A Wiener Lemma for the discrete Heisenberg group: Invertibility criteria and applications to algebraic dynamics

This article contains a Wiener Lemma for the convolution algebra $\ell^1(\mathbb H,\mathbb C)$ and group $C^\ast$-algebra $C^\ast(\mathbb H)$ of the discrete Heisenberg group $\mathbb H$. At first, a short review of Wiener's Lemma in its classical form and general results about invertibility in group algebras of nilpotent groups will be presented. The known literature on this topic suggests that invertibility investigations in the group algebras of $\mathbb H$ rely on the complete knowledge of $\widehat{\mathbb H}$ -- the dual of $\mathbb H$, i.e., the space of unitary equivalence classes of irreducible unitary representations. We will describe the dual of ${\mathbb H}$ explicitly and discuss its structure. Wiener's Lemma provides a convenient condition to verify invertibility in $\ell^1(\mathbb H,\mathbb C)$ and $C^\ast(\mathbb H)$ which bypasses $\widehat{\mathbb H}$. The proof of Wiener's Lemma for $\mathbb H$ relies on local principles and can be generalised to countable nilpotent groups. As our analysis shows, the main representation theoretical objects to study invertibility in group algebras of nilpotent groups are the corresponding primitive ideal spaces. Wiener's Lemma for $\mathbb H$ has interesting applications in algebraic dynamics and Time-Frequency Analysis which will be presented in this article as well.


Motivation
Let Γ be a countably infinite discrete group. The aim of this article is to find a verifiable criterion -a Wiener Lemma -for invertibility in the group algebra 1 (Γ, C) := (f γ ) γ∈Γ : in particular for the case where Γ is the discrete Heisenberg group H.
Our main motivation to study this problem is an application in the field of algebraic dynamics which we introduce first. An algebraic Γ-action is a homomorphism α : Γ −→ Aut(X) from Γ to the group of automorphisms of a compact metrisable abelian group X [34].
We are especially interested in principal actions which are defined as follows. Let f be an element in the integer group ring Z[Γ], i.e., the ring of functions Γ −→ Z with finite support.
The Pontryagin dual of the discrete abelian group Z[Γ]/ Z[Γ]f will be denoted by X f ⊆ T Γ , where T = R/Z (which will be identified with the unit interval (0, 1]). Pontryagin's duality theory of locally compact abelian groups tells us that X f can be identified with the annihilator of the principal left ideal Z[Γ]f , i.e., The left shift-action λ on T Γ is defined by (λ γ x) γ = x γ −1 γ for every x ∈ T Γ and γ, γ ∈ Γ.
Denote by α f the restriction of λ on T Γ to X f . The pair (X f , α f ) forms an algebraic dynamical system which we call principal Γ-action -because it is defined by a principal ideal (cf. (1.1)).
Since a principal Γ-action (X f , α f ) is completely determined by an element f ∈ Z[Γ], one should be able to express its dynamical properties in terms of properties of f . Expansiveness is such a dynamical property which allows a nice algebraic interpretation. Let (X, α) be an algebraic dynamical system and d a translation invariant metric on X. The Γ-action α is expansive if there exists a constant ε > 0 such that sup γ∈Γ d(α γ x, α γ y) > ε , for all pairs of distinct elements x, y ∈ X. We know from [8,Theorem 3.2] that (X f , α f ) is expansive if and only if f is invertible in 1 (Γ, R). This result was proved already in the special cases Γ = Z d and for groups Γ which are nilpotent in [34] and in [9], respectively. Although, this result is a complete characterisation of expansiveness, it is in general hard to check whether f is invertible in 1 (Γ, R) or not.
1.1. Outline of the article. In Section 2 we will recall known criteria for invertibility in symmetric unital Banach algebras A. The most important result links invertibility investigations in A to the representation theory of A. More precisely, the existence of an inverse a −1 of a ∈ A is equivalent to the invertibility of the operators π(a) for every irreducible unitary representation π of A. The representation theory of H is unmanageable as we will demonstrate in Section 3.
Theorem 3.8 -Wiener's Lemma for the discrete Heisenberg group -is the main result of this paper and allows one to restrict the attention to certain 'nice' and canonical irreducible representations for questions concerning invertibility in the group algebra of the discrete Heisenberg group H. The proof of Theorem 3.8 can be found in Section 4. Moreover, as will be shown in Section 4 as well, invertibility of f ∈ Z[H] in 1 (H, R) can be verified with the help of the finite-dimensional irreducible unitary representations of H.
In Section 5 we generalise Theorem 3.8 to countable discrete nilpotent groups Γ. This result says that an element a in C * (Γ) is invertible if and only if for every primitive ideal I of C * (Γ) the projection of a onto the quotient space C * (Γ)/I is invertible. As we will see, the primitive ideal space is more accessible than the space of irreducible representations and easy to determine. Moreover, this Wiener Lemma for nilpotent groups can be converted to a statement about invertibility of evaluations of irreducible monomial representations.
In Section 6 we will explore a connection to Time-Frequency Analysis. Allan's local principle (cf. Section 4) directly links localisations of 1 (H, C) to twisted convolution algebras and hence, the representations of H and the relevant representation theory in the field of Time-Frequency Analysis coincide. In order to highlight this connection even more, Time-Frequency Analysis might be interpreted as the Fourier theory on the discrete Heisenberg group H; due to the striking similarities to the Fourier Analysis of the additive group Z and its group algebras. Moreover, we give an alternative proof of Wiener's Lemma for twisted convolution algebras, which only uses the representation theory of H. Theorem 6.8 -which is based on a result of Linnell (cf. [25]) -gives a full description of the spectrum of the operators π(f ) acting on L 2 (R, C), where π is a Stone-von Neumann representation (cf. (6.1) for a definition) and f ∈ Z[H].
Section 7 contains applications of Theorem 3.8 and Wiener's Lemma for twisted convolution algebras, in particular, conditions for non-invertibility for 'linear' elements in f ∈ Z[H].

