A Wiener Lemma for the discrete Heisenberg group
Abstract
This article contains a Wiener Lemma for the convolution algebra \(\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})\) and group \(C^*\)algebra \(C^*({{\mathrm{\mathbb {H}}}})\) of the discrete Heisenberg group \({{\mathrm{\mathbb {H}}}}\). 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 \({{\mathrm{\mathbb {H}}}}\) rely on the complete knowledge of \(\widehat{{{\mathrm{\mathbb {H}}}}}\)—the dual of \({{\mathrm{\mathbb {H}}}}\), i.e., the space of unitary equivalence classes of irreducible unitary representations. We will describe the dual of \({{{\mathrm{\mathbb {H}}}}}\) explicitly and discuss its structure. Wiener’s Lemma provides a convenient condition to verify invertibility in \(\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})\) and \(C^*({{\mathrm{\mathbb {H}}}})\) which bypasses \(\widehat{{{\mathrm{\mathbb {H}}}}}\). The proof of Wiener’s Lemma for \({{\mathrm{\mathbb {H}}}}\) 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 \({{\mathrm{\mathbb {H}}}}\) has interesting applications in algebraic dynamics and timefrequency analysis which will be presented in this article as well.
Keywords
Invertibility Wiener’s Lemma Discrete Heisenberg groupMathematics Subject Classification
54H20 37A45 22D10 37C85 47B381 Motivation
Our main motivation to study this problem is an application in the field of algebraic dynamics which we introduce first. An algebraic \(\varGamma \) action is a homomorphism \(\alpha : \varGamma \longrightarrow \,\text {Aut}\, (X)\) from \(\varGamma \) to the group of automorphisms of a compact metrisable abelian group X [33].
1.1 Outline of the article
In Sect. 2 we will recall known criteria for invertibility in symmetric unital Banach algebras \(\mathcal A\). The most important result links invertibility investigations in \(\mathcal A\) to the representation theory of \(\mathcal A\). More precisely, the existence of an inverse \(a^{1}\) of \(a \in \mathcal A\) is equivalent to the invertibility of the operators \(\pi (a)\) for every irreducible unitary representation \(\pi \) of \(\mathcal A\). The representation theory of \({{\mathrm{\mathbb {H}}}}\) is unmanageable as we will demonstrate in Sect. 3.
Theorem 11—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 \({{\mathrm{\mathbb {H}}}}\). The proof of Theorem 11 can be found in Sect. 4. Moreover, as will be shown in Sect. 4 as well, invertibility of \(f\in {{\mathrm{\mathbb {Z}}}}[{{\mathrm{\mathbb {H}}}}]\) in \(\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {R}}}})\) can be verified with the help of the finitedimensional irreducible unitary representations of \({{\mathrm{\mathbb {H}}}}\).
In Sect. 5 we generalise Theorem 11 to countable discrete nilpotent groups \(\varGamma \). This result says that an element a in \(C^*(\varGamma )\) is invertible if and only if for every primitive ideal \(\mathtt I\) of \(C^*(\varGamma )\) the projection of a onto the quotient space \(C^*(\varGamma )/\mathtt I\) 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 Sect. 6 we will explore a connection to timefrequency analysis. Allan’s local principle (cf. Sect. 4) directly links localisations of \({{\mathrm{\ell ^1(\mathbb {H},\mathbb {C})}}}\) to twisted convolution algebras and hence, the representations of \({{\mathrm{\mathbb {H}}}}\) and the relevant representation theory in the field of timefrequency analysis coincide. In order to highlight this connection even more, timefrequency analysis might be interpreted as the Fourier theory on the discrete Heisenberg group \({{\mathrm{\mathbb {H}}}}\); due to the striking similarities to the Fourier analysis of the additive group \({{\mathrm{\mathbb {Z}}}}\) 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 \({{\mathrm{\mathbb {H}}}}\). Theorem 22—which is based on a result of Linnell (cf. [25])—gives a full description of the spectrum of the operators \(\pi (f)\) acting on \(L^2({{\mathrm{\mathbb {R}}}},{{\mathrm{\mathbb {C}}}})\), where \(\pi \) is a Stonevon Neumann representation [cf. (23] for a definition) and \(f\in {{\mathrm{\mathbb {Z}}}}[{{\mathrm{\mathbb {H}}}}]\).
Section 7 contains applications of Theorem 11 and Wiener’s Lemma for twisted convolution algebras, in particular, conditions for noninvertibility for ‘linear’ elements in \(f\in {{\mathrm{\mathbb {Z}}}}[{{\mathrm{\mathbb {H}}}}]\).
2 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 \(\varGamma \). First of all we refer to the article [14] by 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 \(\ell ^p\)spaces (cf. Sect. 2.2 and in particular Theorem 7) and inverseclosedness. Moreover, Gröchenig explains how these topics are related to questions on invertibility in timefrequency 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 8 which is yet another result which relates invertibility in \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) to invertibility of convolution operators. This result is completely independent of Theorem 7 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 \(\varGamma \) is of importance for invertibility investigations in the group algebras of \(\varGamma \). 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 \(\mathcal A ({{\mathrm{\mathbb {T}}}})\) the Banach algebra of functions with absolutely convergent Fourier series on \({{\mathrm{\mathbb {T}}}}\).
Theorem 1
(Wiener’s Lemma) An element \(F\in \mathcal A ({{\mathrm{\mathbb {T}}}})\) is invertible, i.e. \(1/F \in \mathcal A ({{\mathrm{\mathbb {T}}}})\), if and only if \(F(s)\not = 0\) for all \(s \in {{\mathrm{\mathbb {T}}}}\).
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 \(\mathcal A\) to the study of its irreducible representations and the compact space of maximal ideals \(\text {Max} (\mathcal A)\). We collect in the following theorem several criteria for invertibility in unital commutative Banach algebras.
Theorem 2
 1.
\(a \in \mathcal A\) is invertible;
 2.
\(a \not \in m\) for all \(m \in \, \text {Max}\, (\mathcal A)\);
 3.
\(\varPhi _m(a)\) is invertible in \(\mathcal A /m\) for all \(m \in \text {Max} (\mathcal A)\), where \(\varPhi _m: \mathcal A \longrightarrow \mathcal A /m \cong {{\mathrm{\mathbb {C}}}}\) is the canonical projection map;
 4.
\(\varPhi _m(a) \not = 0\) for all \(m \in \text {Max} (\mathcal A)\);
 5.
\(\pi (a)v \not = 0\) for every onedimensional irreducible unitary representation \(\pi \) of \(\mathcal A\) and \(v\in {{\mathrm{\mathbb {C}}}}{\backslash } \{ 0\}\) (definitions can be found in Sect. 2.1).
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 \(\varGamma \) furnished with the discrete topology. Beside \({{\mathrm{\mathbb {Z}}}}[\varGamma ]\) and \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) we are interested in \(C^*(\varGamma )\), the group \(C^*\) algebra of \(\varGamma \), i.e., the enveloping \(C^*\)algebra of \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\).
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.
