_{2}/H_{1} in (G/H_{1}) , or equivalently with the annihilator of H_{2}/H_{1} in H_{1}^{⊥}, which immediately implies the desired relation.

Proposition 3.4. Let G be a locally compact Abelian group. Suppose that G contains a closed subgroup H satisfying H^{(2)} = G^{(2)} and such that Φ(Ĥ) is non-empty. Then Φ(Ĝ) is non-empty.

Proof. Let ψ ∈ Φ(Ĥ). Since Ĝ/H^{⊥} ≅ Ĥ, we may assume that ψ is a function on Ĝ/H^{⊥}. Let π be the canonical homomorphism from Ĝ onto Ĝ/H^{⊥}. We have

π^{−1}((Ĝ/H^{⊥})_{(2)}) = (Ĝ)_{(2)}.  (3.4)

Indeed, a moment’s reflection shows that an element χ of Ĝ belongs to π^{−1}((Ĝ/H^{⊥})_{(2)}) if and only if 2χ ∈ H^{⊥} or, equivalently, if χ ∈ (H^{(2)})^{⊥}. Since H^{(2)} = G^{(2)}, the latter condition is satisfied if and only if χ ∈ (G^{(2)})^{⊥}, which in turn is equivalent to χ ∈ (Ĝ)_{(2)}.

Let φ = ψ ∘ π. Taking into account (3.4), we find that φ is an odd function on Ĝ such that φ(G \ (Ĝ)_{(2)}) ⊂ {−1, 1} and φ((Ĝ)_{(2)}) = {0}. Given h ∈ H, let η_{h} be a measure in M_{a}(H) such that

η̂_{h}(γ) = ψ(γ)(1 − (2h, −γ))

for each γ ∈ Ĝ/H^{⊥}, and let ν_{h} be the unique measure in M_{a}(G), concentrated on H, whose restriction to H coincides with η_{h}. Then, for each h ∈ H and each χ ∈ Ĝ,

ν̂_{h}(χ) = φ(χ)(1 − (2h, −χ))  (3.5)

(cf. [13, §31.46]). Now, given a ∈ G, choose h ∈ H so that 2a = 2h and set μ_{a} = ν_{h}. Using (3.5), one verifies at once that (3.1) holds for each a ∈ G. Consequently, φ is a member of Φ(Ĝ). □

Proposition 3.5. Let G be a locally compact Abelian group such that Φ(Ĝ) is nonempty, and let K be a locally compact Abelian group such that K^{(2)} is a singleton. Then Φ((G × K) ) is non-empty.

Proof. The result follows immediately from the foregoing proposition upon taking G × {0} for H. □

For any function f on a group G, we denote by If the function on G given by

If(a) = f(−a) (a ∈ G),

and, for any Borel measure μ on G, we designate by Iμ the Borel measure on G given by

Iμ(E) = μ(−E) (E ∈ 𝔅(G)).

Proposition 3.6. Let G be a countable torsion Abelian group with the discrete topology. Then Φ(Ĝ) is non-empty.

Proof. Let {F_{n}}_{n∈ℕ} be a collection of finite subgroups of G such that F_{n} ⊂ F_{n+1} for each n ∈ ℕ and ⋃_{n∈ℕ} F_{n} = G. Then {F_{n}^{⊥}}_{n∈ℕ} is a collection of closed subgroups of Ĝ such that F_{n}^{⊥} ⊃ F_{n+1}^{⊥} for each n ∈ ℕ and ⋂_{n∈ℕ} F_{n}^{⊥} = {0}. Given n ∈ ℕ, F̂_{n} is topologically isomorphic with Ĝ/F_{n}^{⊥} and has the same cardinality as F_{n}. It follows that F_{n}^{⊥} has finite index in Ĝ, and is therefore an open subgroup of Ĝ. Taking into account that, for each χ ∈ Ĝ, χ and −χ belong to the same coset of F_{1}^{⊥} if and only if χ ∈ m_{2}^{−1}(F_{1}^{⊥}), one can easily define a function f_{1} : Ĝ → {−1, 0, 1} such that:

(i) f_{1}(Ĝ \ m_{2}^{−1}(F_{1}^{⊥})) ⊂ {−1, 1} and f_{1}(m_{2}^{−1}(F_{1}^{⊥})) = {0}; (ii) f_{1} is constant on the cosets of F_{1}^{⊥} in Ĝ; (iii) f_{1} is odd.

Since F_{2}^{⊥} is a subgroup of F_{1}^{⊥} of finite index (for F_{1}^{⊥}/F_{2}^{⊥} ≅ (F_{2}/F_{1}) ), one can now define a function f_{2} : Ĝ → {−1, 0, 1} such that:

(i) f_{2}(Ĝ \ m_{2}^{−1}(F_{2}^{⊥})) ⊂ {−1, 1} and f_{2}(m_{2}^{−1}(F_{2}^{⊥})) = {0}; (ii) f_{2} is constant on the cosets of F_{2}^{⊥} in Ĝ; (iii) f_{2} is odd; (iv) f_{2} = f_{1} on Ĝ \ m_{2}^{−1}(F_{1}^{⊥}).

Continuing the process, we obtain a sequence {f_{n}}_{n∈ℕ} of functions from Ĝ into {−1, 0, 1} such that for each n ∈ ℕ:

(i) f_{n}(Ĝ \ m_{2}^{−1}(F_{n}^{⊥})) ⊂ {−1, 1} and f_{n}(m_{2}^{−1}(F_{n}^{⊥})) = {0}; (ii) f_{n} is constant on the cosets of F_{n}^{⊥} in Ĝ; (iii) f_{n} is odd; (iv) f_{n+1} = f_{n} on Ĝ \ m_{2}^{−1}(F_{n}^{⊥}).