Invertibility in group algebras and Wiener's Lemma: A review
In this section we review known conditions for invertibility in group algebras of nilpotent groups Γ. First of all we refer to the article [14] by K. Gröchenig for a modern survey of Wiener's Lemma and its variations. Gröchenig's survey focuses on two main topics, namely on invertibility of convolution operators on p -spaces (cf. Subsection 2.2 and in particular Theorem 2.8) and inverse-closedness. Moreover, Gröchenig explains how these topics are related to questions on invertibility in Time-Frequency analysis and invertibility in group algebras. Although, Wiener's Lemma for convolution operators is stated here as well it will play an insignificant role in the rest of the paper. However, we would like to bring the reader's attention to Theorem 2.9 which is yet another result which relates invertibility in 1 (Γ, C) to invertibility of convolution operators. This result is completely independent of Theorem 2.8 and holds in much greater generality.
In this review we will explain why a detailed understanding of the space of irreducible representations of a nilpotent group Γ is of importance for invertibility investigations in the group algebras of Γ. Furthermore, we will present Gelfand's results on invertibility in commutative Banach algebras in the form of local principles; which will be discussed in greater detail in later sections of this article.
We start the discussion with Wiener's Lemma in its classical form. Let us denote by A(T) the Banach algebra of functions with absolutely convergent Fourier series on T. Before we start our review of more general results let us mention the concept of inverseclosedness which originates from Wiener's Lemma as well. The convolution algebra 1 (Z, C) is isomorphic to A(T) and hence 1 (Z, C) can be embedded in the larger Banach algebra of continuous functions C(T, C) in a natural way. The fact that F ∈ A(T) is invertible in A(T) if and only if F is invertible in C(T, C) leads to the question: for which pairs of nested unital Banach algebras A, B with A ⊆ B and with the same multiplicative identity element does the following implication hold: In the literature a pair of Banach algebras which fulfils (2.1) is called a Wiener pair.
Wiener's Lemma was the starting point of Gelfand's study of invertibility in commutative Banach algebras. Gelfand's theory links the question of invertibility in a commutative Banach algebra A to the study of its irreducible representations and the compact space of maximal ideals Max(A). We collect in the following theorem several criteria for invertibility in unital commutative Banach algebras.
Theorem 2.2 (cf. [11]). Suppose A is a unital commutative Banach algebra. The set of irreducible representations of A is isomorphic to the compact space of maximal ideals Max(A). Furthermore, the following statements are equivalent The main goal of this article is to prove that similar results hold for group algebras of nilpotent groups and, in particular, for the discrete Heisenberg group.
In this article we concentrate on the harmonic analysis of rings associated with a countably infinite group Γ furnished with the discrete topology. Beside Z[Γ] and 1 (Γ, C) we are interested in C * (Γ), the group-C * -algebra of Γ, i.e., the enveloping C * -algebra of 1 (Γ, C).
Let ∞ (Γ, C) be the space of bounded complex-valued maps. We write a typical element The product of f ∈ 1 (Γ, C) and g ∈ ∞ (Γ, C) is given by convolution For 1 ≤ p < ∞ we set 2.1. Representation theory. We recall at this point some relevant definitions and results from representation theory, which will be used later. Moreover, we will state results for symmetric Banach- * -algebras which are in the spirit of Wiener's Lemma.
Unitary Representations. Let H be a complex Hilbert space with inner product ·, · . We denote by B(H) the algebra of bounded linear operators on H, furnished with the strong operator topology. Further, denote by U(H) ⊂ B(H) the group of unitary operators on H. If Γ is a countable group, a unitary representation π of Γ is a homomorphism γ → π(γ) from Γ into U(H) for some complex Hilbert space H. Every unitary representation π of Γ extends to a * -representation of 1 (Γ, C), which is again denoted by π, and which is given by the formula Clearly, π(f * ) = π(f ) * . The following theorem was probably first published in [12] but we refer to [30,Theorem 12 • the class of non-degenerate 1 * -representations of 1 (Γ, C); • the class of non-degenerate * -representations of C * (Γ).
Moreover, these bijections respect unitary equivalence and irreducibility.
Hence the representation theories of Γ, 1 (Γ, C) and C * (Γ) coincide. In consideration of this result we will use the same symbol for a unitary representation of Γ and its corresponding * -representations of the group algebras 1 (Γ, C) and C * (Γ).
States and the GNS construction. Suppose that A is a unital C * -algebra. A positive linear functional φ : A −→ C is a state if φ(1 A ) = 1. We denote by S(A) the space of states of A, which is a weak * -compact convex subset of the dual space of A. The extreme points of S(A) are called pure states.
A representation π of A is cyclic if there exists a vector v ∈ H π such that the set {π(a)v : a ∈ A} is dense in H π , in which case v is called a cyclic vector. The Gelfand-Naimark-Segal (GNS) construction links the cyclic representations of A and the states of A in the following way. If π is a cyclic representation with a cyclic unit vector v, then φ π,v , defined by for every a ∈ A, is a state of A. If π is irreducible, then φ π,v is a pure state. Moreover, for every state φ of A there is a cyclic representation (π φ , H φ ) and a cyclic unit vector v φ ∈ H φ such that φ(a) = π φ (a)v φ , v φ for every a ∈ A. The pure states of A correspond to irreducible representations of A (up to unitary equivalence) via the GNS construction. 1 A representation π of a Banach * -algebra A is called non-degenerate if there is no non-zero vector v ∈ Hπ such that π(a)v = 0 for every a ∈ A. A von Neumann algebra N is a * -subalgebra of bounded operators on some Hilbert space H which fulfils N = (N ) . The von Neumann algebra N π generated by a unitary representation π of a group Γ, is the smallest von Neumann algebra which contains π(Γ).
We call a representation π a factor if N π ∩ N π = C ·1 B(Hπ) . A group is of Type I if every factor representation is a direct sum of copies of an irreducible representation.
Induced and monomial representations. Let H be a subgroup of a countably infinite group Γ. Suppose σ is a unitary representation of H with representation space H σ . A natural way to extend the representation σ of H to a representation of Γ is as follows: consider the Hilbert space H Γ σ consisting of all maps F ∈ L 2 (Γ, H σ ) which satisfy Hence, Ind Γ H (σ) can be viewed as the right regular representation of Γ acting on the Hilbert space H Γ σ . This construction will become more transparent when we discuss specific examples below. A representation of Γ is called monomial if it is unitarily equivalent to a representation induced from a one-dimensional representation of a subgroup of Γ.
Theorem 2.4 ( [16]). If Γ is a nilpotent group of Type I, then all its irreducible representations are monomial.