2.1.1 Unitary representations
Let \(\mathcal {H}\) be a complex Hilbert space with inner product \(\langle \cdot ,\cdot \rangle \). We denote by \(\mathcal {B}(\mathcal {H})\) the algebra of bounded linear operators on \(\mathcal {H}\), furnished with the strong operator topology. Further, denote by \(\mathcal {U}(\mathcal {H})\subset \mathcal {B}(\mathcal {H})\) the group of unitary operators on \(\mathcal {H}\). If \(\varGamma \) is a countable group, a unitary representation \(\pi \) of \(\varGamma \) is a homomorphism \(\gamma \mapsto \pi (\gamma )\) from \(\varGamma \) into \(\mathcal {U}(\mathcal {H})\) for some complex Hilbert space \(\mathcal {H}\). Every unitary representation \(\pi \) of \(\varGamma \) extends to a \(^*\)representation of \(\ell ^1(\varGamma ,\mathbb {C})\), which is again denoted by \(\pi \), and which is given by the formula \(\pi (f)=\sum _{\gamma \in \varGamma }f_\gamma \pi (\gamma )\) for \(f=\sum _{\gamma \in \varGamma }f_\gamma \cdot \gamma \in \ell ^1(\varGamma ,\mathbb {C})\). Clearly, \(\pi (f^*)=\pi (f)^*\). The following theorem was probably first published in [12] but we refer to [30, Theorem 12.4.1].
Theorem 3

the class of unitary representations of \(\varGamma \);

the class of nondegenerate^{1} \(^*\)representations of \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\);

the class of nondegenerate \(^*\)representations of \(C^*(\varGamma )\).
Hence the representation theories of \(\varGamma \), \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) and \(C^*(\varGamma )\) coincide. In consideration of this result we will use the same symbol for a unitary representation of \(\varGamma \) and its corresponding \(^*\)representations of the group algebras \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) and \(C^*(\varGamma )\).
2.1.2 States and the GNS construction
Suppose that \(\mathcal A\) is a unital \(C^*\)algebra. A positive linear functional \(\phi : \mathcal A \longrightarrow {{\mathrm{\mathbb {C}}}}\) is a state if \(\phi (1_{\mathcal A})=1\). We denote by \(\mathcal S (\mathcal A )\) the space of states of \(\mathcal A\), which is a weak\(^*\)compact convex subset of the dual space of \(\mathcal A\). The extreme points of \(\mathcal S (\mathcal A)\) are called pure states.
2.1.3 Type I groups
We call a representation \(\pi \) a factor if \(\mathcal N _\pi \cap \mathcal N^{'} _\pi = {{\mathrm{\mathbb {C}}}}\cdot 1_{\mathcal B (\mathcal H _\pi )}\). A group is of Type I if every factor representation is a direct sum of copies of an irreducible representation.
2.1.4 Induced and monomial representations
This construction will become more transparent when we discuss specific examples below.
A representation of \(\varGamma \) is called monomial if it is unitarily equivalent to a representation induced from a onedimensional representation of a subgroup of \(\varGamma \).
Theorem 4
([16]) If \(\varGamma \) is a nilpotent group of Type I, then all its irreducible representations are monomial.
2.2 Symmetric Banach\(^*\)algebras
Let \(\mathcal A\) be a Banach algebra with multiplicative identity element \(1_{\mathcal A}\). The spectrum of \(a\in \mathcal A\) is the set of elements \(c\in {{\mathrm{\mathbb {C}}}}\) such that \(ac 1_{\mathcal A}\) is not invertible in \(\mathcal A\) and will be denoted by \(\sigma (a)\).
In order to study invertibility in \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) and \(C^*(\varGamma )\) in the nonabelian setting we will try to find criteria similar to those described in Theorem 2. For this purpose the following definition will play a key role.
Definition 1
A unital Banach\(^*\)algebra \(\mathcal A\) is symmetric if for every element \(a\in \mathcal A\) the spectrum of \(a^*a\) is nonnegative, i.e., \(\sigma (a^*a) \subseteq [0,\infty )\).
Typical examples of symmetric Banach*algebras are \(C^*\)algebras.
We turn to the study of nilpotent groups and their associated group algebras.
Theorem 5
([19]) Let \(\varGamma \) be a countably infinite discrete nilpotent group. Then the Banach\(^*\)algebra \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) is symmetric.
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, respectively.
For the class of symmetric group algebras one has the following important result on inverseclosedness.
Theorem 6
 1.
\(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) is semisimple, i.e., the intersection of the kernels of all the irreducible representations of \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) is trivial.
 2.
\(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) and its enveloping \(C^*\)algebra \(C^*(\varGamma )\) form a Wiener pair.
Next we are discussing spectral invariance of convolution operators. It is a well known fact (cf. [14]) that invertibility of \(f\in \ell ^1({{\mathrm{\mathbb {Z}}}},{{\mathrm{\mathbb {C}}}})\) can be validated by studying invertibility of the convolution operator \(\text {C}_f\) acting on the Hilbert space \(\ell ^2({{\mathrm{\mathbb {Z}}}},{{\mathrm{\mathbb {C}}}})\). Moreover, the spectrum of \(\text {C}_f\) is independent of the domain, i.e., the spectrum of the operator \(\text {C}_f: \ell ^p({{\mathrm{\mathbb {Z}}}},{{\mathrm{\mathbb {C}}}})\longrightarrow \ell ^p({{\mathrm{\mathbb {Z}}}},{{\mathrm{\mathbb {C}}}})\) is the same for all \(p\in [1,\infty ]\). As the following theorem shows, this result is true for a large class of groups, in particular, for all finitely generated nilpotent groups.
Theorem 7
([3]) Let \(f\in \ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) and \(\text {C}_f\) the associated convolution operator on \(\ell ^p(\varGamma ,{{\mathrm{\mathbb {C}}}})\). For all \(1\le p \le \infty \) one has \(\sigma _{\mathcal B (\ell ^p(\varGamma ,{{\mathrm{\mathbb {C}}}}))}(\text {C}_f)=\sigma _{\mathcal B (\ell ^2(\varGamma ,{{\mathrm{\mathbb {C}}}}))}(\text {C}_f)\) if and only if \(\varGamma \) is amenable and \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) is a symmetric Banach\(^*\)algebra.
In particular, for a nilpotent group \(\varGamma \), \(f\in \ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) is invertible in \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) if and only if \(0 \notin \sigma _{\mathcal B (\ell ^p(\varGamma ,{{\mathrm{\mathbb {C}}}}))}(\text {C}_f)\) for any \(p\in [1,\infty ]\).
Let us now give a condition for invertibility of an element \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\), where \(\varGamma \) is an arbitrary discrete countably infinite group, in terms of the point spectrum of the corresponding convolution operator.
Theorem 8
This theorem says that it is enough to check if 0 is an eigenvalue of the left convolution operator \(\text {C}_f: \ell ^\infty (\varGamma ,{{\mathrm{\mathbb {C}}}})\longrightarrow \ell ^\infty (\varGamma ,{{\mathrm{\mathbb {C}}}})\) in order to determine whether f is invertible or not [cf. (3)].