For each n ∈ ℕ, f_{n} can be written as f_{n} = g_{n} ∘ π_{n}, where π_{n} is the canonical homomorphism from Ĝ onto Ĝ/F_{n}^{⊥}, and g_{n} is a unique function on Ĝ/F_{n}^{⊥}. Since Ĝ/F_{n}^{⊥} ≅ F̂_{n} and since F_{n} is finite, ĝ_{n} can be identified with an element of M_{a}(F_{n}). Let ν_{n} be the unique measure in M_{a}(G), concentrated on F_{n}, such that the restriction of ν_{n} to F_{n} coincides with Iĝ_{n}. Applying the Fourier inversion theorem, it is easy to see that f_{n} = ν̂_{n}.

Let

V = ⋃_{n∈ℕ} f_{n}^{−1}({1}).

As all the f_{n} (n ∈ ℕ) are odd, we have

−V = ⋃_{n∈ℕ} f_{n}^{−1}({−1}),

and so V ∩ (−V) = ∅. Moreover,

Ĝ \ (V ∪ (−V)) = ⋂_{n∈ℕ} f_{n}^{−1}({0}) = ⋂_{n∈ℕ} m_{2}^{−1}(F_{n}^{⊥}) = m_{2}^{−1} ( ⋂_{n∈ℕ} F_{n}^{⊥}) = (Ĝ)_{(2)},

showing that V ∪ (−V) = Ĝ \ (Ĝ)_{(2)}. With this information about V, we can now define a function φ on Ĝ as follows:

φ(χ) = \begin{cases}1 & if χ ∈ V, \\ 0 & if χ ∈ (Ĝ)_{(2)}, \\ −1 & if χ ∈ −V. \end{cases}

It is clear that, for each n ∈ ℕ, φ = f_{n} on Ĝ \ m_{2}^{−1}(F_{n}^{⊥}).

If a ∈ G, then a ∈ F_{n} for some n ∈ ℕ. We shall verify that (3.1) holds if we take ν_{n} − ν_{n} ∗ δ_{2a} for μ_{a}, and thereby shall show that φ ∈ Φ(Ĝ). To this end, note that

μ̂_{a}(χ) = f_{n}(χ)(1 − (2a, −χ))  (3.6)

for each χ ∈ Ĝ. Now, if χ ∈ Ĝ \ m_{2}^{−1}(F_{n}^{⊥}), then φ(χ) = f_{n}(χ); this together with (3.6) yields (3.1) for χ ∈ Ĝ \ m_{2}^{−1}(F_{n}^{⊥}). On the other hand, if χ ∈ m_{2}^{−1}(F_{n}^{⊥}), then (2a, −χ) = (−a, 2χ) = 1, and hence

f_{n}(χ)(1 − (2a, −χ)) = φ(χ)(1 − (2a, −χ)) = 0;

this, by (3.6), implies (3.1) for χ ∈ Ĝ \ m_{2}^{−1}(F_{n}^{⊥}). The result follows. □

Proposition 3.7. Let G be a locally compact Abelian group such that G^{(2)} is a countable torsion group. Then Φ(Ĝ) is non-empty.

Proof. By virtue of Proposition 3.3, we may assume (without loss of generality) that the topology of G is discrete. Since G/G_{(2)} ≃ G^{(2)}, it follows that G/G_{(2)} is countable. Let S be a complete set of representatives modulo G_{(2)}, and let H be the smallest subgroup of G containing S. Clearly, S has the same cardinality as G/G_{(2)}, so it is countable. Accordingly, H is countable too. It is evident that H^{(2)} = G^{(2)}. Since G^{(2)} is a torsion group, so too is H. Now, by Proposition 3.6, Φ(Ĥ) is non-empty and, by Proposition 3.4, also Φ(Ĝ) is non-empty. □

Proposition 3.8. Let G be a locally compact Abelian group such that Φ(Ĝ) is non-empty, and let F be a finite Abelian group. Then Φ((G × F) ) is non-empty.

Proof. Let φ_{1} ∈ Φ(Ĝ), let φ_{2} be the characteristic function of (F̂)_{(2)} defined on F̂, and let φ_{3} be an odd function on F̂ mapping F̂ \ (F̂)_{(2)} into {−1, 1} and (F̂)_{(2)} onto {0}. Let μ be the measure in M_{a}(F), concentrated on F^{(2)}, whose restriction to F^{(2)} coincides with the Haar measure of F^{(2)}. Since (F̂)_{(2)} = (F^{(2)})^{⊥}, it follows that

φ_{2} = μ̂.  (3.7)

Moreover, for each y ∈ F and each β ∈ F̂,

φ_{2}(β) = φ_{2}(β)(2y, −β).  (3.8)

Obviously, since F is finite, there exists ρ ∈ M_{a}(F) such that

φ_{3} = ρ̂.  (3.9)

For each χ = (α, β) ∈ Ĝ × F̂, let

φ(χ) = φ_{1}(α)φ_{2}(β) + φ_{3}(β).

If a = (x, y) ∈ G × F, then 2a = (2x, 2y) and, by (3.8),

φ(χ)(1 − (2a, −χ))

= φ_{1}(α)(1 − (2x, −α))φ_{2}(β) + φ_{3}(β) − (2x, −α)φ_{3}(β)(2y, −β)). (3.10)

For each x ∈ G, let μ_{x} be the measure in M_{a}(G) such that

μ̂_{x}(α) = φ_{1}(α)(1 − (2x, −α)) (α ∈ Ĝ).