2.2.
Symmetric Banach- * -algebras. Let A be a Banach algebra with multiplicative identity element 1 A . The spectrum of a ∈ A is the set of elements c ∈ C such that a − c1 A is not invertible in A and will be denoted by σ(a).
In order to study invertibility in 1 (Γ, C) and C * (Γ) in the non-abelian setting we will try to find criteria similar to those described in Theorem 2.2. For this purpose the following definition will play a key role. Typical examples of symmetric Banach-*-algebras are C * -algebras. We turn to the study of nilpotent groups and their associated group algebras.
The reason why it is convenient to restrict to the study of invertibility in symmetric unital Banach- * -algebra is demonstrated by the following theorems, which show similarities to Wiener's Lemma and Theorem 2.2, respectively.
For the class of symmetric group algebras one has the following important result on inverseclosedness.
Next we are discussing spectral invariance of convolution operators. It is a well known fact (cf. [14]) that invertibility of f ∈ 1 (Z, C) can be validated by studying invertibility of the convolution operator C f acting on the Hilbert space 2 (Z, C). Moreover, the spectrum of C f is independent of the domain, i.e., the spectrum of the operator C f : p (Z, C) −→ p (Z, C) is the same for all p ∈ [1, ∞]. As the following theorem shows, this result is true for a large class of groups, in particular, for all finitely generated nilpotent groups.
Let us now give a condition for invertibility of an element 1 (Γ, C), where Γ is an arbitrary discrete countably infinite group, in terms of the point spectrum of the corresponding convolution operator. Theorem 2.9 ([8, Theorem 3.2]). An element f ∈ 1 (Γ, C) is invertible in 1 This theorem says that it is enough to check if 0 is an eigenvalue of the left convolution operator C f : ∞ (Γ, C) −→ ∞ (Γ, C) in order to determine whether f is invertible or not (cf. (2.2)).
Finally, we present a condition for invertibility in a symmetric unital Banach- * -algebra A which links invertibility in A to its representation theory.
Theorem 2.10 ( [28]). An element a in a symmetric unital Banach- * -algebra A is not left invertible in A if and only if there exists a pure state φ with φ(a * a) = 0. Equivalently, a is not left invertible if and only if there exists an irreducible representation π of A and a unit vector u ∈ H π such that π(a)u = 0. This result should be compared with Gelfand's theory for commutative Banach algebras. Wiener's Lemma for 1 (Z, C) says that an element f ∈ 1 (Z, C) is invertible if and only if the Fourier-transform of f does not vanish on T, i.e., (Ff )(s) = 0 for all s ∈ T. 2 The Fourier-transform of f , evaluated at the point θ ∈ T, can be viewed as the evaluation of the one-dimensional irreducible unitary representation π θ : n → e 2πinθ of Z at f , i.e., (Ff )(θ) = n∈Z f n π θ (n) 1 = π θ (f )1 .
We will explain in the next section that it is not feasible to describe explicitly the space of unitary equivalence classes of irreducible representations of a non-Type I group. Hence, Theorem 2.10 seems to be of limited use for investigating invertibility of an element f ∈ 1 (Γ, C) for a non-Type I nilpotent group Γ. However, as we will see later, it is one of the key results for obtaining a Wiener Lemma for 1 (Γ, C).

The dual of the discrete Heisenberg group and a Wiener Lemma
In this section we explain how results from ergodic theory give insight into the space of irreducible representations of the discrete Heisenberg group, but that this space has no reasonable parametrisation and is therefore not useful for determining invertibility in the corresponding group algebras (cf. Theorem 2.10). At the end of this section, we will state our main result -a Wiener Lemma for the discrete Heisenberg group H -which allows one to restrict the attention to certain canonical representations of H which can be parametrised effectively and used for solving the invertibility problem.
3.1. The dual of a discrete group. Let Γ be a countable discrete group. Denote by Γ the dual of Γ, i.e., the set of all unitary equivalence classes of irreducible unitary representations of Γ.
Definition 3.1. Let A be a C * -algebra. A closed two-sided ideal I of A is primitive if there exists an irreducible representation π of A such that ker(π) = I. The set of primitive ideals of A is denoted by Prim(A).
Suppose that the group Γ is not of Type I. Then certain pathologies arise: • The map Γ −→ Prim(C * (Γ)) given by π → ker(π) is not injective. In other words, if π 1 , π 2 ∈ Γ, then ker(π 1 ) = ker(π 2 ) does not necessarily imply that π 1 and π 2 are unitarily equivalent. • Γ is not behaving nicely neither as a topological space nor as a measurable space in its natural topology or Borel structure, respectively (cf. [11,Chapter 7] for an overview).  The centre of H is generated by The elements x, y, z satisfy the following commutation relations The discrete Heisenberg group is nilpotent and hence amenable.
Since H does not possess an abelian normal subgroup of finite index it is not a group of Type I (cf. [35]), and hence the space of irreducible representations does not have any nice structure as discussed above. As we will show below, one can construct uncountably many unitarily inequivalent irreducible representations of H for every irrational θ ∈ T. These representations arise from certain singular measures on T. This fact is well-known to specialists, but details are not easily accessible in the literature. Since these results are important for our understanding of invertibility, we present this construction in some detail for the convenience of the reader. We would like to mention first that Moran announced in [27] a construction of unitary representations of H using the same approach as presented here. These results were not published as far as we know. Moreover, Brown [6] gave examples of unitary irreducible representations of the discrete Heisenberg group which are not monomial.
Let (X, B, µ) be a measure space, where X is a compact metric space, B is a Borel σalgebra, and µ a finite measure.
In [23] uncountably many inequivalent ergodic quasi-invariant measures for every irrational rotation of the circle were constructed. Later it was shown in [22] that a homeomorphism φ on a compact metric space X has uncountably many inequivalent non-atomic ergodic quasiinvariant measures if and only if φ has a recurrent point x, i.e., φ n (x) returns infinitely often to any punctured neighbourhood of x.
Let Z act on T via rotation by an irrational angle θ ∈ T.
Theorem 3.3. For each irrational θ ∈ T there is a bijection between the set of ergodic R θquasi-invariant probability measures on T and the set of irreducible representations π of H with π(z) = e 2πiθ .
We use the measures found in [23] to construct unitary irreducible representations of H.
for every F ∈ L 2 (T, µ) and t ∈ T. The operator T θ,µ is well-defined because of the quasiinvariance of µ. Consider also the unitary operator M µ defined by for every F ∈ L 2 (T, µ) and t ∈ T. We will show that the representation π θ,µ of H defined by Lemma 3.4. The unitary representation π θ,µ of H given by (3.5) is irreducible.
Proof. Every element in L 2 (T, µ) can be approximated by linear combinations of elements in the set , which commutes with all operators of the form M n µ , n ∈ Z, and hence with multiplication with any L ∞ -function, must be a multiplication operator, i.e., say. Denote by · op the operator norm, then which implies that G ∈ L ∞ (T, µ) (otherwise one would be able to find a measurable set B with positive measure on which G is strictly larger than O op , and the indicator function 1 B would lead to a contradiction with (3.6)). The ergodicity of µ with respect to R θ implies that only constant functions in L ∞ (T, µ) are R θ -invariant µ-a.e.. Hence, if O commutes with T θ,µ as well, then we can conclude that O is multiplication by a constant c ∈ C. By Schur's Lemma, the operators T θ,µ , M µ ∈ B(L 2 (T, µ)) define an irreducible representation π θ,µ of H.
Suppose that θ ∈ T is irrational, and that µ and ν are two ergodic R θ -quasi-invariant measures on T. Let π θ,µ and π θ,ν be the corresponding irreducible unitary representations constructed above.
Proof. Assume π θ,µ and π θ,ν are unitarily equivalent. Then there exists a unitary operator U : Denote multiplication by a function H ∈ C(T, C) by O H . The set of trigonometric polynomials, which is spanned by {M n µ 1 : n ∈ Z}, is dense in C(T, C). This implies that (3.7) holds for all H ∈ C(T, C), i.e., that UO H = O H U for any H ∈ C(T, C).
Since U is an isometry we get that where ·, · σ is the standard inner product on the Hilbert space L 2 (T, σ). Using the same argument for U −1 we get, for every H ∈ C(T, C), Define, for every positive finite measure σ on T, a linear functional Since Conversely, if µ and ν are equivalent, then the linear operator , is unitary and satisfies that Uπ θ,µ (γ) = π θ,ν (γ)U for every γ ∈ H.
In this way one obtains uncountably many inequivalent irreducible unitary representation of H for a given irrational rotation number θ ∈ T.
In fact, every irreducible unitary representation π of H with π(z) = e 2πiθ , θ irrational, is unitarily equivalent to π θ,µ for some probability measure µ on T which is quasi-invariant and ergodic with respect to an irrational circle rotation. For convenience of the reader we sketch a proof of this fact based on elementary spectral theory of unitary operators.
Let π be an irreducible unitary representation of H with representation space H π such that π(x)π(y) = e 2πiθ π(y)π(x). Let v ∈ H π be a normalised cyclic vector, put U = π(y) and denote by H v the closure of the subspace generated by {U n v : n ∈ Z}. The GNS-construction tells us that a n = U n v, v Hπ , n ∈ Z, forms a positive-definite sequence. Due to Herglotz' (or, more generally, Bochner's) representation theorem there exists a probability measure µ v on Z T whose Fourier-Stieltjes transform µ v fulfils One easily verifies that there exists an isometric isomorphism φ : consisting of multiplication by e 2πit . In other words, the unitary operators U v and M v are unitarily equivalent.
Put S = π(x) and consider the cyclic normalised vector w = Sv of the representation π. By replacing v by w in the construction above one can define the corresponding objects Lemma 3.6. The measures µ v and µ w are equivalent.
Proof. First note that U v and U w are unitarily equivalent. From this fact and the discussion preceding the lemma one concludes that M v and M w are unitarily equivalent as well, i.e., that there exists a unitary operator O : By arguing as in the first part of the proof of Lemma 3.5 one gets that O is a multiplication operator. Put G = O1 L 2 (T,µv) . Since O is an isometry one gets for all µ v -measurable sets B By repeating these arguments with v and w interchanged one concludes that µ v and µ w are equivalent.
Proof. Note that µ w (n) = S −1 U n Sv, v Hπ = e −2πiθn µ v (n) for every n ∈ Z. As one can easily verify, for every probability measure µ on T, multiplying µ with a character e −2πiθn is the same as the Fourier-Stieltjes transform of µ • R θ . Hence, we obtain that µ w = µ v • R θ . As µ v and µ w are equivalent, µ v is a R θ -quasi-invariant probability measure on T.
Completion of the proof of Theorem 3.3. The preceding discussion allows us to define an irreducible representation π v of H acting on L 2 (T, µ v ) which is unitarily equivalent to π. The evaluation of π v at y is given by M v and T v = π v (x) acts as composition of a translation operator by an angle θ and multiplication by some function D v ∈ L ∞ (T, µ v ). Due to the fact that T v has to be a unitary (and hence, an isometric) operator on Since π v is irreducible only those multiplication operators in B(L 2 (T, µ v )) which act via multiplication by a constant function c ∈ C will commute with multiples of the modified translation operator T v . This implies the ergodicity of µ v and completes the proof of the theorem.