Finally, we present a condition for invertibility in a symmetric unital Banach\(^*\)algebra \(\mathcal A\) which links invertibility in \(\mathcal A\) to its representation theory.
Theorem 9
([28]) An element a in a symmetric unital Banach\(^*\)algebra \(\mathcal A\) is not left invertible in \(\mathcal A\) if and only if there exists a pure state \(\phi \) with \(\phi (a^*a)=0\). Equivalently, a is not left invertible if and only if there exists an irreducible representation \(\pi \) of \(\mathcal A\) and a unit vector \(u \in \mathcal H _\pi \) such that \(\pi (a)u=0\).
3 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 9). At the end of this section, we will state our main result—a Wiener Lemma for the discrete Heisenberg group \({{\mathrm{\mathbb {H}}}}\)—which allows one to restrict the attention to certain canonical representations of \({{\mathrm{\mathbb {H}}}}\) which can be parametrised effectively and used for solving the invertibility problem.
3.1 The dual of a discrete group
Let \(\varGamma \) be a countable discrete group. Denote by \(\widehat{\varGamma }\) the dual of \(\varGamma \), i.e., the set of all unitary equivalence classes of irreducible unitary representations of \(\varGamma \).
Definition 2
Let \(\mathcal A\) be a \(C^*\)algebra. A closed twosided ideal \(\mathtt I\) of \(\mathcal A\) is primitive if there exists an irreducible representation \(\pi \) of \(\mathcal A\) such that \(\ker (\pi )=\mathtt I\). The set of primitive ideals of \(\mathcal A\) is denoted by \(\text {Prim}(\mathcal A)\).

The map \(\widehat{\varGamma } \longrightarrow \text {Prim}(C^*(\varGamma ))\) given by \(\pi \mapsto \ker (\pi )\) is not injective. In other words, if \(\pi _1, \pi _2 \in \widehat{\varGamma }\), then \(\ker (\pi _1 )=\ker (\pi _2 )\) does not necessarily imply that \(\pi _1\) and \(\pi _2\) are unitarily equivalent.

\(\widehat{\varGamma }\) 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).
3.2 The discrete Heisenberg group and its dual
Since \({{\mathrm{\mathbb {H}}}}\) does not possess an abelian normal subgroup of finite index it is not a group of Type I (cf. [34]), 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 \({{\mathrm{\mathbb {H}}}}\) for every irrational \(\theta \in \mathbb {T}\). These representations arise from certain singular measures on \({{\mathrm{\mathbb {T}}}}\). This fact is wellknown 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 \({{\mathrm{\mathbb {H}}}}\) 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 \((X,\mathfrak {B},\mu )\) be a measure space, where X is a compact metric space, \(\mathfrak {B}\) is a Borel \(\sigma \)algebra, and \(\mu \) a finite measure.
Definition 3
In [23] uncountably many inequivalent ergodic quasiinvariant measures for every irrational rotation of the circle were constructed. Later it was shown in [22] that a homeomorphism \(\phi \) on a compact metric space X has uncountably many inequivalent nonatomic ergodic quasiinvariant measures if and only if \(\phi \) has a recurrent point x, i.e., \(\phi ^n (x)\) returns infinitely often to any punctured neighbourhood of x.
Theorem 10
For each irrational \(\theta \in {{\mathrm{\mathbb {T}}}}\) there is a bijection between the set of ergodic \(\text {R}_\theta \)quasiinvariant probability measures on \({{\mathrm{\mathbb {T}}}}\) and the set of irreducible representations \(\pi \) of \({{\mathrm{\mathbb {H}}}}\) with \(\pi (z)=e^{2\pi i \theta }\).
Lemma 1
The unitary representation \(\pi _{\theta ,\mu }\) of \({{\mathrm{\mathbb {H}}}}\) given by (8) is irreducible.
Proof
The ergodicity of \(\mu \) with respect to \(\text {R}_\theta \) implies that only constant functions in \(L^\infty ({{\mathrm{\mathbb {T}}}},\mu )\) are \(\text {R}_\theta \)invariant \(\mu \)a.e.. Hence, if \(\text {O}\) commutes with \(\text {T}_{\theta ,\mu }\) as well, then we can conclude that \(\text {O}\) is multiplication by a constant \(c\in {{\mathrm{\mathbb {C}}}}\). By Schur’s Lemma, the operators \(\text {T}_{\theta ,\mu },\text {M}_\mu \in \mathcal B (L^2({{\mathrm{\mathbb {T}}}},\mu ))\) define an irreducible representation \(\pi _{\theta ,\mu }\) of \({{\mathrm{\mathbb {H}}}}\). \(\square \)
Suppose that \(\theta \in {{\mathrm{\mathbb {T}}}}\) is irrational, and that \(\mu \) and \(\nu \) are two ergodic \(\text {R}_\theta \)quasiinvariant measures on \({{\mathrm{\mathbb {T}}}}\). Let \(\pi _{\theta ,\mu }\) and \( \pi _{\theta ,\nu }\) be the corresponding irreducible unitary representations constructed above.
Lemma 2
The representations \(\pi _{\theta ,\mu }\) and \(\pi _{\theta ,\nu }\) are unitarily equivalent if and only if \(\mu \) and \(\nu \) are equivalent.
Proof
Denote multiplication by a function \(H\in C(\mathbb {T},\mathbb {C})\) by \(\text {O}_H\). The set of trigonometric polynomials, which is spanned by \(\{\text {M}_\mu ^n 1\,:\,n\in {{\mathrm{\mathbb {Z}}}}\}\), is dense in \(C({{\mathrm{\mathbb {T}}}},{{\mathrm{\mathbb {C}}}})\). This implies that (10) holds for all \(H\in C({{\mathrm{\mathbb {T}}}},{{\mathrm{\mathbb {C}}}})\), i.e., that \(\text {U} O _H = \text {O}_H \text {U}\) for any \(H \in C({{\mathrm{\mathbb {T}}}},{{\mathrm{\mathbb {C}}}})\).
In this way one obtains uncountably many inequivalent irreducible unitary representation of \({{\mathrm{\mathbb {H}}}}\) for a given irrational rotation number \(\theta \in \mathbb {T}\).
In fact, every irreducible unitary representation \(\pi \) of \({{\mathrm{\mathbb {H}}}}\) with \(\pi (z)=e^{2\pi i \theta }\), \(\theta \) irrational, is unitarily equivalent to \(\pi _{\theta ,\mu }\) for some probability measure \(\mu \) on \(\mathbb {T}\) which is quasiinvariant 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.
Put \(\text {S} =\pi (x)\) and consider the cyclic normalised vector \(w=\text {S} v\) of the representation \(\pi \). By replacing v by w in the construction above one can define the corresponding objects \(\mathcal H _w, \text {U}_w, \mu _w, L^2 ({{\mathrm{\mathbb {T}}}},\mu _w), \text {M}_w\).