Comparing the latter equality with (3.7), (3.9), and (3.10), and identifying in a standard way (G×F) with Ĝ×F̂, we see that the function χ ↦ φ(χ)(1−(2a, −χ)) is the Fourier transform of the measure ν_{a} given by

ν_{a} = μ_{x} ⊗ μ + δ_{0} ⊗ ρ − δ_{2x} ⊗ (ρ ∗ δ_{2y}).

Now, if β ∉ (F̂)_{(2)}, then φ_{2}(β) = 0 and hence φ(χ) = φ_{3}(β). Similarly, if β ∈ (F̂)_{(2)}, then φ_{2}(β) = 1 and φ_{3}(β) = 0, and so φ(χ) = φ_{1}(α). These observations combined with the fact that χ ∈ (Ĝ × F̂)_{(2)} if and only if α ∈ (Ĝ)_{(2)} and β ∈ F̂_{(2)} show that φ is an odd function mapping Ĝ × F̂ \ (Ĝ × F̂)_{(2)} into {−1, 1} and (Ĝ × F̂)_{(2)} onto {0}. Consequently, φ is an element of Φ((G × F) ). □

For each n ∈ ℕ, we denote by ℤ(n) the cyclic group of order n.

Proposition 3.9. Let G be an Abelian group such that G^{(2)} is finite. Then G ≃ ℤ(2)^{𝔪∗} × F, where 𝔪 is a cardinal number and F is a finite Abelian group.

Proof. Clearly, G is of bounded order. Hence, by a theorem of Prüfer-Baer [17, 1] (see also [13, §A.25] and [10, Theorem 17.2]), G is isomorphic with ∏_{i∈I}^{∗} ℤ(p_{i}^{r_{i}}), where only finitely many distinct primes p_{i} and positive integers r_{i} occur. Since G^{(2)} is finite, among the factors ℤ(p_{i}^{r_{i}}) there are only finitely many ones with p_{i} = 2 and r_{i} > 1, and with p_{i} > 2. The proposition follows. □

Note that, for any Abelian group G, G_{(2)} is isomorphic with ℤ(2)^{𝔪∗}, where 𝔪 is a cardinal number. This follows immediately from G_{(2)} being in a natural way a vector space over the field ℤ(2). One might for a moment think that perhaps Proposition 3.9 can be established by showing that G is a direct product of G_{(2)} and a group isomorphic with G^{(2)}. Taking ℤ(4) for G shows, however, that G_{(2)} may fail to be a direct factor of G.

Theorem 3.1. Let G be a locally compact Abelian group such that G^{(2)} is either a countable torsion group or a group isomorphic with ℤ × F, where F is a finite Abelian group. Then Φ(Ĝ) is non-empty.

Proof. In view of Proposition 3.3, we may assume that G is discrete. If G^{(2)} is a countable torsion group, then the theorem immediately follows upon applying Proposition 3.7. Suppose then that G^{(2)} ≃ ℤ×F, where F is a finite Abelian group. Let π be the canonical homomorphism from G onto G/G_{(2)}. Identifying G^{(2)} with G/G_{(2)} and F with a suitable subgroup of G^{(2)}, we find that G/π^{−1}(F) ≃ ℤ. By a theorem on direct factorisation (cf. [13, §25.30(a)]), π^{−1}(F) is a direct factor of G, and so G ≃ ℤ × π^{−1}(F). Clearly, (π^{−1}(F))^{(2)} is isomorphic with F, and hence it is finite. By Proposition 3.9, π^{−1}(F) ≃ ℤ(2)^{𝔪∗} × F′, where 𝔪 is a cardinal number and F′ is a finite Abelian group. Now the theorem follows upon applying Propositions 3.2, 3.5 and 3.8. □

As an immediate consequence of Proposition 3.1 and Theorem 3.1, we obtain the sufficiency part of Theorem 2.1 which for clarity we state below as a separate proposition.

Proposition 3.10. Let G be a locally compact Abelian group G such that G^{(2)} is either a countable torsion group or a group isomorphic with ℤ × F, where F is a finite Abelian group. Then G is a c-group.

For a locally compact Abelian group G, we denote by A_{a}(Ĝ) the space of all Fourier transforms of measures in M_{a}(G).

Theorem 3.2. Let G be a locally compact Abelian group such that G^{(2)} is finite. Then Φ(Ĝ) ∩ A_{a}(Ĝ) is non-empty.

Proof. By Proposition 3.9, G is isomorphic with ℤ(2)^{𝔪∗} × F, where 𝔪 is a cardinal number and F is a finite Abelian group. Let ν be a measure in M_{a}(F) such that ν̂ is an odd function mapping F̂ \ (F̂)_{(2)} into {−1, 1} and (F̂)_{(2)} onto {0}. Set μ = δ_{0} ⊗ ν. Then μ is an element of M_{a}(G) and it is easy to verify that μ̂ belongs to Φ(Ĝ) ∩ A_{a}(Ĝ). □

As before, we see that Proposition 3.1 and Theorem 3.2 imply the sufficiency part of Theorem 2.2 which can be stated in the following form:

Proposition 3.11. Let G be a locally compact Abelian group such that G^{(2)} is finite. Then G is a bc-group.

