# The ideal structure of measure algebras and asymptotic properties of group
representations

Jared T. White  Jared T. White, 31 Warham Road, London N4 1AR, United Kingdom
<EMAIL_ADDRESS>
(Date: 2021; Date: 2021)

###### Abstract.

We classify the weak*-closed maximal left ideals of the measure algebra $M(G)$
for certain Hermitian locally compact groups $G$ in terms of the irreducible
representations of $G$ and their asymptotic properties. In particular, we
obtain a classification for connected nilpotent Lie groups, and the Euclidean
rigid motion groups. We also prove a version of this result for certain
weighted measure algebras. We apply our classification to obtain an analogue
of Barnes’ Theorem on integrable representations for representations vanishing
at infinity. We next study the relationship between weak*-closedness and
finite generation, proving that in many cases $M(G)$ has no finitely-generated
maximal left ideals. We also show that $M(\mathbb{R}^{2}\rtimes SO(2))$ has a
weak*-closed maximal left ideal that is not generated by a projection, and
investigate whether or not it has any weak*-closed left ideals which are not
finitely-generated.

###### Key words and phrases:

locally compact group, measure algebra, irreducible representation, vanishing
at infinity, dual Banach algebra.

###### 2010 Mathematics Subject Classification:

43A05, 43A65 (primary); 43A20, 46H10 (secondary)

Open University

##  1\. Introduction

In this article we shall study the weak*-closed ideal structure of the measure
algebra of a locally compact group, and use the understanding gained to prove
new results about representations vanishing at infinity and about finitely-
generated left ideals.

In [21] the author proved that for a compact group $G$ there is a bijective
correspondence between the closed left ideals of $L^{1}(G)$ and the
weak*-closed left ideals of $M(G)$, and used this to give a classification of
the weak*-closed left ideals of $M(G)$ in terms of the representation theory
of $G$. In the present article we seek similar results that go beyond the
compact setting. For a non-compact locally compact group $G$ there is no
description available of the closed left ideals of $L^{1}(G)$ (and finding any
description seems to be an intractable problem), and as such we shall focus on
maximal left ideals. For a Hermitian locally compact group $G$ it is known
that the maximal modular left ideals of $L^{1}(G)$ all have the form
$\mathcal{I}_{\pi,\xi}:=\\{f\in L^{1}(G):\pi(f)\xi=0\\}$, for some irreducible
representation $\pi$ and some unit vector $\xi\in\mathcal{H}_{\pi}$ (although
it is unclear exactly which left ideals of this form are maximal modular). We
shall study the following conjecture. We define $\mathcal{J}_{\pi,\xi}$ to be
the closed left ideal of the measure algebra given by $\\{\mu\in
M(G):\pi(\mu)\xi=0\\}$.

######  Conjecture 1.1.

Let $G$ be a Hermitian locally compact group. The weak*-closed maximal left
ideals of $M(G)$ are given exactly by $\mathcal{J}_{\pi,\xi}$, where $\pi$ is
an irreducible representation vanishing at infinity, and $\xi$ is a unit
vector in $\mathcal{H}_{\pi}$ with the property that $\mathcal{I}_{\pi,\xi}$
is a maximal modular left ideal of $L^{1}(G)$.

Our main result is Theorem 4.11, which states that our conjecture is true for
a large class of Hermitian locally compact groups, which includes connected
nilpotent Lie groups, the Euclidean rigid motion groups $\mathbb{R}^{n}\rtimes
SO(n)$, and the Fell groups $\mathbb{Q}_{p}\rtimes\mathcal{O}_{p}$. We shall
also prove in Theorem 4.12 that certain weighted measure algebras enjoy a
bijective correspondence between their weak*-closed maximal left ideals, and
the maximal modular left ideals of the weighted $L^{1}$-algebra (without any
constraint on the representation), similar to the compact case.