3.3.
Wiener's Lemma for the discrete Heisenberg group. Theorem 2.10 states that in order to decide on invertibility of f ∈ 1 (H, C), one has to check invertibility of π(f ) for every irreducible representations π of H, and hence for every π θ,µ as above, where µ is a probability measure on T on T which is quasi-invariant and ergodic with respect to a circle rotation R θ .
The problem of deciding on invertibility of f ∈ 1 (H, C) via Theorem 2.10 becomes much more straightforward if one is able to restrict oneself to unitary representations arising from rotation invariant probability measures. This is exactly our main result.
Before formulating this result we write down the relevant representations explicitly. Take θ ∈ T, and consider the corresponding rotation R θ : T −→ T given by (3.2). If θ is irrational, the Lebesgue measure λ = λ T on T is the unique R θ -invariant probability measure, and the representation π θ,λ on L 2 (T, λ) defined in (3.5) is irreducible. One can modify this representation by setting, for every s, t ∈ T, Then π (s,t) θ is obviously again an irreducible unitary representation of H on H π (s,t) If θ is rational we write it as θ = p/q where p, q are integers with the properties 0 ≤ p < q and gcd(p, q) = 1 and consider the unitary representation π with s, t ∈ T, where I q−1 is the (q −1)×(q −1) identity matrix. Every R θ -invariant and ergodic probability measure µ on T is uniformly distributed on the set {t, 1/q+t, . . . , t+(q−1)/q} ⊂ T for some t ∈ T; if we denote this measure by µ t then µ t = µ t+k/q for every k = 0, . . . , q − 1.
With this notation at hand we can state our main result, the proof of which will be given in Section 4. The main advantage of Theorem 3.8 over Theorem 2.10 is that it is not necessary to check invertibility of π(f ) for every irreducible representation of H, but that one can restrict oneself for this purpose to the 'nice' part of the dual of the non-Type I group H. As we shall see later, one can make a further reduction if θ is irrational: in this case one only has to check invertibility of π θ (f ) = π