4. Decomposable groups

In this section, we single out a class of locally compact Abelian groups each member of which is decomposable in a certain sense, and characterise compact groups in this class. This characterisation will be of direct relevance in the subsequent section.

A locally compact Abelian group G will be called decomposable if there exists an open subset U of G such that U ∪ (−U) = G \ G_{(2)} and U ∩ (−U) = ∅.

Proposition 4.1. Any decomposable connected compact Abelian group different from a singleton is topologically isomorphic with 𝕋.

Proof. Let G be a decomposable connected compact Abelian group different from a singleton. By the connectedness of G, Ĝ is torsion free (cf. [13, §24.25]). Let {χ_{i}}_{i∈I} be a maximal indexed collection of independent elements of Ĝ, and let 𝔪 be the cardinality of the index set I. As is known, 𝔪 does not depend on the particular choice of the maximal family of independent elements of Ĝ and defines the so-called torsion-free rank of Ĝ. By the maximality of {χ_{i}}_{i∈I}, for each χ ∈ Ĝ, there exist integers n(χ) and n_{i}(χ) (i ∈ I) such that n_{i}(χ) ≠ 0 for only finitely many i ∈ I, and n(χ)χ = ∑_{i∈I} n_{i}(χ)χ_{i}. By the independency of the χ_{i} (i ∈ I), n(χ) can be taken to be non-zero so that—in particular—for each i ∈ I the rational number n_{i}(χ)/n(χ) makes sense; moreover, this number depends only on χ. One verifies at once that, for each i ∈ I, the function ρ_{i}: χ ↦ n_{i}(χ)/n(χ) is a homomorphism from Ĝ into the group ℚ of rational numbers. We claim that, for each i ∈ I, ρ_{i}(Ĝ) is isomorphic with ℤ.

In fact, fixing i ∈ I arbitrarily and, for a given homomorphism f, letting ker f denote the kernel of f, observe first that (ρ_{i}(Ĝ)) ≅ (ker ρ_{i})^{⊥}, where the annihilator

is taken in ˆ Ĝ identified with G. Being a subgroup of G, (ker ρ_{i})^{⊥} is decomposable. Let U be an open subset of (ker ρ_{i})^{⊥} such that U ∪(−U) = (ker ρ_{i})^{⊥} \((ker ρ_{i})^{⊥})_{(2)} and U ∩ (−U) = ∅. Since ρ_{i}(Ĝ) ⊂ ℚ, ρ_{i}(Ĝ) can be regarded as a subgroup of ℝ. Let α_{i} be the embedding of ρ_{i}(Ĝ) into ℝ, and let α̂_{i} be the dual homomorphism from ℝ̂ (≅ ℝ) onto a dense subgroup of (ker ρ_{i})^{⊥} given by

(r, α̂_{i}(t)) = e^{itα_{i}(r)} (r ∈ ρ_{i}(Ĝ), t ∈ ℝ̂).

Let S be the closure of α̂_{i}((0, +∞)) in (ker ρ_{i})^{⊥}. Clearly, S is a compact cancellative semigroup. By an elementary result from the theory of topological semigroups (cf. [13, §9.16] and [2, Theorem 1.10]), S is a group. Hence S contains α̂_{i}(ℝ̂), and, since α̂_{i}(ℝ̂) is dense in (ker ρ_{i})^{⊥}, S coincides with (ker ρ_{i})^{⊥}. Now, proceeding by reductio ad absurdum, suppose that ρ_{i}(Ĝ) is not isomorphic with ℤ. Then ρ_{i}(Ĝ) is a dense subgroup of ℝ, and so α̂_{i} is one-to-one. Consequently, α̂_{i}^{−1}(((ker ρ_{i})^{⊥})_{(2)}) = (α̂_{i}^{−1}((ker ρ_{i})^{⊥}))_{(2)} = {0} so that α̂_{i}^{−1}(U) and α̂_{i}^{−1}(−U) are disjoint, open and closed subsets of ℝ \ {0} whose union coincides with ℝ \ {0}. From this it follows that α̂_{i}^{−1}(U) coincides either with (0, +∞) or with (−∞, 0). Substituting −U for U if necessary, we may assume that α̂_{i}^{−1}(U) = (0, +∞). Then α̂_{i}((0, +∞)) ⊂ U, and so, if we let \overline{U} denote the closure of U in (ker ρ_{i})^{⊥}, we see that S ⊂ \overline{U}, whence \overline{U} = (ker ρ_{i})^{⊥}. On the other hand, it is clear that −U ⊂ (ker ρ_{i})^{⊥} \ \overline{U}. This contradiction establishes the claim.

To complete the proof, it suffices to show that 𝔪 = 1. Suppose the contrary. Let i_{1} and i_{2} be two distinct elements of I. Let ρ: Ĝ → ℚ^{2} be the homomorphism defined by ρ = (ρ_{i_{1}}, ρ_{i_{2}}). Then, in view of the fact that χ_{i_{1}} and χ_{i_{2}} are independent and both ρ_{i_{1}}(Ĝ) and ρ_{i_{2}}(Ĝ) are isomorphic with ℤ, we see that ρ(Ĝ) is isomorphic with ℤ^{2}. Since (ρ(Ĝ)) ≅ (ker ρ)^{⊥}, it follows that (ker ρ)^{⊥} ≅ 𝕋^{2}. On the other hand, (ker ρ)^{⊥} as a subgroup of G is decomposable. But clearly 𝕋^{2} is not decomposable, as 𝕋^{2}\(𝕋^{2})_{(2)} = 𝕋^{2}\{(1, 1), (−1, 1), (1, −1), (−1, −1)} is connected. This contradiction completes the proof. □

Proposition 4.2. Any decomposable compact Abelian group that is not totally disconnected is topologically isomorphic with 𝕋 × ℤ(2)^{𝔪} × F, where 𝔪 is a cardinal number and F is a finite Abelian group.