We shall now explain some of the applications of our classification result. In
his seminal work of 1980 [1], Barnes established a correspondence between
integrable irreducible representations of a unimodular locally compact group,
and minimal projections in $L^{1}(G)$. In this context “minimal” means that
the left ideal generated by the projection is a minimal left ideal, which is a
stronger property than being minimal with respect to the partial order on
projections. Another important asymptotic property of a representation of a
locally compact group is that of vanishing at infinity. In this article we
make a conjecture relating weak*-closed maximal left ideals of the measure
algebra $M(G)$ of a Hermitian locally compact group $G$ and irreducible
representations vanishing at infinity that we believe should be thought of as
the correct analogue of Barnes’ Theorem in this setting.

######  Conjecture 1.2.

Let $G$ be a Hermitian locally compact group, and let $\pi\in\widehat{G}$.
Then there exists a unit vector $\xi\in\mathcal{H}_{\pi}$ for which
$\mathcal{J}_{\pi,\xi}$ is a weak*-closed maximal left ideal of $M(G)$ if and
only if $\pi$ vanishes at infinity.

In Proposition 5.2 we shall rephrase Barnes’ Theorem in terms of maximal left
ideals of the measure algebra, so that the analogy with our conjecture is more
transparent. In Theorem 5.4 we shall prove the conjecture for those groups
satisfying the hypothesis of Theorem 4.11 that are additionally CCR (which
still includes connected nilpotent Lie groups and the Euclidean rigid motion
groups).

Another application is to the theory of finitely-generated maximal left ideals
of measure algebras and group algebras, a topic previously studied by the
author in [19]. It was proved there that, when $G$ is not discrete, $L^{1}(G)$
has no finitely-generated maximal modular left ideals, but the case of
discrete $G$ remains open. The connection with our classification result is
that finitely-generated norm-closed left ideals of a measure algebra are
automatically weak*-closed. A consequence of our investigations shall be that,
for some locally compact groups, the measure algebra has no finitely-generated
maximal left ideals at all. In particular, when $G$ is finitely-generated and
virtually nilpotent $L^{1}(G)=M(G)$ has no finitely-generated maximal
(modular) left ideals, just like for non-discrete groups.

######  Theorem 1.3.

Let $G$ be a locally compact group which is any of the following:

  1. (i)

a non-compact Moore group;

  2. (ii)

a group with non-compact centre;

  3. (iii)

an infinite finitely-generated virtually nilpotent group;

  4. (iv)

a group of the form $\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$, for some $A\in
M_{2}(\mathbb{R})$;

Then $M(G)$ has no weak*-closed, and hence no finitely-generated, maximal left
ideals.

In case (iv), $\mathbb{R}$ acts on $\mathbb{R}^{2}$ via $t\colon
x\rightarrow{\rm e}^{tA}x\ (t\in\mathbb{R},\ x\in\mathbb{R}^{2}).$

We now discuss our final topic. We wish to better understand the relationship
between the following properties of a closed left ideal of a measure algebra:
(1) being generated by a projection, (2) being finitely-generated, and (3)
being weak*-closed. Each of these notions is implied by the previous one, and
it would be interesting to know whether or not they are all distinct. We are
able to partially resolve this question by looking at the two-dimensional
Euclidean rigid motion group in detail.

######  Theorem 1.4.

Let $G=\mathbb{R}^{2}\rtimes SO(2)$, and let
$U=\operatorname{ind}_{\mathbb{R}^{2}}^{G}\chi$, where $\chi$ is the character
on $\mathbb{R}^{2}$ given by $\chi(x)={\rm e}^{ix_{1}}\ (x\in\mathbb{R}^{2})$.
Then

  1. (i)

There exists a unit vector $\xi\in\mathcal{H}_{\pi}$ such that
$\mathcal{J}_{U,\xi}$ is a weak*-closed maximal left ideal that is not
generated by a projection.

  2. (ii)

The left ideal $\mathcal{J}_{U,1}$ is weak*-closed, and is not generated by
finitely many compactly supported measures.

We have been unable to resolve whether $\mathcal{J}_{U,1}$ is maximal, and
whether it is finitely-generated.