Lemma 3
The measures \(\mu _v\) and \(\mu _w\) are equivalent.
Proof
Lemma 4
The measure \(\mu _v\) is \(\text {R} _\theta \)quasiinvariant.
Proof
Note that \(\widehat{\mu _w}(n)=\langle \text {S} ^{1}\text {U} ^n \text{ S } v,v\rangle _{\mathcal H _\pi } = e^{2 \pi i \theta n} \widehat{\mu _v}(n)\) for every \(n\in \mathbb {Z}\). As one can easily verify, for every probability measure \(\mu \) on \({{\mathrm{\mathbb {T}}}}\), multiplying \(\widehat{\mu }\) with a character \(e^{2 \pi i \theta n}\) is the same as the FourierStieltjes transform of \(\mu \circ \text {R} _\theta \). Hence, we obtain that \(\mu _w=\mu _v \circ \text {R} _\theta \). As \(\mu _v\) and \(\mu _w\) are equivalent, \(\mu _v\) is a \(\text {R} _\theta \)quasiinvariant probability measure on \({{\mathrm{\mathbb {T}}}}\). \(\square \)
Proof
(Completion of the proof of Theorem 10) The preceding discussion allows us to define an irreducible representation \(\pi _v\) of \({{\mathrm{\mathbb {H}}}}\) acting on \( L^2 ({{\mathrm{\mathbb {T}}}},\mu _v)\) which is unitarily equivalent to \(\pi \). The evaluation of \(\pi _v\) at y is given by \(\text {M}_v\) and \(\text {T}_v= \pi _v(x)\) acts as composition of a translation operator by an angle \(\theta \) and multiplication by some function \(D_v\in L^\infty ({{\mathrm{\mathbb {T}}}},\mu _v)\). Due to the fact that \(\text {T}_v\) has to be a unitary (and hence, an isometric) operator on \( L^2 ({{\mathrm{\mathbb {T}}}},\mu _v)\) the form of \(D_v\) is fully determined (cf. the definition of \(\text {T} _{\theta ,\mu }\) in (6) for a \(\text {R} _\theta \)quasiinvariant measure \(\mu \)). Since \(\pi _v\) is irreducible only those multiplication operators in \(\mathcal B ( L^2 ({{\mathrm{\mathbb {T}}}},\mu _v))\) which act via multiplication by a constant function \(c\in {{\mathrm{\mathbb {C}}}}\) will commute with multiples of the modified translation operator \(\text {T}_v\). This implies the ergodicity of \(\mu _v\) and completes the proof of the theorem. \(\square \)
3.3 Wiener’s Lemma for the discrete Heisenberg group
Theorem 9 states that in order to decide on invertibility of \(f \in \ell ^1 ({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})\), one has to check invertibility of \(\pi (f)\) for every irreducible representations \(\pi \) of \({{\mathrm{\mathbb {H}}}}\), and hence for every \(\pi _{\theta ,\mu }\) as above, where \(\mu \) is a probability measure on \(\mathbb {T}\) on \(\mathbb {T}\) which is quasiinvariant and ergodic with respect to a circle rotation \(R_\theta \).
The problem of deciding on invertibility of \(f\in {{\mathrm{\ell ^1(\mathbb {H},\mathbb {C})}}}\) via Theorem 9 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.
With this notation at hand we can state our main result, the proof of which will be given in Sect. 4.
Theorem 11
An element \(f\in \ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})\) is invertible if and only if the linear operators \(\pi _{\theta }^{(s,t)}(f)\) are invertible on the corresponding Hilbert spaces \(\mathcal {H}_{\pi _{\theta }^{(s,t)}}\) for every \(\theta ,s,t\in {{\mathrm{\mathbb {T}}}}\).
The main advantage of Theorem 11 over Theorem 9 is that it is not necessary to check invertibility of \(\pi (f)\) for every irreducible representation of \({{\mathrm{\mathbb {H}}}}\), but that one can restrict oneself for this purpose to the ‘nice’ part of the dual of the nonType I group \({{\mathrm{\mathbb {H}}}}\). As we shall see later, one can make a further reduction if \(\theta \) is irrational: in this case one only has to check invertibility of \(\pi _{\theta }(f)= \pi _{\theta }^{(1,1)}(f)\) on \(L^2(\mathbb {T},\lambda )\).
4 Wiener’s Lemma for the discrete Heisenberg group: a proof and a first application

Allan’s local principle, which reduces the problem of verifying invertibility in \(\ell ^1({{\mathrm{\mathbb {H}}}}, {{\mathrm{\mathbb {C}}}})\) and \(C^*({{\mathrm{\mathbb {H}}}})\) to the study of invertibility in rotation algebras.

The fact that irrational rotation algebras are simple will eliminate the ‘nonType I problem’ for questions about invertibility in \(\ell ^1({{\mathrm{\mathbb {H}}}}, {{\mathrm{\mathbb {C}}}})\) and \(C^*({{\mathrm{\mathbb {H}}}})\) discussed in the previous section.
4.1 Local principles
Let \(\mathcal {A}\) be a unital Banach algebra and \(a \in \mathcal A\). Local principles are based on the following idea: one checks invertibility of projections of a onto certain quotient algebras of \(\mathcal A\) in order to conclude from this information whether a is invertible or not. Therefore, the main task is to find a sufficient family \(\mathfrak J\) of ideals of \(\mathcal A\) such that one can deduce the invertibility of a from the invertibility of the projections of a on \(\mathcal A/\mathtt I\) for all \(\mathtt I \in \mathfrak J\).
Allan’s local principle provides us with such a sufficient family of ideals in case the centre of \(\mathcal A\) is large enough. We have used Allan’s local principle already in [13] to study invertibility in \(\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})\). However, in that paper we were not able to prove Theorem 11 with this approach.
Theorem 12
([1] Allan’s local principle) An element \(a \in \mathcal {A}\) is invertible in \(\mathcal {A}\) if and only if \(\varPhi _m(a)\) is invertible in \(\mathcal {A}/\mathtt {J}_m\) for every \(m\in \text {Max} (\mathcal C(\mathcal {A}))\).
We would like to mention already here that in Sect. 7 Allan’s local principle will appear a second time and will link invertibility of \(f\in {{\mathrm{\ell ^1(\mathbb {H},\mathbb {C})}}}\) to the invertibility of the evaluations of Stonevon Neumann representations at f.
Let us now prove our main theorem.
4.2 Proof of Wiener’s Lemma
We apply the general observations made in the previous subsection to explore invertibility in \({{\mathrm{\ell ^1(\mathbb {H},\mathbb {C})}}}\) and \(C^*({{\mathrm{\mathbb {H}}}})\). Since \({{\mathrm{\ell ^1(\mathbb {H},\mathbb {C})}}}\) is inverseclosed in \(C^*({{\mathrm{\mathbb {H}}}})\) we can focus on the study of invertibility in \(C^*({{\mathrm{\mathbb {H}}}})\).