Wiener's Lemma for the discrete Heisenberg group: A proof and a first application
In this section we will present one possible proof of Wiener's Lemma for 1 (H, C) and C * (H). This proof has two main ingredients, namely: • Allan's local principle, which reduces the problem of verifying invertibility in 1 (H, C) and C * (H) to the study of invertibility in rotation algebras. • The fact that irrational rotation algebras are simple will eliminate the 'non-Type I problem' for questions about invertibility in 1 (H, C) and C * (H) discussed in the previous section.
These results will be generalised in Section 5 to group algebras of nilpotent groups.
4.1. Local principles. Let A be a unital Banach algebra and a ∈ A. Local principles are based on the following idea: one checks invertibility of projections of a onto certain quotient algebras of A in order to conclude from this information whether a is invertible or not. Therefore, the main task is to find a sufficient family J of ideals of A such that one can deduce the invertibility of a from the invertibility of the projections of a on A/I for all I ∈ J. Allan's local principle provides us with such a sufficient family of ideals in case the centre of A is large enough. We have used Allan's local principle already in [13] to study invertibility in 1 (H, C). However, in that paper we were not able to prove Theorem 3.8 with this approach.
Suppose A is a unital Banach algebra with non-trivial centre  We would like to mention already here that in Section 7 Allan's local principle will appear a second time and will link invertibility of f ∈ 1 (H, C) to the invertibility of the evaluations of Stone-von Neumann representations at f .
Let us now prove our main theorem. Since J θ = (z − e 2πiθ )C * (H) is a two-sided closed ideal we know that the quotient Q θ is a C * -algebra and hence symmetric for each θ ∈ T.
If θ is rational the irreducible unitary representations of H vanishing on J θ are given by (3.13) -(3.14) and were determined in [5]. Due to the fact that Q θ is symmetric we can apply Theorem 2.10 in order to study invertibility in Q θ via the representations (3.13) -(3.14). Now suppose θ is irrational. In order to study the representation theory of the C * -algebra Q θ we have to understand the link to one of the most studied non-commutative C * -algebras -the irrational rotation algebras.
We call a C * -algebra an irrational rotation algebra if it is generated by two unitaries U, V which fulfil the commutation relation for some irrational θ ∈ T. We already saw examples of irrational rotation algebras above, namely, the C * -subalgebras of B(L 2 (T, µ)) which are generated by M µ and T θ,µ , where µ is a R θ -quasi-invariant and ergodic measure. The reason why we call all C * -algebras which fulfil (4.2) irrational rotation algebras with parameter θ is the following striking result which can be found in [7, Theorem VI.1.4] (and which was already proved in the 1960s, cf. [32] for a list of references).
We will denote the irrational rotation algebra with parameter θ by R θ and will not distinguish between the different realisations of R θ because of the universal property described in Theorem 4.2. Let us further note that the proof of Theorem 4.2 is deduced from the simplicity of the universal irrational rotation algebra.
The C * -algebra Q θ is clearly a rotation algebra with parameter θ. The simplicity of R θ implies that J θ is a maximal two-sided ideal of C * (H). Hence, there exists an irreducible representation π of H such that ker(π) = J θ , since every two-sided maximal ideal is primitive (cf. [29,Theorem 4.1.9]). Moreover, all the irreducible representations π vanishing on J θ have the same kernel: otherwise we would get a violation of the maximality of J θ . These representations are not all in the same unitary equivalence class (as we saw in Section 3), which is an indication of the fact that H is not of Type I.
Proof of Theorem 3.8. First of all recall that 1 (H, C) is inverse-closed in C * (H). By applying Allan's local principle for C * (H) the problem of verifying invertibility in 1 (H, C) and C * (H) reduces to the study of invertibility in the C * -algebras Q θ , with θ ∈ T.
The rational case is trivial and was already treated at the beginning of the discussion. If θ is irrational, any irreducible representation π of H which vanishes on J θ can be used to check invertibility in Q θ . Indeed, since for an arbitrary unital C * -algebra A and an irreducible representation π of A, the C * -algebras π(A) and A/ ker(π) are isomorphic one gets π(C * (H)) C * (H)/ ker(π) = C * (H)/J θ = Q θ due to Theorem 4.2. In particular we may use the representations π (1,1) θ as in (3.12).
Remark 4.3. We should note here that for all realisations of the irrational rotation algebra the spectrum of a ∈ R θ is the same as a set. But this does not imply that an eigenvalue (or an element of the continuous spectrum) of a in one realisation is an eigenvalue (or an element of the continuous spectrum) of a in all the other realisations.     for every H ∈ L 2 (T, λ T ) and t ∈ T. For rational θ of the form θ = p/q with (p, q) = 1 we replace the Lebesgue measure λ = λ T in (4.3) by the uniform probability measure ν q concentrated on the cyclic group {1/q, . . . , (q − 1)/q, 1} ⊂ T.
Proof. One direction is obvious. For the converse, assume that α f is non-expansive, but that there exists a constant c > 0 such that π(f ) is invertible and π(f ) −1 ≤ c for every finitedimensional irreducible representation π of H. Since α f is non-expansive, there exists an irrational θ (by our assumption) such that the operator π (1,1) θ (f ) has no bounded inverse due to Theorem 3.8 and its proof. Therefore, π (1,1) θ (f ) is either not bounded from below or its range is not dense in the representation space or both.

Invertibility in group algebras of discrete nilpotent groups
In this section we aim to find more evident conditions for invertibility in group algebras for discrete countable nilpotent groups than the one given in Theorem 2.10. The main objects of our investigations are the primitive ideal space and the class of irreducible monomial representations of the group.