Let us compare the previous result with the situation when $G$ is compact. In
that case it follows from the classification theorem [21, Theorem 6.2] that
the weak*-closed maximal left ideals of $M(G)$ are given exactly by
$\mathcal{J}_{\pi,\xi}$, for $\pi\in\widehat{G}$ and $\xi\in\mathcal{H}_{\pi}$
of norm $1$. Moreover, it is routinely verified that $\mathcal{J}_{\pi,\xi}$
is equal to $M(G)*(\delta_{e}-p)$, where $p$ is the projection in $L^{1}(G)$
defined by
$p(t)=\frac{1}{\dim\mathcal{H}_{\pi}}\langle\pi(t^{-1})\xi,\xi\rangle\ (t\in
G)$. In particular, in this setting, a maximal left ideal of $M(G)$ is
weak*-closed if and only if it is generated by a projection.

The paper is organised as follows. In Section 2 we recall some background and
fix our notation. In Section 3 we prove that $M(G)$ cannot have a proper
weak*-closed left ideal of finite codimension unless $G$ is compact, and
derive some consequences of this. We begin Section 4 by proving some general
results that relate maximal modular left ideals of a Banach algebra $A$ to the
weak*-closed left ideals of its multiplier algebra $M(A)$ in the special case
that $M(A)$ is a dual Banach algebra. We then use these results to prove our
main classification result, Theorem 4.11, which establishes Conjecture 1.1 in
many cases, as well as Theorem 4.12, which addresses the weighted setting. In
Section 5 we discuss Barnes’ Theorem and prove Conjecture 1.2 for many
Hermitian locally compact groups. Next, in Section 6, we give a detailed
description of the weak*-closed maximal left ideals for some chosen examples.
Section 7 is concerned with proving Theorem 1.3 and Theorem 1.4. Finally, in
Section 8 we list some open problems.

##  2\. Background and Notation

For us $\mathbb{N}=\\{1,2,\ldots\\}$. Let $X$ be a set, and let $Y\subset X$
be any subset. We denote by $\mathbbm{1}_{Y}\colon X\rightarrow\\{0,1\\}$ the
characteristic function of $Y$.

Let $E$ be a Banach space. We define $B_{E}$ to be the unit ball of $E$ and
$S_{E}$ to be the unit sphere. We denote the dual space of $E$ by
$E^{\prime}$, and given a bounded linear map $\varphi\colon E\rightarrow F$
between Banach spaces, we write $\varphi^{\prime}\colon F^{\prime}\rightarrow
E^{\prime}$ for the dual map. Given subsets $X\subset E$ and $Y\subset
E^{\prime}$ we shall write

$X^{\perp}=\\{\lambda\in E^{\prime}:\langle x,\lambda\rangle=0\ (x\in
X)\\}\qquad\text{and}\qquad Y_{\perp}=\\{x\in E:\langle x,\lambda\rangle=0\
(\lambda\in Y)\\}.$

Let $A$ be an algebra, and let $I$ be a left ideal of $A$. We say that $u\in
A$ is a modular right identity for $I$ if $a-au\in I$ for all $a\in A$. In the
case that $I$ has a modular right identity we say that $I$ is a modular left
ideal. By a maximal modular left ideal we mean a maximal left ideal that is
modular, or equivalently a left ideal that is maximal subject to being
modular. We say that $A$ is semisimple if the intersection of the maximal
modular left ideals of $A$ is $\\{0\\}$.

Let $G$ be a locally compact group. We write $\int\cdots{\rm\,d}t$ for
integrals with respect to the left Haar measure on $G$, which we always take
to be normalised when $G$ is compact.