Proof. Let G be a decomposable compact Abelian group that is not totally disconnected. Let G_{0} be the component of 0 in G. Since G is not totally disconnected, G_{0} is not a singleton. Being a closed subgroup of G, G_{0} is compact and decomposable. By Proposition 4.1, G_{0} is topologically isomorphic with 𝕋. Now it follows from a theorem on direct factorisation (cf. [13, §25.31(a)]) that G ≅ G_{0} × (G/G_{0}). The proof will be complete once we show that G/G_{0} ≅ ℤ(2)^{𝔪} × F for some cardinal number 𝔪 and some finite Abelian group F. We first show that (G/G_{0})^{(2)} is finite.

Suppose, on the contrary, that (G/G_{0})^{(2)} is infinite. Then, since (G/G_{0})^{(2)} ≃ (G/G_{0})/((G/G_{0})_{(2)}), (G/G_{0})_{(2)} is a subgroup of G/G_{0} of infinite index. Since G/G_{0} is compact and (G/G_{0})_{(2)} is closed in G/G_{0}, it follows that (G/G_{0})_{(2)} is a subset of G/G_{0} of null Haar measure. Consequently, there is a net {x_{α}}_{α∈A} in (G/G_{0})\(G/G_{0})_{(2)} such that lim_{A} x_{α} = 0. Let U be an open subset of G such that U ∪(−U) = G\G_{(2)} and U ∩(−U) = ∅. Since, for each α ∈ A, G_{0} +x_{α} is connected and does not intersect G_{(2)}, it follows that either G_{0} + x_{α} ⊂ U or G_{0} + x_{α} ⊂ −U. Substituting −x_{α} for x_{α} if necessary, we may assume that G_{0} + x_{α} ⊂ U for each α ∈ A. Since x_{α} → 0 as α tends to infinity along A, we see that G_{0} is contained in the intersection of the closures in G of U and −U, and as such it is a subset of G_{(2)}. But the latter is incompatible with the fact that G_{0} is isomorphic with 𝕋. This contradiction establishes the finiteness of (G/G_{0})^{(2)}.

If H is a compact group, then H^{(2)} is a closed subgroup of H, and (H^{(2)}) ≅ Ĥ/(H^{(2)})^{⊥}. But (H^{(2)})^{⊥} = (Ĥ)_{(2)}, and Ĥ/(Ĥ)_{(2)} ≃ (Ĥ)^{(2)}. Thus (H^{(2)}) ≃ (Ĥ)^{(2)}. Now, if H^{(2)} is finite, then, since H^{(2)} and (H^{(2)}) have equal cardinalities, (Ĥ)^{(2)} is also finite. Applying these observations to H = G/G_{0} and using Proposition 3.9, we see that (G/G_{0}) ≃ ℤ(2)^{𝔪∗} ×F, where 𝔪 is a cardinal number and F is a finite Abelian group. By duality, it follows that G/G_{0} is topologically isomorphic with ℤ(2)^{𝔪} × F̂. □

Proposition 4.3. If G is a decomposable totally disconnected compact Abelian group, then (Ĝ)^{(2)} is countable.

Proof. Let G be a decomposable totally disconnected compact Abelian group. Proceeding by reductio ad absurdum, suppose that (Ĝ)^{(2)} is uncountable. Let U be an open subset of G such that U ∪ (−U) = G \ G_{(2)} and U ∩ (−U) = ∅. Since G is totally disconnected, it has a basis 𝔙 of neighbourhoods of 0 consisting of compact open subgroups of G (cf. [13, §7.7]). Choose a_{1} ∈ U. Then a_{1} + V_{1} ⊂ U for some V_{1} ∈ 𝔙. Note that V_{1} \ G_{(2)} ≠ ∅. Indeed, otherwise V_{1} would be contained in G_{(2)} so that G_{(2)} would be an open subgroup of G, and, since G is compact, G_{(2)} would have finite index in G. Now, since G/G_{(2)} ≃ G^{(2)}, G^{(2)} would be finite. But then, as we saw while establishing the foregoing proposition, (Ĝ)^{(2)} would also be finite, a contradiction. Let a_{2} ∈ V_{1} \ G_{(2)}. Passing if necessary to −a_{2}, we may assume that a_{2} ∈ V_{1} ∩ U. Now, since V_{1} ∩ U is open, there exists V_{2} ∈ 𝔙 such that V_{2} ⊂ V_{1} and a_{2} + V_{2} ⊂ V_{1} ∩ U. Continuing the process, we obtain a sequence {V_{n}}_{n∈ℕ} in 𝔙 and a sequence {a_{n}}_{n∈ℕ} in G such that, for each n ∈ ℕ,

V_{n+1} ⊂ V_{n}  (4.1)

and

a_{n+1} + V_{n+1} ⊂ V_{n} ∩ U.  (4.2)

Since 0 ∈ V_{n}, we have G_{(2)} ⊂ V_{n} + G_{(2)} for each n ∈ ℕ, and so

G_{(2)} ⊂ ⋂_{n∈ℕ}(V_{n} + G_{(2)}).

We claim that

( ⋂_{n∈ℕ}(V_{n} + G_{(2)})) \ G_{(2)} ≠ ∅.