5.1.
Wiener's Lemma for nilpotent groups. Let Γ be a countable discrete nilpotent group. As we have seen earlier, 1 (Γ, C) is inverse-closed in C * (Γ). Hence we concentrate on the group C * -algebra C * (Γ).
In order to establish a Wiener Lemma in this more general setting we are first going to reinterpret Wiener's Lemma for the discrete Heisenberg group. From the discussion in Subsection 4.2 one can easily see that the irreducible unitary representations π (s,t) θ , θ, s, t ∈ T, those representations which correspond to ergodic R θ -invariant measures on T, generate the primitive ideal space Prim(C * (H)). Moreover, since π(C * (H)) C * (H)/ ker(π) for every π ∈ H the study of invertibility is directly linked to invertibility of projections onto the primitive ideals. We may interpret this as a localisation principle.
Before formulating a Wiener Lemma for an arbitrary discrete nilpotent group let us fix some notation. Let A be a unital C * -algebra. For every two-sided closed ideal J of A, denote by Φ J the canonical projection from A onto the C * -algebra A/J. This theorem links questions about invertibility in 1 (Γ, C) and C * (Γ) to their representation theory and, to be more specific, to the primitive ideal space Prim(C * (Γ)). At the same time this result provides us with a sufficient family of ideals in order to study invertibility and hence, Wiener's Lemma for nilpotent groups describes a localisation principle. We will learn in the next subsection that for discrete nilpotent groups Γ the class of irreducible representation which are induced by one-dimensional representations of subgroups of Γ provide us with an effective tool to generate Prim(C * (Γ)). In other words, it is a feasible task to determine the primitive ideal space Prim(C * (Γ)).
Theorem 5.1 can be generalised to all unital C * -algebras. Moreover, we provide a sufficient condition for a family of ideals in order to check invertibility via localisations.
Suppose I is a collection of ideals of a C * -algebra A, such that (i) every I ∈ I is closed and two-sided, (ii) for any primitive ideal J ∈ Prim(A) there exists I ∈ I such that I ⊆ J. Proof. If a ∈ A is not invertible, then by Theorem 2.10 there exists an irreducible unitary representation π of A such that π(a)v = 0 for some non-zero vector v ∈ H π . Moreover, for every two-sided closed ideal I ⊆ ker(π) of A the representation π induces a well-defined irreducible representation π I of the C * -algebra A/I, i.e., π I (Φ I (a)) = π(a) .
Hence, for every two-sided closed ideal I ⊆ ker(π) of C * (H), the element Φ I (a) is not invertible in A/I, since the operator π I (Φ I (a)) has a non-trivial kernel in H π . Let us assume now that Φ I (a) is not invertible in the C * -algebra A/I for some I ∈ I. Hence, there exists an irreducible representation ρ of A/I such that ρ(Φ I (a))v = 0 for some vector v ∈ H ρ . The irreducible representation ρ can be extended to an irreducible representationρ of A which vanishes on I and which is given byρ = ρ • Φ I . Therefore, a is not invertible in A.
From the proof of Theorem 5.2 we get the following corollary. For the convenience of the reader we will write down Ind H N (σ θ,s ) for every θ, s ∈ T explicitly for all k, l, m, n ∈ Z and F ∈ 2 (Z, C). It is easy to see that for irrational θ the representation π (1,1) θ in (3.12) is unitarily equivalent (via Fourier transformation) to the representation Ind H N (σ θ,1 ). Moreover, every irreducible finite dimensional representation of a nilpotent group Γ is induced from a one dimensional representation of a subgroup of Γ (cf. [6, Lemma 1]). Therefore, the monomial representations contain all representations involved in validating invertibility via Theorem 3.8.
The natural question arises, whether one can always restrict oneself to the class of monomial representations of Γ when analysing invertibility in the corresponding group algebras, in case Γ is a countable discrete nilpotent group. We will show that the answer is positive.
The general case. Let Γ be a countable discrete nilpotent group. Define an equivalence relation on Γ as follows: where π 1 , π 2 are irreducible unitary representations of Γ. This equivalence relation is the same as the notion of weak equivalence according to [10]. The next theorem was established by Howe in [17, Proposition 5].
Theorem 5.5. Suppose that Γ is a countable discrete nilpotent group. Then every irreducible unitary representation is weakly equivalent to an irreducible monomial representation of Γ.
In other words the map from the subclass of irreducible monomial representations to the primitive ideal space is surjective and as a conclusion the monomial representations generate the primitive ideal space. It is therefore not surprising that the class of irreducible monomial representations contains all the information which is necessary in order to study invertibility in the group algebras. As we will show, combining Theorem 5.5 with Theorem 2.10 leads to another Wiener Lemma: Theorem 5.6. An element a ∈ C * (Γ) is non-invertible if and only if there exists an irreducible monomial representation π such that π(a) has no bounded inverse.
For convenience of the reader we explain the ideas once more.

5.3.
Maximality of primitive ideals. In the previous subsection we saw that we can restrict our attention to irreducible monomial representations for questions about invertibility. Unfortunately, this subclass of irreducible representations might still be quite big. We will use another general result about the structure of Prim(C * (Γ)) to make the analysis of invertibility in C * (Γ) easier. i.e., every primitive ideal of C * (Γ) is maximal.
The simplification in the study of invertibility in C * (H) was due to the simplicity of the irrational rotation algebras R θ , which is equivalent to the maximality of the two-sided closed ideal J θ . We should note here that Theorem 4.2 is usually proved by the construction of a unique trace on R θ , which is rather complicated. Alternatively, let θ ∈ T be irrational. Then it easily follows from Theorem 5.7 and the fact that π (s,t) θ is an irreducible representation (cf. Lemma 3.4) with ker(π (s,t) θ ) = J θ that J θ is maximal. This is exactly the statement of Theorem 4.2. In the next subsection we will see applications of Theorem 5.7. It turns out that this representation theoretical result will eliminate the 'non-Type I issues' exactly as the simplicity of irrational rotation algebras did for the group algebras of the discrete Heisenberg group.

5.4.
Examples. The first example shows how to establish a Wiener Lemma for H from the general observation made in this section.
Example 5.8. Consider the monomial representations Ind H N (σ θ,s ) of H as defined in (5.1) for irrational θ and arbitrary s ∈ T. Obviously, one has for every s ∈ T that ker(Ind H N (σ θ,s )) = J θ . We will show that there is no bounded operator on 2 (Z, C) which commutes with the operators Ind H N (σ θ,s )(x) and Ind H N (σ θ,s )(y) except multiples of the identity operator. Let {δ k : k ∈ Z} be the standard basis of 2 (Z, C) and C = (c n,k ) n,k∈Z an operator which commutes with Ind H N (σ θ,s )(x) and Ind H N (σ θ,s )(y). From the equations n∈Z c n,k δ n and the fact that θ is irrational we can conclude that c n,k = 0 for all n, k ∈ Z with n = k. On the other hand, for k ∈ Z Therefore, the only operators in the commutant of Ind H N (σ θ,s )(H) are scalar multiples of the identity, which is equivalent to the irreducibility of the representation Ind H N (σ θ,s ) by Schur's Lemma. Hence, the kernel of the irreducible monomial representation Ind H N (σ θ,s ) is a maximal two-sided ideal (cf. Theorem 5.7) given by J θ .
For every irreducible representation π of H with J θ ⊆ ker(π) one has ker(π) = J θ due to the maximality of J θ which we deduce from the irreducibility of Ind H N (σ θ,s ). Consider θ = n d with relatively prime n, d ∈ N. We note that analysing invertibility in Q θ reduces to the study of monomial representations as well. Set Note that in the general study of invertibility in this example we have not used Allan's local principle or any results from Section 4 explicitly.
We give another example of a group where Theorem 5.7 simplifies the analysis. Example 5.9. Let us denote by G the multiplicative group given by One can easily verify that the centre of this group is given by and hence the corresponding central sub-algebra 1 (Z) is exactly the same as it was in the case of the discrete Heisenberg group. It is therefore not surprising that the invertibility problem can be addressed in a similar fashion.
Let us construct monomial representations, which will be sufficient to check global invertibility (cf. Theorem 5.6).
First note that G can be identified with the semi-direct product G 1 G 2 of the normal abelian subgroup and the abelian subgroup In such a situation the construction of induced representations becomes very easy. We refer to [21,Section 2.4] for all the details. Now let σ θ b ,θc,θe,θ f be the one-dimensional representation of G 1 given by The inclusion map from G 2 to G will serve as a cross-section. The induced representation Ind G G 1 (σ θ b ,θc,θe,θ f ) (is unitarily equivalent to a representation which) acts on the Hilbert space 2 (Z 2 , C) and is given by for every a, b, c, d, e, f, k, l ∈ Z and F ∈ 2 (Z 2 , C). The localisation fibres are indexed by θ f . It is clear that for every irrational θ f and arbitrary θ b , θ c , θ e , ker Ind G G 1 (σ θ b ,θc,θe,θ f ) = J θ f . In the case of irrational θ f , the commutant of Ind G G 1 (σ θ b ,θc,θe,θ f )(G) in B( 2 (Z 2 , C)) is trivial which is equivalent to irreducibility by Schur's Lemma. Hence, for irrational θ f the twosided closed ideal J θ f is maximal by Theorem 5.7 and one has to consider only a single representation, e.g., the one given in (5.2) for fixed parameters θ b , θ c , θ e , to study invertibility on these fibres.