By a representation of $G$ we shall always mean a strongly continuous unitary
representation of $G$ on a Hilbert space. Typically a representation will be
denoted by $(\pi,\mathcal{H}_{\pi})$, where $\pi$ is a homomorphism from $G$
to the group of unitaries on the Hilbert space $\mathcal{H}_{\pi}$. We shall
write $\widehat{G}$ for the unitary dual of $G$, that is the collection of
irreducible representations of $G$ up to unitary equivalence. Given a
representation $(\pi,\mathcal{H}_{\pi})$, and $\xi,\eta\in\mathcal{H}_{\pi}$
we shall write $\xi*_{\pi}\eta$ for the function $G\rightarrow\mathbb{C}$
given by $(\xi*_{\pi}\eta)(t)=\langle\pi(t)\xi,\eta\rangle\ (t\in G)$. We
shall also write $\operatorname{pker}\pi=\\{t\in
G:\pi(t)\in\mathbb{C}\operatorname{id}_{\mathcal{H}_{\pi}}\\}$, the projective
kernel of $\pi$, which is a closed normal subgroup of $G$. Note that, given
vectors $\xi,\eta$, the function
$G/\operatorname{pker}\pi\rightarrow[0,\infty)$ given by
$t\operatorname{pker}\pi\mapsto|\langle\pi(t)\xi,\eta\rangle|\ (t\in G)$ is
well-defined and continuous, and we say that $\pi$ vanishes at infinity modulo
its projective kernel if this function belongs to
$C_{0}(G/\operatorname{pker}\pi)$ for each $\xi,\eta\in\mathcal{H}_{\pi}$.

We say that a group $G$ is CCR if, for every $\pi\in\widehat{G}$ and every
$f\in L^{1}(G)$, the operator $\pi(f)$ is compact; this is equivalent to the
Fell topology on $\widehat{G}$ being $T_{1}$. See [6, Chapter 7] for more
details.

Let $(\pi,\mathcal{H}_{\pi})$ be a representation of the locally compact group
$G$. We recall the definitions of the following asymptotic properties that
$\pi$ could have.

######  Definition 2.1.

We say that

  1. (i)

$\pi$ vanishes at infinity if, for every $\xi,\eta\in\mathcal{H}_{\pi}$ we
have $\xi*_{\pi}\eta\in C_{0}(G)$;

  2. (ii)

$\pi$ is integrable if $\xi*_{\pi}\eta\in L^{1}(G)$ for some
$\xi,\eta\in\mathcal{H}_{\pi}\setminus\\{0\\}$;

For an irreducible representation $\pi$ it is known that $\pi$ vanishes at
infinity if and only if there exist some
$\xi,\eta\in\mathcal{H}_{\pi}\setminus\\{0\\}$ such that $\xi*_{\pi}\eta\in
C_{0}(G)$. For background on asymptotic properties of representations see [5].

The main focus of this article will be the measure algebra of a locally
compact group, but we shall also prove a couple of results in the weighted
setting, which we recall now. By a weight on $G$ we mean a continuous function
$\omega\colon G\rightarrow[1,\infty)$ such that $\omega(e)=1$, and
$\omega(st)\leq\omega(s)\omega(t)\ (s,t\in G)$. Many authors do not insist
that weights need to be continuous, but we find this assumption convenient for
the present article. We then define

$L^{1}(G,\omega)=\left\\{f\in
L^{1}(G):\int_{G}|f(t)|\omega(t){\rm\,d}t<\infty\right\\},$

and

$M(G,\omega)=\left\\{\mu\in
M(G):\int_{G}\omega(t){\rm\,d}|\mu|(t)<\infty\right\\}.$

We define a norm on $M(G,\omega)$ by
$\|\mu\|_{\omega}=\int_{G}\omega(t){\rm\,d}|\mu|(t)$, and this makes it into a
Banach algebra, with multiplication given by convolution. It has a predual
given by

$C_{0}(G,1/\omega)=\\{f\colon G\rightarrow\mathbb{C}:f/\omega\in C_{0}(G)\\}.$

Moreover, $L^{1}(G,\omega)$ is a closed ideal of $M(G,\omega)$. When
$\omega(t)=\omega(t^{-1})$ for all $t\in G$ both $M(G,\omega)$ and
$L^{1}(G,\omega)$ are Banach *-algebras, with the involution given by

$f^{*}(t)=\Delta(t^{-1})\overline{f(t^{-1})}\qquad(t\in G),$

where $\Delta$ is the modular function of $G$.