Since \({\mathtt J} _\theta =\overline{(ze^{2\pi i\theta })C^*({{\mathrm{\mathbb {H}}}})} \) is a twosided closed ideal we know that the quotient \(\mathcal Q_\theta \) is a \(C^*\)algebra and hence symmetric for each \(\theta \in {{\mathrm{\mathbb {T}}}}\).
By Schur’s Lemma, if \(\pi \) is an irreducible unitary representation of \({{\mathrm{\mathbb {H}}}}\), then \(\pi (z)= e^{2\pi i\theta }1_{\mathcal B (\mathcal H _\pi )}\) for some \(\theta \in \mathbb {T}\). Hence, \(\mathtt J_\theta \) is a subset of \(\ker (\pi )\) for every irreducible unitary representation \(\pi \) of \({{\mathrm{\mathbb {H}}}}\) with \(\pi (z)= e^{2\pi i\theta }1_{\mathcal B (\mathcal H _\pi )}\).
If \(\theta \) is rational the irreducible unitary representations of \({{\mathrm{\mathbb {H}}}}\) vanishing on \(\mathtt J _\theta \) are given by (16)–(17) and were determined in [5]. Due to the fact that \(\mathcal Q_\theta \) is symmetric we can apply Theorem 9 in order to study invertibility in \(\mathcal Q_\theta \) via the representations (16)–(17).
Now suppose \(\theta \) is irrational. In order to study the representation theory of the \(C^*\)algebra \(\mathcal Q_\theta \) we have to understand the link to one of the most studied noncommutative \(C^*\)algebras—the irrational rotation algebras.
Theorem 13
If \(\theta \in \mathbb {T}\) is irrational, then all \(C^*\)algebras which are generated by two unitaries \(\text {U},\text {V}\) satisfying (19), are \(^*\)isomorphic.
We will denote the irrational rotation algebra with parameter \(\theta \) by \(\mathcal R _\theta \) and will not distinguish between the different realisations of \(\mathcal R _\theta \) because of the universal property described in Theorem 13. Let us further note that the proof of Theorem 13 is deduced from the simplicity of the universal irrational rotation algebra.
The \(C^*\)algebra \(\mathcal Q_\theta \) is clearly a rotation algebra with parameter \(\theta \). The simplicity of \(\mathcal R _\theta \) implies that \(\mathtt J _\theta \) is a maximal twosided ideal of \(C^*({{\mathrm{\mathbb {H}}}})\). Hence, there exists an irreducible representation \(\pi \) of \({{\mathrm{\mathbb {H}}}}\) such that \(\ker (\pi )=\mathtt J _\theta \), since every twosided maximal ideal is primitive (cf. [29, Theorem 4.1.9]). Moreover, all the irreducible representations \(\pi \) vanishing on \(\mathtt J _\theta \) have the same kernel: otherwise we would get a violation of the maximality of \(\mathtt J _\theta \). These representations are not all in the same unitary equivalence class (as we saw in Sect. 3), which is an indication of the fact that \({{\mathrm{\mathbb {H}}}}\) is not of Type I.
Proof
(Proof of Theorem 11) First of all recall that \({{\mathrm{\ell ^1(\mathbb {H},\mathbb {C})}}}\) is inverseclosed in \(C^*({{\mathrm{\mathbb {H}}}})\). By applying Allan’s local principle for \(C^*({{\mathrm{\mathbb {H}}}})\) the problem of verifying invertibility in \({{\mathrm{\ell ^1(\mathbb {H},\mathbb {C})}}}\) and \(C^*({{\mathrm{\mathbb {H}}}})\) reduces to the study of invertibility in the \(C^*\)algebras \(\mathcal Q_\theta \), with \(\theta \in {{\mathrm{\mathbb {T}}}}\).
The rational case is trivial and was already treated at the beginning of the discussion.
Remark 1
We should note here that for all realisations of the irrational rotation algebra the spectrum of \(a \in \mathcal R_\theta \) 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.
4.3 Finitedimensional approximation
The following proposition follows from Theorem 11 and might be useful for checking invertibility of \(f\in {{\mathrm{\mathbb {Z}}}}[{{\mathrm{\mathbb {H}}}}]\) via numerical simulations.
Proposition 1
Let \(f\in {{\mathrm{\mathbb {Z}}}}[{{\mathrm{\mathbb {H}}}}]\). Then \(\alpha _f\) is expansive if and only if there exists a constant \(c>0\) such that \(\pi (f)\) is invertible and \(\Vert \pi (f)^{1}\Vert \le c\) for every finitedimensional irreducible representation \(\pi \) of \({{\mathrm{\mathbb {H}}}}\).
Proof
One direction is obvious. For the converse, assume that \(\alpha _f\) is nonexpansive, but that there exists a constant \(c>0\) such that \(\pi (f)\) is invertible and \(\Vert \pi (f)^{1}\Vert \le c\) for every finitedimensional irreducible representation \(\pi \) of \({{\mathrm{\mathbb {H}}}}\).
Since \(\alpha _f\) is nonexpansive, there exists an irrational \(\theta \) (by our assumption) such that the operator \(\pi _{\theta }^{(1,1)}(f)\) has no bounded inverse due to Theorem 11 and its proof. Therefore, \(\pi _{\theta }^{(1,1)}(f)\) is either not bounded from below or its range is not dense in the representation space or both.
We consider first the case where \(\pi _{\theta }^{(1,1)}(f)\) is not bounded from below. Then there exists, for every \(\varepsilon >0\), an element \(H_\varepsilon \in L^2(\mathbb {T},\lambda _\mathbb {T})\) with \(\Vert H_\varepsilon \Vert _2 =1\) and \(\Vert \pi _{\theta }^{(1,1)}(f)H_\varepsilon \Vert _2 < \varepsilon \). By approximating the \(H_\varepsilon \) by continuous functions we may obviously assume that each \(H_\varepsilon \) is continuous.
Finally, assume that \(\pi _{\theta }^{(1,1)}(f)\) has no dense image in \(L^2({{\mathrm{\mathbb {T}}}},\lambda )\). In that case the adjoint operator \((\pi _{\theta }^{(1,1)}(f))^*=\pi _{\theta }^{(1,1)}(f^*)\) is not injective.^{3} Furthermore, by our assumptions, \(\Vert \pi (f^*)^{1}\Vert \le c\) for every finitedimensional irreducible representation \(\pi \) of \({{\mathrm{\mathbb {H}}}}\). The same arguments as in the first part of the proof lead to a contradiction. \(\square \)
5 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 9. 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 \(\varGamma \) be a countable discrete nilpotent group. As we have seen earlier, \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) is inverseclosed in \(C^*(\varGamma )\). Hence we concentrate on the group \(C^*\)algebra \(C^*(\varGamma )\).