5.5.
A kernel condition and finite-dimensional representations. The proof of Proposition 4.4 provides an approximation argument which allows restricting oneself to finitedimensional representations for the purpose of proving invertibility. This result can be reinterpreted as a density argument. The finite-dimensional irreducible representations of H correspond to 'rational' points in the dual of H. We know that the intersection of all irreducible representations π with π(z) = e 2πiθ coincides with J θ . In the same way as one shows that no non-zero element in C(T) (which is isomorphic to C * (Z)) vanishes at all rational points, one can prove that rational θ∈T We will show that this empty-intersection condition actually implies that for checking invertibility of a ∈ C * (H) it is enough to check the invertibility of the evaluations π(a) for all finite-dimensional irreducible representations of C * (H) provided that the inverses π(a) −1 of these elements are uniformly bounded in norm.
Let A and B t , for all t ∈ I for some index set I, be C * -algebras. Let us denote by F a family of * -morphisms φ t : A −→ B t , t ∈ I, which fulfils • for all t ∈ I one has J t = ker(φ t ) is a two-sided closed ideal of A, Furthermore, let us denote by A I the set of elements a = (a t ) t∈I ∈ t∈I A t with a I := sup t∈I a t t < ∞; A I together with the norm · I forms a C * -algebra. Let Φ : A −→ A I be defined by a → (φ t (a)) t∈I . Since one has that Φ is a bijective map from A to Φ(A). The C * -algebras Φ(A) and A I form a Wiener pair. Hence, a ∈ A is invertible if and only if φ t (a) is invertible for all t ∈ I and φ t (a) −1 t is uniformly bounded in t.
Example 5.10. We apply these ideas to C * (H) and set F H = {π ∈ H : π a finite dimensional representation} in order to get an algebraic interpretation of Proposition 4.4.
Example 5.11. Definitions and results that are used in this example can be found in [18] by Howe. Consider a finitely-generated nilpotent torsion-free group Γ. The set of kernels of finite-dimensional representations forms a dense subset of Prim(C * (Γ)) with respect to the hull-kernel topology. Since all C * -algebras are semi-simple one gets that for every dense subset J ⊆ Prim(C * (Γ)) the following holds Hence, for verifying invertibility in C * (Γ), the study of the evaluations of the finite-dimensional representations -combined with a boundedness condition -is sufficient. Suppose that Γ is also elementarily-exponentiable -Howe labels such groups to have a well-defined 'Lie-algebra', say L. Then the finite dimensional representations correspond to finite quasi-orbits of a canonical action of Γ on L and the representation theory of Γ is closely related to the one of its Mal'cev completion.
A systematic treatment of group-C * -algebras C * (Γ) whose finite-dimensional representations separate points of C * (Γ) can be found in Section 4 of [4].

A connection to Time-Frequency-Analysis via localisations
In this section we formulate yet another Wiener Lemma for 1 (H, C) which involves Stonevon Neumann representations. These unitary representations of the discrete Heisenberg group are highly reducible and therefore, not the first choice for invertibility investigations (cf. Theorem 2.10). However, by exploring a connection from localisations of 1 (H, C) to twisted convolution algebras we establish a link to Time-Frequency-Analysis. In this discipline of mathematics Stone-von Neumann representations are of great importance.
6.1. Localisations and twisted convolution algebras. In [13] we determined the explicit form of the localisation ideals J m in order to formulate Allan's local principle for the group algebra 1 (H, C) of the discrete Heisenberg group. Let us recall this result.
We write a typical element f ∈ 1 (H, C) in the form: with f (k,l,m) ∈ C and (k,l,m)∈Z 3 |f (k,l,m) | < ∞. We identify the centre of 1 (H, C) with 1 (Z, C) since the centre of the group H is generated by powers of z. The maximal ideal space Max( 1 (Z, C)) is canonically homeomorphic to Z ∼ = T. The smallest closed two-sided ideal in 1 (H, C) containing m θ ∈ Max( 1 (Z, C)), θ ∈ T, is given by the subset J θ ⊂ 1 (H, C) which consists of all elements f ∈ 1 (H, C) such that The next definition plays an important role in the field of Time-Frequency-Analysis. Fix θ ∈ T. The twisted convolution θ on 1 (Z 2 , C) is defined as follows. Let f, g ∈ 1 (Z 2 , C), then (f θ g) m,n = k,l∈Z f k,l g m−k,n−l e 2πi(m−k)lθ .
The representations π α,β appear in the literature under various names: Stone-von Neumann, Weyl-Heisenberg or Schrödinger representations.
As an immediate corollary of Theorem 6.1 one obtains the following Wiener Lemma for the discrete Heisenberg group. Proof. The result follows by combining Allan's local principle with Wiener's Lemma for twisted convolution algebras.
Proof. Let θ be irrational and suppose a ∈ Q θ ( 1 (Z 2 , C), θ , * ) is not invertible. We just have to show that the non-invertibility of the element a implies that a is not invertible in the irrational rotation algebra R θ and, in particular, not in its realisation π α,β (C * (H)) with αβ = θ mod 1. Since Q θ is symmetric (cf. Lemma 6.3), there exists an irreducible unitary representation π of H such that π vanishes on J θ and π(Φ θ (a))v = 0 for some non-zero vector v ∈ H π . This implies that a is not invertible in R θ .
The proof of Lemma 6.4 basically says that for irrational θ the Banach algebra Q θ is inverse-closed in R θ .
We will show that Lemma 6.4 holds for rational θ as well. The representation π α,β of H can be decomposed in the following way (cf. [2]). Let ν be the Haar measure on (0, θ], where θ ∈ T with θ = αβ mod 1. There exists a unitary operator and a family of representations {Ind H N (σ θ,s ) : s ∈ (0, θ]} such that π α,β is unitarily equivalent via U to the direct integral Since unitary equivalence of two representations implies weak equivalence one gets that ker(π α,β ) = ker ker Ind H N (σ θ,t ) = J θ and hence that π α,β ( 1 (H, C)) 1 (H, C)/J θ = Q θ . From this observation we get the following lemma.
Lemma 6.5. Let θ ∈ T be rational. Then a ∈ Q θ is invertible if and only if π α,β (a) is invertible in B(L 2 (R, C)).
Remark 6.6. The decomposition (6.2) of π α,β depends only on the product αβ = θ mod 1 and is thus independent of the particular choice of α and β. Hence, in Theorem 6.1 and Theorem 6.2 one has to consider, e.g., α = θ and β = 1 only.