Given $\pi\in\widehat{G}$ and $\xi\in\mathcal{H}_{\pi}$, we shall write

$\mathcal{I}_{\pi,\xi}=\\{f\in
L^{1}(G):\pi(f)\xi=0\\}\qquad\text{and}\qquad\mathcal{J}_{\pi,\xi}=\\{\mu\in
M(G):\pi(\mu)\xi=0\\}.$

Moreover, given a weight $\omega$ on $G$, we shall write
$\mathcal{I}_{\pi,\xi,\omega}=\mathcal{I}_{\pi,\xi}\cap L^{1}(G,\omega)$ and
$\mathcal{J}_{\pi,\xi,\omega}=\mathcal{J}_{\pi,\xi}\cap M(G,\omega).$ These
are always closed left ideals of $L^{1}(G,\omega)$ and $M(G,\omega)$
respectively.

Let $A$ be a Banach *-algebra. By a projection we mean a self-adjoint
idempotent. We say that $p\in A$ is a minimal projection if $Ap$ is a minimal
left ideal in $A$. When $A$ is a C*-algebra or has the form $L^{1}(G)$ for
some locally compact group $G$, a projection $p\in A$ is minimal if and only
if $pAp=\mathbb{C}p$. However, this is in general a stronger condition than
being minimal with respect to the usual partial order on projections.

There has long been known a procedure for constructing an irreducible
*-representation $\pi$ from a minimal projection $p$ in a Banach *-algebra
$A$, summarised by Barnes in [1]. Another important contribution to this
theory is Valette’s paper [22]. Following Barnes, we say that $\pi$ is
determined by $p$. Barnes [1, Theorem 1] proved the following theorem,
relating integrable representations with minimal projections.

######  Theorem 2.2 (Barnes).

Let $G$ be a unimodular locally compact group and let $\pi\in\widehat{G}$.
Then $\pi$ is integrable if and only if $\pi$ is determined by a minimal
projection in $L^{1}(G)$.

Since minimal projections are defined as those that generate minimal left
ideals, Barnes’ Theorem tacitly establishes a link between left ideals and
integrable representations. In Proposition 5.2 below we shall reformulate this
result in terms of maximal left ideals of the measure algebra of the group.

We say that a Banach *-algebra $A$ is Hermitian if every self-adjoint element
of $A$ has real spectrum. A locally compact group $G$ is called Hermitian if
$L^{1}(G)$ is Hermitian. Examples of Hermitian locally compact groups include
compact extensions of nilpotent locally compact groups and Moore groups [16].
It has recently been shown by Samei and Weirsma [18] that Hermitian locally
compact groups are amenable, solving a long-standing open problem.

Our interest in Hermitian Banach *-algebras is due to Palmer’s Theorem [14],
which states that a Banach *-algebra $A$ is Hermitian if and only if every
maximal modular left ideal of $A$ has the form $\\{a\in A:\psi(a^{*}a)=0\\}$,
for some pure state $\psi$ on $A$. For the special case of a Hermitian group
algebra, this theorem states the following.

######  Theorem 2.3 (Palmer).

Let $G$ be a Hermitian locally compact group. The maximal modular left ideals
of $L^{1}(G)$ all have the form $\mathcal{I}_{\pi,\xi},$ where
$\pi\in\widehat{G}$, and $\xi$ is a unit vector in $\mathcal{H}_{\pi}$.

Unfortunately, Palmer’s Theorem does not tell us which of the left ideals
$\mathcal{I}_{\pi,\xi}$ are maximal modular, and it seems to us that this is
generally difficult to determine. Since $L^{1}(G)$ is semisimple, we know that
it has enough maximal modular left ideals to separate points. We shall prove a
more precise result for Hermitian CCR locally compact groups in Proposition
5.3.