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 Sect. 4.2 one can easily see that the irreducible unitary representations \(\pi _\theta ^{(s,t)}\), \(\theta ,s,t\in {{\mathrm{\mathbb {T}}}}\), those representations which correspond to ergodic \(\text {R}_\theta \)invariant measures on \({{\mathrm{\mathbb {T}}}}\), generate the primitive ideal space \(\text {Prim}(C^*({{\mathrm{\mathbb {H}}}}))\). Moreover, since \(\pi (C^*({{\mathrm{\mathbb {H}}}})) \simeq C^*({{\mathrm{\mathbb {H}}}}) / \ker (\pi )\) for every \(\pi \in \widehat{{\mathrm{\mathbb {H}}}}\) 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 \(\mathcal A\) be a unital \(C^*\)algebra. For every twosided closed ideal \(\mathtt J\) of \(\mathcal A\), denote by \(\varPhi _{\mathtt J}\) the canonical projection from \(\mathcal A\) onto the \(C^*\)algebra \(\mathcal A/ \mathtt J\).
Theorem 14
(Wiener’s Lemma for nilpotent groups) If \(\varGamma \) is a discrete nilpotent group, then \(a \in C^*(\varGamma )\) is invertible if and only if \(\varPhi _\mathtt {I}(a)\) is invertible for every \(\mathtt I \in \text {Prim}(C^*(\varGamma ))\).
This theorem links questions about invertibility in \(\ell ^1(\varGamma ,{{\mathrm{\mathbb {C}}}})\) and \(C^*(\varGamma )\) to their representation theory and, to be more specific, to the primitive ideal space \(\text {Prim}(C^*(\varGamma ))\). 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 \(\varGamma \) the class of irreducible representation which are induced by onedimensional representations of subgroups of \(\varGamma \) provide us with an effective tool to generate \(\text {Prim}(C^*(\varGamma ))\). In other words, it is a feasible task to determine the primitive ideal space \(\text {Prim}(C^*(\varGamma ))\).
Theorem 14 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.
 (i)
every \(\mathtt I\in \mathfrak I\) is closed and twosided,
 (ii)
for any primitive ideal \(\mathtt J\in \text {Prim}(\mathcal A)\) there exists \(\mathtt I\in \mathfrak I\) such that \(\mathtt I\subseteq \mathtt J\).
Theorem 15
Let \(\mathcal A\) be a unital \(C^*\)algebra. Suppose \(\mathfrak I\) satisfies conditions (i) and (ii) above. Then an element a in \(\mathcal A\) is invertible if and only if for every \( \mathtt I\in \mathfrak I\) the projection of a on \(\mathcal A/\mathtt I\) is invertible.
By setting \(\mathfrak I=\text {Prim}(C^*(\varGamma ))\), Theorem 14 just becomes a particular case of Theorem 15.
Proof
From the proof of Theorem 15 we get the following corollary.
Corollary 1
If \(\pi (a)\) is not invertible for an irreducible representation \(\pi \), then for every twosided closed ideal \(\mathtt I \subseteq \ker (\pi )\) of \(C^*({{\mathrm{\mathbb {H}}}})\), the element \(\varPhi _{\mathtt I}(a)\) is noninvertible in \(\mathcal A/ \mathtt I\).
Example 1
5.2 Monomial representations
5.2.1 The Heisenberg group
It is easy to see that for irrational \(\theta \) the representation \(\pi _{\theta }^{(1,1)}\) in (15) is unitarily equivalent (via Fourier transformation) to the representation \(\text {Ind}^{{{\mathrm{\mathbb {H}}}}}_N(\sigma _{\theta ,1})\). Moreover, every irreducible finite dimensional representation of a nilpotent group \(\varGamma \) is induced from a one dimensional representation of a subgroup of \(\varGamma \) (cf. [6, Lemma 1]).
Therefore, the monomial representations contain all representations involved in validating invertibility via Theorem 11.
The natural question arises, whether one can always restrict oneself to the class of monomial representations of \(\varGamma \) when analysing invertibility in the corresponding group algebras, in case \(\varGamma \) is a countable discrete nilpotent group. We will show that the answer is positive.
5.2.2 The general case
The next theorem was established by Howe in [17, Proposition 5].
Theorem 16
Suppose that \(\varGamma \) is a countable discrete nilpotent group. Then every irreducible unitary representation is weakly equivalent to an irreducible monomial representation of \(\varGamma \).
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 16 with Theorem 9 leads to another Wiener Lemma:
Theorem 17
An element \(a\in C^*(\varGamma )\) is noninvertible if and only if there exists an irreducible monomial representation \(\pi \) such that \(\pi (a)\) has no bounded inverse.
For convenience of the reader we explain the ideas once more.
Proof
On the other hand, if \(\pi (a)\) is not invertible for an irreducible monomial representation \(\pi \), then \(\varPhi _{\ker (\pi ) }(a)\) is not invertible in the \(C^*\)algebra \(C^*(\varGamma ) / \ker (\pi )\). Hence there exists an irreducible representation \(\rho \) of \(C^*(\varGamma ) / \ker (\pi )\) such that \(\rho (\varPhi _{\ker (\pi )}(a))\) has a nontrivial kernel. Moreover, \(\rho \) can be extended to a representation \(\tilde{\rho }\) of \(C^*(\varGamma )\) vanishing on \(\ker (\pi )\). Therefore, a is not invertible. \(\square \)
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 \(\text {Prim}(C^*(\varGamma ))\) to make the analysis of invertibility in \(C^*(\varGamma )\) easier.
Theorem 18
The simplification in the study of invertibility in \(C^*({{\mathrm{\mathbb {H}}}})\) was due to the simplicity of the irrational rotation algebras \(\mathcal R _\theta \), which is equivalent to the maximality of the twosided closed ideal \(\mathtt J _\theta \). We should note here that Theorem 13 is usually proved by the construction of a unique trace on \(\mathcal R _\theta \), which is rather complicated. Alternatively, let \(\theta \in {{\mathrm{\mathbb {T}}}}\) be irrational. Then it easily follows from Theorem 18 and the fact that \(\pi _{\theta }^{(s,t)}\) is an irreducible representation (cf. Lemma 1) with \(\ker (\pi _{\theta }^{(s,t)}) = \mathtt J_\theta \) that \(\mathtt J_\theta \) is maximal. This is exactly the statement of Theorem 13. In the next subsection we will see applications of Theorem 18. It turns out that this representation theoretical result will eliminate the ‘nonType 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 \({{\mathrm{\mathbb {H}}}}\) from the general observation made in this section.
Example 2
Consider the monomial representations \(\text {Ind}_N^{{{\mathrm{\mathbb {H}}}}}(\sigma _{\theta , s})\) of \({{\mathrm{\mathbb {H}}}}\) as defined in (21) for irrational \(\theta \) and arbitrary \(s\in {{\mathrm{\mathbb {T}}}}\). Obviously, one has for every \(s\in {{\mathrm{\mathbb {T}}}}\) that \(\ker (\text {Ind}_N^{{{\mathrm{\mathbb {H}}}}}(\sigma _{\theta , s})) =\mathtt J_\theta \).
For every irreducible representation \(\pi \) of \({{\mathrm{\mathbb {H}}}}\) with \(\mathtt J _\theta \subseteq \ker (\pi )\) one has \(\ker (\pi ) =\mathtt J _\theta \) due to the maximality of \(\mathtt J _\theta \) which we deduce from the irreducibility of \(\text {Ind}_N^{{{\mathrm{\mathbb {H}}}}}(\sigma _{\theta , s})\).