6.
3. An application to algebraic dynamical systems. As already mentioned in the first section of this article the problem of deciding on the invertibility in 1 (H, C) has an application in algebraic dynamics. The following result is important to check invertibility for f ∈ Z[H] in the group algebra 1 (H, C) because it tells us that π α,β (f ) has a trivial kernel in L 2 (R, C) for α, β = 0.
Theorem 6.7 ([25]). Let G be a non-zero element in L 2 (R, C), then for every finite set A ⊆ Z 2 the set {T k α M l β G : (k, l) ∈ A} is linear independent over C.
The following result is a reformulation of Theorem 6.7 and gives an exact description of the spectrum of an operator π α,β (f ), for α, β ∈ R {0} and f ∈ C[H], where C[H] is the ring of functions H −→ C with finite support. Theorem 6.8. Let f ∈ C[H] with f θ = 0 for θ = αβ = 0, α, β ∈ R, then for all c ∈ σ(π α,β (f )) the operators π α,β (c − f ) are injective and have dense range in L 2 (R, C) but are not bounded from below.
Suppose that the range of c − π α,β (f ) is not dense in L 2 (R, C). Then is not injective (cf. the footnote on page 17) which is a contradiction to Theorem 6.7 because (c−f ) * ∈ C[H]. Hence, c−π α,β (f ) not being invertible on L 2 (R, C) is equivalent to c−π α,β (f ) not being bounded from below.
Therefore, non-expansiveness of α f can be checked via two different approaches: • The dual of H: there is an irreducible representation π of H such that 0 is an eigenvalue of π(f ).

Examples
We now demonstrate how to apply Wiener's Lemma to obtain easily verifiable sufficient conditions for non-expansivity of a principal algebraic action.
Let f ∈ Z[H] be of the form Moreover, for a polynomial h ∈ C[Z d ] define the logarithmic Mahler measure m(h) by the integral In [24] (cf. [13, Theorem 2.6] for a proof) the following result was established: for linear f ∈ Z[H] of the form (7.1) with U(g i ) = ∅ for i = 0, 1, the action α f is expansive if and only if m(g 0 ) = m(g 1 ) .
In this section we use results on invertibility to derive criteria for non-expansiveness of principal actions of elements f in Z[H] of the form f = g 1 (y, z)x − g 0 (y, z) in cases when the unitary varieties U(g 0 ) and U(g 1 ) are not necessarily empty.
For every χ ∈ S, consider the rational function ψ χ on S: and consider the map ψ : N × S −→ C given by 7.1. Either U(g 0 ) or U(g 1 ) is a non-empty set. We fix the following notation. For every χ ∈ S and i = 0, 1, put U χ (g i ) = {ζ ∈ S : g i (ζ, χ) = 0} , and g i,χ (y) = g i (y, χ) , which we will view as a Laurent polynomial in y with complex coefficients, i.e., g i,χ ∈ C[Z] for every χ and i = 0, 1. Note also that the set U χ (g i ) is infinite if and only if g i,χ is the zero polynomial.
be of the form f = g 1 (y, z)x − g 0 (y, z). Suppose there exists an element χ ∈ S of infinite order which satisfies either of the following conditions.
Then α f is non-expansive.
Proof. We will prove only the first case, the second case can be proved similarly.
Suppose f is such that (X f , α f ) is expansive and the conditions in (i) are satisfied. We will now show that certain consequences of expansivity of α f are inconsistent with the conditions in (i). Hence, by arriving to a contradiction, we will prove that under (i) α f is not expansive.
We know that (X f , α f ) is expansive if and only if f is invertible in 1 (H, C). Hence (X f , α f ) is expansive if and only if there exists a w ∈ 1 (H, C), Suppose θ ∈ (0, 1] is irrational and that χ = e 2πiθ ∈ S satisfies condition (i). Consider the following representation π 1,θ of 1 (H, C) on L 2 (R, C), defined by and (π 1,θ (z)F )(t) = e 2πiθ F (t) .
is finite if and only if |P 0,θ (t)| = 0. Combining this fact with the uniform bound (7.2) -(7.3), we are able to conclude that (7.10) P 0,θ (t) = 0 on a set of full measure in R. The function P 0,θ (t) is continuous and therefore, P 0,θ must be the identically zero function on R.
The assumption that φ χ dλ S < 0 cannot be dropped in (i) of Theorem 7.1 as the following simple minded example shows.
If m < 0, then we choose π such that (π(x)F )(n) = F (n − 1) and (π(y)F )(n) = ξχ −n F (n) for each F ∈ 2 (Z, C) and n ∈ Z. Exactly the same arguments as in the case m ≥ 0 can be used to get a contradiction to the invertibility of π(f ).
Corollary 7.4. Let f be of the form (7.1) which satisfies the conditions of Theorem 7.3. Then α f defined by f = g 0 (y, z)x − g 1 (y, z) is non-expansive.
Then α f is non-expansive.
Fix ξ ∈ U χ (g 1 ), which is a non-empty set by the assumptions of the theorem. Since one has (π (ξ,χ) (f )F )(0) is equal to 0 for every F ∈ 2 (Z, C), 0 is an element of σ(π (ξ,χ) (f )) and hence f is not invertible. The same conclusions can be drawn for the cases g 1,χ = 0 C[Z] and g 0,χ = g 1,χ = 0 C[Z] . Next consider the case where g 0,χ and g 1,χ are not the zero elements in C[Z], which implies that m(g 0,χ ) and m(g 1,χ ) are finite and moreover U χ (g 0 ) and U χ (g 1 ) are finite sets. Suppose that m(g 0,χ ) < m(g 1,χ ). We follow the line of arguments in the proof of Theorem 7.1. The only adaption one has to make is to take the countable set B = t ∈ R : e 2πit χ k ∈ U χ (g 0 ) for some k ∈ Z into consideration, i.e., to exclude points in B in the equations (7.7) -(7.10).