We next recall some background on dual Banach algebras. A dual Banach algebra
is a Banach algebra $A$ that has a Banach space predual $X$ in such a way that
the multiplication on $A$ is separately weak*-continuous. For example, the
measure algebra of a locally compact group $G$ is a dual Banach algebra with
predual $C_{0}(G)$; more generally, given a weight $\omega$ on $G$, the
weighted measure algebra $M(G,\omega)$ is a dual Banach algebra with predual
$C_{0}(G,1/\omega)$ (as verified in [20]). This is where we use the assumption
that $\omega$ is continuous. Other examples include von Neumann algebras, and
the Fourier-Stieltjes algebra $B(G)$ of a locally compact group $G$.

Given a Banach algebra $A$ we shall write $M(A)$ for its multiplier algebra.
Recall that $A$ is faithful if $Ax=\\{0\\}$ implies that $x=0\ (x\in A)$, and
also $xA=\\{0\\}$ implies that $x=0\ (x\in A)$. The map canonical
$A\rightarrow M(A)$ is injective whenever $A$ is faithful, and we shall
typically identify a faithful Banach algebra with its image inside $M(A)$.
When $A$ has a bounded approximate identity of bound 1, the canonical map is
an isometry. This is the case for Banach algebras of the form
$L^{1}(G,\omega)$, and in this case the multiplier algebra may be identified
isometrically with $M(G,\omega)$ [3, Theorem 7.14]. In [4, 8, 21] multiplier
algebras that are also dual Banach algebras were studied, and in [21]
correspondences between the ideal structure of a Banach algebra and that of
its multiplier algebra were explored. The present work could be seen as a
continuation of these investigations.

##  3\. Left Ideals of Finite Codimension

In this section we shall prove that, for a locally compact group $G$, the
measure algebra $M(G)$ can never have a weak*-closed left ideal of finite
codimension unless $G$ is compact, and note some of the immediate consequences
of this fact. Our strategy is to prove the corresponding result for the
predual, namely that, when $G$ is not compact, $C_{0}(G)$ has no non-zero
finite-dimensional linear subspaces which are invariant under left
translation.

We begin by proving a short lemma.

######  Lemma 3.1.

Let $n\in\mathbb{N}$, and let $\lambda_{0},\ldots,\lambda_{n}\in\mathbb{C}$,
and suppose that

$|\lambda_{j}|\leq 2^{-n}\sum_{i\neq j}|\lambda_{i}|\quad(j=0,\ldots,n).$

Then $\lambda_{0}=\cdots=\lambda_{n}=0$.

###### Proof.

We proceed by induction on $n$, with the base case $n=0$ being trivial (with
empty sums interpreted as zero). Suppose that $n\geq 1$. Fix
$j\in\\{0,\ldots,n\\}$. We may choose $k\in\\{0,\ldots,n\\}$ with $j\neq k$.
Our hypothesis implies that, for every $p\neq k$, we have

$\displaystyle|\lambda_{p}|$ $\displaystyle\leq 2^{-n}\sum_{i\neq
p}|\lambda_{i}|=2^{-n}\left(|\lambda_{k}|+\sum_{i\neq
p,k}|\lambda_{i}|\right)\leq 2^{-n}\left(2^{-n}\sum_{i\neq
k}|\lambda_{i}|+\sum_{i\neq p,k}|\lambda_{i}|\right)$
$\displaystyle=2^{-2n}|\lambda_{p}|+(2^{-n}+2^{-2n})\sum_{i\neq
p,k}|\lambda_{i}|.$

Rearranging, we obtain

$|\lambda_{p}|\leq\frac{2^{-n}+2^{-2n}}{1-2^{-2n}}\sum_{i\neq
p,k}|\lambda_{i}|=\frac{2^{-n}}{1-2^{-n}}\sum_{i\neq p,k}|\lambda_{i}|\leq
2^{-(n-1)}\sum_{i\neq p,k}|\lambda_{i}|$

for every $p\neq k$. We may now apply the induction hypothesis to conclude
that $\lambda_{p}=0$ for every $p\neq k$, so that in particular
$\lambda_{j}=0.$ As $j$ was arbitrary, this gives the result. ∎

In the next proposition, given $t\in G$ and $\phi\in C_{0}(G)$, we write
$(L_{t}\phi)(s):=\phi(ts)\ (s\in G)$.