A Wiener Lemma can now be deduced from Theorem 18.
Note that in the general study of invertibility in this example we have not used Allan’s local principle or any results from Sect. 4 explicitly.
We give another example of a group where Theorem 18 simplifies the analysis.
Example 3
Let us construct monomial representations, which will be sufficient to check global invertibility (cf. Theorem 17).
5.5 A kernel condition and finitedimensional representations

for all \(t\in I\) one has \(\mathtt J_t = \ker (\phi _t )\) is a twosided closed ideal of \(\mathcal A\), hence \(\mathcal A _t= \mathcal A / \mathtt J_t \) is a \(C^*\)algebra with quotient norm \(\Vert \cdot \Vert _t\);

\(\bigcap _{t \in I} \mathtt J _t = \{0\}\).
Example 4
Example 5
Suppose that \(\varGamma \) is also elementarilyexponentiable—Howe labels such groups to have a welldefined ‘Liealgebra’, say \(\mathcal L\). Then the finite dimensional representations correspond to finite quasiorbits of a canonical action of \(\varGamma \) on \(\mathcal L\) and the representation theory of \(\varGamma \) is closely related to the one of its Mal’cev completion.
A systematic treatment of group\(C^*\)algebras \(C^*(\varGamma )\) whose finitedimensional representations separate points of \(C^*(\varGamma )\) can be found in Section 4 of [4].
6 A connection to timefrequencyanalysis via localisations
In this section we formulate yet another Wiener Lemma for \(\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})\) 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 9). However, by exploring a connection from localisations of \(\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})\) to twisted convolution algebras we establish a link to TimeFrequencyAnalysis. In this discipline of mathematics Stonevon Neumann representations are of great importance.
6.1 Localisations and twisted convolution algebras
In [13] we determined the explicit form of the localisation ideals \(\mathtt J _m\) in order to formulate Allan’s local principle for the group algebra \(\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})\) of the discrete Heisenberg group. Let us recall this result.
6.2 Wiener’s Lemma for twisted convolution algebras
Theorem 19
([15, Lemma 3.3]) Suppose that \(\theta \in {{\mathrm{\mathbb {T}}}}\), \(\alpha \beta =\theta \mod 1\), and that \(f\in \ell ^1 ({{\mathrm{\mathbb {Z}}}}^2,{{\mathrm{\mathbb {C}}}})\) and \(\pi _{\alpha ,\beta }(f)\) is invertible on \(L^2({{\mathrm{\mathbb {R}}}},{{\mathrm{\mathbb {C}}}})\). Then f is invertible in \((\ell ^1 ({{\mathrm{\mathbb {Z}}}}^2,{{\mathrm{\mathbb {C}}}}),\natural _{\theta },^*)\).
As an immediate corollary of Theorem 19 one obtains the following Wiener Lemma for the discrete Heisenberg group.
Theorem 20
Let \(f\in {{\mathrm{\ell ^1(\mathbb {H},\mathbb {C})}}}\), then f is invertible if and only if \(\pi _{\alpha ,\beta }(f)\) is invertible for each nonzero pair \(\alpha ,\beta \in {{\mathrm{\mathbb {R}}}}\).
Proof
The result follows by combining Allan’s local principle with Wiener’s Lemma for twisted convolution algebras. \(\square \)
Finally, we give an alternative proof of Wiener’s Lemma for the twisted convolution algebra which relies on the representation theory of \({{\mathrm{\mathbb {H}}}}\) only. We start with the following lemmas.
Lemma 5
The twisted convolution algebra \((\ell ^1 ({{\mathrm{\mathbb {Z}}}}^2,{{\mathrm{\mathbb {C}}}}),\natural _{\theta },^*)\) is symmetric.
Proof
Lemma 6
Consider an irrational \(\theta \in {{\mathrm{\mathbb {T}}}}\). Then \(f \in (\ell ^1 ({{\mathrm{\mathbb {Z}}}}^2,{{\mathrm{\mathbb {C}}}}),\natural _{\theta },^*)\) is invertible if and only if \(\pi _{\alpha ,\beta }(f)\) has a bounded inverse, where \(\alpha \beta =\theta \mod 1\).
Proof
Let \(\theta \) be irrational and suppose \(a \in \mathcal Q _\theta \simeq (\ell ^1 ({{\mathrm{\mathbb {Z}}}}^2,{{\mathrm{\mathbb {C}}}}),\natural _{\theta },^*)\) is not invertible. We just have to show that the noninvertibility of the element a implies that a is not invertible in the irrational rotation algebra \(\mathcal R_\theta \) and, in particular, not in its realisation \(\pi _{\alpha ,\beta }(C^*({{\mathrm{\mathbb {H}}}}))\) with \(\alpha \beta =\theta \mod 1\). Since \(\mathcal Q_\theta \) is symmetric (cf. Lemma 5), there exists an irreducible unitary representation \(\pi \) of \({{\mathrm{\mathbb {H}}}}\) such that \(\pi \) vanishes on \(\mathtt J_\theta \) and \(\pi (\varPhi _\theta (a))v =0\) for some nonzero vector \(v\in \mathcal H _\pi \). This implies that a is not invertible in \(\mathcal R_\theta \). \(\square \)
The proof of Lemma 6 basically says that for irrational \(\theta \) the Banach algebra \(\mathcal Q _\theta \) is inverseclosed in \(\mathcal R_\theta \).
Lemma 7
Let \(\theta \in {{\mathrm{\mathbb {T}}}}\) be rational. Then \(a\in \mathcal Q _\theta \) is invertible if and only if \(\pi _{\alpha ,\beta }(a)\) is invertible in \(\mathcal B (L^2({{\mathrm{\mathbb {R}}}},{{\mathrm{\mathbb {C}}}}))\).
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 \(\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})\) has an application in algebraic dynamics. The following result is important to check invertibility for \(f \in {{\mathrm{\mathbb {Z}}}}[{{\mathrm{\mathbb {H}}}}]\) in the group algebra \(\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})\) because it tells us that \(\pi _{\alpha ,\beta }(f)\) has a trivial kernel in \(L^2({{\mathrm{\mathbb {R}}}},{{\mathrm{\mathbb {C}}}})\) for \(\alpha ,\beta \not = 0\).
Theorem 21
([25]) Let G be a nonzero element in \( L^2({{\mathrm{\mathbb {R}}}},{{\mathrm{\mathbb {C}}}})\), then for every finite set \(A \subseteq {{\mathrm{\mathbb {Z}}}}^2\) the set \(\{\text {T}_\alpha ^k\text {M}_\beta ^l G \,:\, (k,l)\in A\}\) is linear independent over \({{\mathrm{\mathbb {C}}}}\).