Indeed, otherwise {V_{n} + G_{(2)}}_{n∈ℕ} can serve as a basis of neighbourhoods of 0 in G/G_{(2)}, and so G/G_{(2)} is metrisable. It is easy to see that G/G_{(2)} is topologically isomorphic with G^{(2)}. In fact, since G is compact, G/G_{(2)} is also compact. The mapping m_{2} : G → G^{(2)} is continuous, and so, by a fundamental property of the quotient topology, the induced isomorphism m_{2}^{∗}: G/G_{(2)} → G^{(2)} is also continuous. Now, in view of the compactness of G/G_{(2)}, m_{2}^{∗} is a homeomorphism. Having established the relation G/G_{(2)} ≅ G^{(2)}, we now infer that G^{(2)} itself is metrisable. Since G^{(2)} is also compact, (G^{(2)}) is countable. But, as we saw while proving the foregoing proposition, (G^{(2)}) is isomorphic with (Ĝ)^{(2)}. Thus (Ĝ)^{(2)} is countable. This contradiction establishes the claim.

Let U be a non-free ultrafilter on ℕ, and let a_{∞} = lim_{n→U} a_{n} (cf. [4, §11.8]; we resort to the notion of a limit with respect to an ultrafilter in order to be able to handle simultaneously and in a smooth way several accumulation points of sequences; the use of ultrafilters and respective limits is a mere technicality and can be avoided at the expense of a lengthening of the argument). In view of (4.1) and the fact that, for each n ∈ ℕ, V_{n} is closed and a_{n+1} ∈ V_{n}, it follows that

a_{∞} ∈ ⋂_{n∈ℕ} V_{n}.  (4.3)

We now show that, by modifying the sequence {a_{n}}_{n∈ℕ} if necessary, we may always assume that a_{∞} ∉ G_{(2)}.

Suppose that initially a_{∞} ∈ G_{(2)}. Let

b ∈ ( ⋂_{n∈ℕ}(V_{n} + G_{(2)})) \ G_{(2)}.

For each n ∈ ℕ, let v_{n} ∈ V_{n} and g_{n} ∈ G_{(2)} be such that b = v_{n} + g_{n}, and set a′_{n} = a_{n} +v_{n}. Since v_{n+1} +V_{n+1} ⊂ V_{n+1}, it follows from (4.2) that a′_{n+1} +V_{n+1} ⊂ V_{n} ∩U, and so {a′_{n}}_{n∈ℕ} satisfies (4.2). Let a′_{∞} = lim_{n→U} a′_{n} and g_{∞} = lim_{n→U} g_{n}. Clearly, a′_{∞} = a_{∞} + b − g_{∞}. Since both a_{∞} and g_{∞} belong to G_{(2)}, we have 2a′_{∞} = 2b, and further, since b ∉ G_{(2)}, we see that a′_{∞} ∉ G_{(2)}. Thus {a′_{n}}_{n∈ℕ} is a desired modification.

Assuming now—as we may—that a_{∞} ∉ G_{(2)}, let Ũ be that of the sets U and −U which contains a_{∞}. Let W ∈ 𝔙 be such that

a_{∞} + W ⊂ Ũ.  (4.4)

Choose W′ ∈ 𝔙 so that W′ + W′ ⊂ W. Since a_{∞} is a cluster point of {a_{n}}_{n∈ℕ}, there exist k_{1}, k_{2} ∈ ℕ such that k_{1} < k_{2} and a_{k_{i}} − a_{∞} ∈ W′ (i = 1, 2). We see that both a_{k_{1}} − a_{k_{2}} and a_{k_{2}} − a_{k_{1}} belong to W, and, on account of (4.4), we have

a_{∞} + a_{k_{1}} − a_{k_{2}} ∈ Ũ  (4.5)

and

a_{∞} + a_{k_{2}} − a_{k_{1}} ∈ Ũ,

the latter relation implying that

a_{k_{1}} − a_{∞} − a_{k_{2}} ∈ −Ũ.  (4.6)

On the other hand, by (4.1) and (4.2), a_{k_{2}} ∈ V_{k_{2}−1} ⊂ V_{k_{1}}, and, by (4.3), a_{∞} ∈ V_{k_{1}}. Since V_{k_{1}} is a group, it follows that a_{∞} − a_{k_{2}} ∈ V_{k_{1}} and −a_{∞} − a_{k_{2}} ∈ V_{k_{1}}. Now, in view of (4.2), we have

a_{k_{1}} + a_{∞} − a_{k_{2}} ∈ U

and

a_{k_{1}} − a_{∞} − a_{k_{2}} ∈ U.

But these last relations are incompatible with (4.5) and (4.6). This contradiction establishes the proposition. □

Now we are in a position to state the main conclusion of this section.

Theorem 4.1. Let G be a compact Abelian group. Then G is decomposable if and only if either (Ĝ)^{(2)} is a countable torsion group or G is topologically isomorphic with 𝕋 × ℤ(2)^{𝔪} × F, where 𝔪 is a cardinal number and F is a finite Abelian group.

Proof. Necessity follows from Propositions 4.2 and 4.3.

To prove sufficiency, suppose that G is such that either (Ĝ)^{(2)} is a countable torsion group or G is topologically isomorphic with 𝕋 × ℤ(2)^{𝔪} × F, where 𝔪 is a cardinal number and F is a finite Abelian group. Note that in the latter case (Ĝ)^{(2)} ≃ ℤ×(F̂)^{(2)}. By Theorem 3.1, in either case the set Φ(G) is non-empty. Let φ ∈ Φ(G) and U = φ^{−1}({1}). Then, clearly, U is an open subset of G such that U ∪ (−U) = G \ G_{(2)} and U ∩ (−U) = ∅. The result follows. □

5. Characterising c-groups and bc-groups: necessary conditions

The aim of this section is to prove the necessity parts of Theorems 2.1 and 2.2. We shall formulate these parts as separate propositions.