######  Proposition 3.2.

Let $G$ be a locally compact group. Then $C_{0}(G)$ contains a non-zero
finite-dimensional linear subspace invariant under left translation if and
only if $G$ is compact.

###### Proof.

If $G$ is compact, then $\mathbb{C}1$ is such a subspace. Suppose that $G$ is
not compact, and assume towards a contradiction that there is a left-
translation-invariant linear subspace $F\subset C_{0}(G)$ of dimension
$n\in\mathbb{N}$.

Choose $\phi\in F$ of norm $1$, and let $x_{0}\in G$ satisfy
$1=|\phi(x_{0})|$. Let $K\subset G$ be a compact subset such that
$|\phi(t)|<2^{-n}$ whenever $t\in G\setminus K$. Inductively choose elements
$t_{0},t_{1},\ldots,t_{n}\in G$ with $t_{0}=e$, and such that

$t_{j}\notin\left(\bigcup_{i=0}^{j-1}x_{0}K^{-1}t_{i}\right)\cup\left(\bigcup_{i=0}^{j-1}Kx_{0}^{-1}t_{i}\right)\quad(j=1,\ldots,n).$

This is possible since each such set is compact, whereas $G$ is not. Our
sequence $(t_{j})$ satisfies

$|\phi(t_{i}t_{j}^{-1}x_{0})|<2^{-n}\quad(i\neq j,\ i,j=0,\ldots,n).$

Since each function $L_{t_{i}}\phi$ also belongs to $F$, which has dimension
$n$, there must exist $\lambda_{0},\ldots,\lambda_{n}\in\mathbb{C}$, not all
zero, such that

$\lambda_{0}L_{t_{0}}\phi+\cdots+\lambda_{n}L_{t_{n}}\phi=0.$

Given $j\in\\{0,\ldots,n\\}$ we have

$\displaystyle|\lambda_{j}|$
$\displaystyle=|\lambda_{j}\phi(x_{0})|=|\lambda_{j}(L_{t_{j}}\phi)(t_{j}^{-1}x_{0})|=\left|\sum_{i\neq
j}\lambda_{i}(L_{t_{i}}\phi)(t_{j}^{-1}x_{0})\right|=\left|\sum_{i\neq
j}\lambda_{i}\phi(t_{i}t_{j}^{-1}x_{0})\right|\leq 2^{-n}\sum_{i\neq
j}|\lambda_{i}|,$

which implies that $\lambda_{j}=0$ for all $j\in\\{0,\ldots,n\\}$ by Lemma
3.1. This contradiction completes the proof. ∎

######  Corollary 3.3.

Let $G$ be a locally compact group which is not compact. Then $M(G)$ contains
no proper weak*-closed left ideals of finite codimension, and hence no
finitely-generated, closed left ideals of finite codimension.

###### Proof.

Assume towards a contradiction that $M(G)$ has a proper weak*-closed left
ideal $J$ of finite codimension. By [19, Lemma 3.3], $J_{\perp}$ is a non-zero
closed linear subspace of $C_{0}(G)$, invariant under left translation, and
since $(J_{\perp})^{\prime}\cong\frac{M(G)}{J}$, it is finite-dimensional.
This contradicts Proposition 3.2. ∎

One consequence of this corollary is that $M(G)$ has no weak*-closed maximal
left ideals when $G$ is a non-compact Moore (see Theorem 1.3(i), proved in
Section 7 below). Another consequence is the following result, which puts a
structural constraint on those locally compact groups $G$ for which $M(G)$ has
a weak*-closed maximal left ideal.