The following result is a reformulation of Theorem 21 and gives an exact description of the spectrum of an operator \(\pi _{\alpha ,\beta }(f)\), for \(\alpha ,\beta \in {{\mathrm{\mathbb {R}}}}{\backslash }\{0\}\) and \(f\in {{\mathrm{\mathbb {C}}}}[{{\mathrm{\mathbb {H}}}}]\), where \({{\mathrm{\mathbb {C}}}}[{{\mathrm{\mathbb {H}}}}]\) is the ring of functions \({{\mathrm{\mathbb {H}}}}\longrightarrow {{\mathrm{\mathbb {C}}}}\) with finite support.
Theorem 22
Let \(f \in {{\mathrm{\mathbb {C}}}}[{{\mathrm{\mathbb {H}}}}]\) with \(f^\theta \not = 0\) for \(\theta =\alpha \beta \not = 0\), \(\alpha ,\beta \in {{\mathrm{\mathbb {R}}}}\), then for all \(c\in \sigma (\pi _{\alpha ,\beta }(f))\) the operators \( \pi _{\alpha ,\beta }(c f)\) are injective and have dense range in \(L^2({{\mathrm{\mathbb {R}}}},{{\mathrm{\mathbb {C}}}})\) but are not bounded from below.
Proof

The dual of \({{\mathrm{\mathbb {H}}}}\): there is an irreducible representation \(\pi \) of \({{\mathrm{\mathbb {H}}}}\) such that 0 is an eigenvalue of \(\pi (f)\).

Stonevon Neumann representations: For all Stonevon Neumann representations \(\pi _{\alpha ,\beta }\), 0 is an eigenvalue of \(\pi _{\alpha ,\beta }(f)\) if and only if \(\pi _{\alpha ,\beta }(f)=0\); and \(\pi _{\alpha ,\beta }(f)\) is not invertible if and only if \(\pi _{\alpha ,\beta }(f)\) is not bounded from below.
Remark 3
The authors are not aware whether the approach based on Theorem 9 and the construction of the dual of \({{\mathrm{\mathbb {H}}}}\) via ergodic quasiinvariant measures are wellknown results in the field of TimeFrequency Analysis. It would be interesting to investigate whether this eigenvalue approach would simplify the problem of deciding on invertibility—at least—for some examples \(f \in \ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}}) {\backslash } {{\mathrm{\mathbb {C}}}}[{{\mathrm{\mathbb {H}}}}]\).
7 Examples
We now demonstrate how to apply Wiener’s Lemma to obtain easily verifiable sufficient conditions for nonexpansivity of a principal algebraic action.
7.1 Either \(\mathsf {U}(g_0)\) or \(\mathsf {U}(g_1)\) is a nonempty set
Theorem 23
 (i)
\(\mathsf {U}_\chi (g_0)=\varnothing \), \(\mathsf {U}_\chi (g_1)\not =\varnothing \) and \(\int \phi _\chi d\lambda _{{{\mathrm{\mathbb {S}}}}} <0\).
 (ii)
\(\mathsf {U}_\chi (g_0)\not =\varnothing \), \(\mathsf {U}_\chi (g_1)=\varnothing \) and \(\int \phi _\chi d\lambda _{{{\mathrm{\mathbb {S}}}}} >0\).
Proof
We will prove only the first case, the second case can be proved similarly.
Suppose f is such that \((X_f,\alpha _f)\) is expansive and the conditions in (i) are satisfied. We will now show that certain consequences of expansivity of \(\alpha _f\) are inconsistent with the conditions in (i). Hence, by arriving to a contradiction, we will prove that under (i) \(\alpha _f\) is not expansive.
The assumption that \(\int \phi _\chi d\lambda _{{{\mathrm{\mathbb {S}}}}} < 0\) cannot be dropped in (i) of Theorem 23 as the following simple minded example shows.
Example 6
 1.
\(K> \Vert g_1(y,z)\Vert _{\ell ^1({{\mathrm{\mathbb {H}}}},{{\mathrm{\mathbb {C}}}})}\);
 2.
\(\int \phi _\chi d\lambda _{{{\mathrm{\mathbb {S}}}}} > 0\).
7.2 The sets \(\mathsf {U}(g_0)\) and \(\mathsf {U}(g_1)\) are both nonempty
 1.
\(\mathsf {U}(g_0) \not = \varnothing \) and \(\mathsf {U}(g_1) \not = \varnothing \);
 2.and there exists an element \((\zeta ,\chi ) \in \mathsf {U}(g_0)\) with$$\begin{aligned} \{ (\eta ,\chi )\in {{\mathrm{\mathbb {S}}}}^2\,:\, \eta \in \text {Orb}_{\chi }(\zeta )\} \cap \mathsf {U}(g_1) \not = \varnothing . \end{aligned}$$
Theorem 24
Proof
Corollary 2
Let f be of the form (25) which satisfies the conditions of Theorem 24. Then \(\alpha _{f^\diamond }\) defined by \(f^\diamond =g_0(y,z)xg_1(y,z)\) is nonexpansive.
Proof
Example 7
Since the conditions of Theorem 24 and Corollary 2 are satisfied, f and \(f^\diamond \) are not invertible.
The next result can be easily deduced from the proof of Theorem 23.
Theorem 25
Proof
Suppose (46) holds. Let us first treat the trivial cases.
The case \(\mathfrak {m} (g_{0,\chi }) > \mathfrak {m} (g_{1,\chi })\) can be proved analogously. \(\square \)
Footnotes
 1.
A representation \(\pi \) of a Banach \(^*\)algebra \(\mathcal A\) is called nondegenerate if there is no nonzero vector \(v\in \mathcal H_\pi \) such that \(\pi (a)v=0\) for every \(a\in \mathcal A\).
 2.
To fix notation: for \(F\in L^2(\mathbb {T},\lambda _\mathbb {T})\) (where \(\lambda _\mathbb {T}\) is the Lebesgue measure on \(\mathbb {T}\)), the Fourier transform \(\hat{F}: \mathbb {Z}\longrightarrow \mathbb {C}\) is defined by \(\hat{F}_n=\int _\mathbb {T}F(s)e^{2\pi ins}\, d\lambda _\mathbb {T}(s)\). The Fourier transform \((\mathcal F g): \mathbb {T}\longrightarrow \mathbb {C}\) of \(g\in \ell ^2(\mathbb {Z},\mathbb {C})\) is defined by \((\mathcal F g)(s)=\sum _{n\in \mathbb {Z}}g_n e^{2\pi ins}\).
 3.
For an operator \(\text {A}\) acting on a Hilbert space \(\mathcal H\) one has \((\ker \text {A})^\perp = \overline{ \text {im } \text {A}^*}\).
Notes
Acknowledgments
The authors would like to thank Karlheinz Gröchenig, A.J.E.M. Janssen, Hanfeng Li and Doug Lind for helpful discussions and insights. Moreover, we thank Mike Keane for making us aware of the article [6] by Ian Brown. MG gratefully acknowledges support by a Huygens Fellowship from Leiden University. MG and EV would like to thank the Erwin Schrödinger Institute, Vienna, and KS the University of Leiden, for hospitality and support while some of this work was done.
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