Proposition 5.1. If a locally compact Abelian group G is a c-group, then G^{(2)} is either a countable torsion group or a group isomorphic with ℤ × F, where F is a finite Abelian group.

Proof. Let G ∋ a ↦ ν_{a} ∈ M_{ae}(G) be a c-homomorphism of G. Identifying M_{ae}(G) with M_{ae}(G_{d}) and taking account of (2.1), we see that, for every χ ∈ (G_{d}) ≅ b(Ĝ), the function a ↦ ν̂_{a}(χ) is a homomorphism from G_{d} into the multiplicative group ℂ \ {⊬}. Since, by (2.2), for each a ∈ G,

ν̂_{a}(χ) + (ν̂_{a}(χ))^{−1} = (a, χ) + (a, −χ)  (5.1)

and the right-hand side of this equality has modulus no greater than 2, it follows that the homomorphism a ↦ ν̂_{a}(χ) takes on values in a bounded subgroup of ℂ \ {⊬}, and hence is a character of G_{d}. Moreover, (5.1) and the uniqueness of Fourier expansion ensure that, for each χ ∈ b(Ĝ), the character a ↦ ν̂_{a} takes the form

ν̂_{a}(χ) = (a, ϵ(χ)χ) (a ∈ G)

for some function ϵ on b(Ĝ) with values in {−1, 1}. Observe that, in view of the last equality and evenness of the ν_{a} (a ∈ G), the restriction of ϵ to b(Ĝ) \ (b(Ĝ))_{(2)} is odd. Let φ(χ) be the function on b(Ĝ) defined by

φ(χ) = \begin{cases}ϵ(χ) & if χ ∈ b(Ĝ) \ (b(Ĝ))_{(2)}, \\ 0 & if χ ∈ (b(Ĝ))_{(2)}. \end{cases}  (5.2)

For each a ∈ G, let μ_{a} = 2ν_{a} ∗ δ_{a} − δ_{2a} − δ_{0}. Direct computation shows that

μ̂_{a}(χ) = φ(χ)(1 − (2a, −χ))  (a ∈ G, χ ∈ b(Ĝ)).  (5.3)

If χ ∈ b(Ĝ) \ (b(Ĝ))_{(2)}, then there exist a ∈ G and an open neighbourhood V of χ such that (2a, −γ) ≠ 1 for all γ ∈ V. From this and (5.3) it follows that the restriction of φ to b(Ĝ) \ (b(Ĝ))_{(2)} is continuous. Let U = φ^{−1}({1}). Since b(Ĝ)\(b(Ĝ))_{(2)} is an open subset of b(Ĝ) and the restriction of φ to b(Ĝ)\(b(Ĝ))_{(2)} is odd and continuous, it follows that U is an open subset of b(Ĝ) such that U ∪(−U) = b(Ĝ)\(b(Ĝ))_{(2)} and U ∩(−U) = ∅. By Theorem 4.1 and the fact that (b(Ĝ)) ≅ G_{d}, either (G_{d})^{(2)} is a countable torsion group or b(Ĝ) is topologically isomorphic with 𝕋 × ℤ(2)^{𝔪} × F, where 𝔪 is a cardinal number and F is a finite Abelian group, in the latter case G_{d} being isomorphic with ℤ × ℤ(2)^{𝔪∗} × F̂. Hence G^{(2)} is either a countable torsion group or a group isomorphic with ℤ × (F̂)^{(2)}. □

Formulated in a different manner, our next result has already appeared in [3]. The proof given below is much simpler than the corresponding proof in [3] (namely, that accompanying Théorème 3.2).

Proposition 5.2. If a locally compact Abelian group G is a bc-group, then G^{(2)} is finite.

Proof. Let G be a locally compact Abelian group admitting a bounded c-homo-morphism G ∋ a ↦ ν_{a} ∈ M_{ae}(G). Let AP (G) be the space of all complex almost periodic functions on G, and let l^{∞}(G) be the space of all complex bounded functions on G. Let m be a Banach mean on l^{∞}(G), that is, a bounded linear functional on l^{∞}(G) satisfying the following conditions:

(i) ∥m∥ = 1 = m(1);

(ii) m(T_{a}f) = m(f) for each f ∈ l^{∞}(G) and each a ∈ G.

The existence of such an invariant mean is ensured by a theorem of Day [5] (see also [13, §17.5] and [12, Theorem 1.2.1]). For each a ∈ G, set μ_{a} = 2ν_{a} ∗ δ_{a} − δ_{2a} − δ_{0}. Since, of course, sup_{a∈G} ∥μ_{a}∥ < +∞, the mapping

AP (G) ∋ f ↦ m_{a} (∫_{G} f dμ_{a}) ∈ ℂ

is a well-defined functional on AP (G); here the subscript a in m_{a} indicates that the action of m refers to the dummy variable a. Since the space AP (G) can canonically be identified with the space of all complex continuous functions on bG, the above functional can be identified with a uniquely determined measure μ in M(bG) such that

μ̂(χ) = m_{a}(μ̂_{a}(χ))  (5.4)

for each χ ∈ (bG) ≅ (Ĝ)_{d}. Repeating the argument from the proof to the preceding proposition, we see that there exists an odd function φ on Ĝ such that φ(Ĝ \ (Ĝ)_{(2)}) ⊂ {−1, 1}, φ((Ĝ)_{(2)}) = {0}, and

μ̂_{a}(χ) = φ(χ)(1 − (2a, −χ)) (a ∈ G, χ ∈ Ĝ).

Comparing the last equality with (5.4), we see that

μ̂(χ) = φ(χ)m_{a}(1 − (2a, −χ))

for each χ ∈ Ĝ. Taking into account that

m_{a}((2a, −χ)) = \begin{cases}1 & if χ ∈ (Ĝ)_{(2)}, \\ 0 & if χ ∈ Ĝ \ (Ĝ)_{(2)}, \end{cases}

we find that μ̂(χ) = φ(χ) for each χ ∈ Ĝ. Accordingly, the measure μ is odd and

(μ ∗ μ) = (μ̂)^{2} = 1_{Ĝ\(Ĝ)_{(2)}}.  (5.5)

If we identify the Haar measure λ_{(bG)^{(2)}} with a suitable measure in M(bG) concentrated on (bG)^{(2)}, then clearly λ̂_{(bG)^{(2)}} = 1_{(Ĝ)_{(2)}}. This identity together with (5.5) shows that

μ ∗ μ = δ_{0} − λ_{(bG)^{(2)}}.  (5.6)

Now, proceeding by reductio ad absurdum, suppose that the group G^{(2)} is infinite. Then the compact group (bG)^{(2)} is also infinite and, correspondingly, the Haar measure λ_{(bG)^{(2)}} is continuous. Passing to atomic parts in both sides of (5.6), we see that π_{a}(μ) ∗ π_{a}(μ) = δ_{0}. Thus |(π_{a}(μ)) (0)| = 1. On the other hand, since μ is odd, so too is π_{a}(μ), and hence (π_{a}(μ)) (0) = 0. This contradiction completes the proof. □

6. Concluding remarks

The aim of this final section is to present some results contributing to the theory of single operators in Banach spaces. These results will be consequences of the fact that ℤ is a c-group but not a bc-group.

For n ∈ ℕ, let T_{n}(X) denote the Chebychev polynomial of degree n defined as

T_{n}(X) = ∑_{k=0}^{[n/2]} \binom{n}{2k}X^{n−2k}(X^{2} − 1)^{k},

where [a] designates the integral part of the real number a. Set T_{0}(X) = 1 and T_{n}(X) = T_{−n}(X) for each negative integer n. It is straightforward to verify that if A is an algebra with identity, then for each a ∈ A the function ℤ ∋ n ↦ T_{n}(a) ∈ A is an A-valued cosine function on ℤ, and conversely any cosine function C: ℤ → A assumes the form

C(n) = T_{n}(a) (n ∈ ℤ)

for some a ∈ A (it suffices to take C(1) for a). By Theorems 2.1 and 2.3, we have the following:

Theorem 6.1. Let A be a sequentially complete semitopological algebra with identity. If a ∈ A is such that {T_{n}(a)}_{n∈ℕ} is bounded, then there exists an invertible element b of A such that

T_{n}(a) = \frac{1}{2}(b^{n} + b^{−n})  (6.1)

for each n ∈ ℕ.

Inspection of the proofs of Theorems 2.1 and 2.3 (with special attention paid to (3.2) and (3.3)) reveals that (6.1) holds with b equal to

\frac{2i}{π}e + a − \frac{4i}{π} ∑_{k=1}^{∞} \frac{1}{4k^{2} − 1}T_{2k}(a),

where e denotes, of course, the identity of A.

It follows from Theorems 2.2 and 2.4 that there exist Banach spaces E, composed of even sequences indexed by ℤ, and there exist elements a of L(E) satisfying ∥T_{n}(a)∥ = 1 for each n ∈ ℕ such that if an invertible element b of L(E) satisfies (6.1), then sup_{n∈ℤ} ∥b^{n}∥ = +∞. As, by virtue of Theorem 6.1, for any such a there is an invertible element b of L(E) satisfying (6.1), we see that the following holds true:

Theorem 6.2. There exist Banach spaces E and invertible elements b of the corresponding algebras L(E) satisfying ∥b^{n} + b^{−n}∥ = 2 for each n ∈ ℕ and having the following property: If c is an invertible element of L(E) such that

b^{n} + b^{−n} = c^{n} + c^{−n}  (6.2)

for each n ∈ ℕ, then sup_{n∈ℤ} ∥c^{n}∥ = +∞.

It should be stressed that none of the spaces E in the above theorem are isomorphic to a Hilbert space. Indeed, as mentioned in the Introduction, if H is a Hilbert space and b ∈ L(H) is invertible and satisfies sup_{n∈ℕ} ∥b^{n} + b^{−n}∥ < +∞, then there exists an invertible element c of L(H) with sup_{n∈ℤ} ∥c^{n}∥ < +∞ for which (6.2) holds. By a theorem of Wermer [19] (see also [6, Lemma XV.6.1]), such c must then necessarily be similar to a unitary operator.

Acknowledgment

The author would like to thank Professor Horst Leptin and an anonymous referee for valuable suggestions which resulted in an improvement of the manuscript.

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Instytut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland

   E-mail address: wojtekch@appli.mimuw.edu.pl

Department of Computer Science, University of Adelaide, Adelaide, SA 5005, Australia

   E-mail address: wojtek@cs.adelaide.edu.au