######  Proposition 3.4.

Let $G$ be a locally compact group such that $M(G)$ has a weak*-closed maximal
left ideal. Then $G$ has compact centre.

###### Proof.

Let $J$ be a weak*-closed maximal left ideal of $M(G)$, and consider $I=J\cap
M(Z(G))$. Then $I$ is closed in the weak*-topology on $M(Z(G))$. Let $T\colon
M(G)\rightarrow\mathcal{B}(M(G)/J)$ be given by $T(\mu)(\nu+J)=\mu*\nu+J\
(\mu,\nu\in M(G)).$ Then since $T[M(Z(G))]\subset Z(T[M(G)])$, and $M(G)/J$ is
a simple left $M(G)$-module, we must have
$T[M(Z(G))]=\mathbb{C}\operatorname{id}_{M(G)/J}$ by [15, Theorem 4.2.11].
Hence $\ker T|_{M(Z(G))}$ has codimension-1 in $M(Z(G))$, but also clearly
$\ker T|_{M(Z(G))}\subset I$, so that $I$ has codimension-1. It now follows
from Corollary 3.3 that $Z(G)$ is compact, as required. ∎

We shall show in Example 6.3 below that there are locally compact groups $G$
for which $M(G)$ has no weak*-closed maximal left ideals, but for which $G$
has trivial centre. In Corollary 4.9 we shall show that Proposition 3.4 can be
strengthened in the case of a Hermitian group.

##  4\. Classification Results for Weak*-Closed Maximal Left Ideals

In this section we shall give a classification of the weak*-closed maximal
left ideals of $M(G)$, valid for $G$ belonging to a large family of Hermitian
locally compact groups. Our classification shall be in terms of the
irreducible representations of $G$ and the maximal modular left ideals of
$L^{1}(G)$. The latter objects seem very difficult to determine if the group
has infinite dimensional irreducible representations, even when the
representation theory is well understood. But, as we shall see in subsequent
sections, our classification is nonetheless good enough to give us valuable
insights into the ideal structure of $M(G)$. As we shall detail in Section 7,
for many examples our classification will actually show that the measure
algebra has no weak*-closed maximal left ideals, and hence no finitely-
generated maximal left ideals either.

We begin by proving some lemmas in a very general context. We hope that this
will allow us to prove classification results about weak*-closed left ideals
of other multiplier algebras which are also dual Banach algebras, such as the
Fourier–Stieltjes algebra of a locally compact amenable group, in future work.
We shall make use of these general lemmas in this article to study the ideal
structure of measure algebras and their weighted analogues.

The first lemma tells us how to determine maximality of a left ideal of $M(A)$
by looking at a left ideal of $A$, in the special case that $M(A)$ is a dual
Banach algebra and $A$ is faithful and weak*-dense in $M(A)$. Note that, by
[21, Lemma 5.2], $A$ is weak*-dense in $M(A)$ whenever $M(A)$ is a dual Banach
algebra and $A$ has a bounded approximate identity.

######  Lemma 4.1.

Let $B$ be a dual Banach algebra, and let $A$ be a weak*-dense ideal of $B$.

  1. (i)

If $J$ is a weak*-closed left ideal of $B$ such that $J\cap A$ is a maximal
modular left ideal of $A$ then $J$ is maximal.

  2. (ii)

If $I$ is a maximal modular left ideal of $A$, then either
$\overline{I}^{w^{*}}$ is a maximal left ideal in $B$, or else $I$ is
weak*-dense in $B$.

###### Proof.

(i) Set $I=J\cap A$, and let $u\in A$ be a modular right identity for $I$. Let
$b\in B$. By weak*-density of $A$ in $B$ we may take a net $(a_{\alpha})$ in
$A$ which converges to $b$ in the weak*-topology. Then as $B$ is a dual Banach
algebra $a_{\alpha}u$ weak*-converges to $bu$. Since $u$ is a modular right
identity for $I$, for all $\alpha$ we have $a_{\alpha}-a_{\alpha}u\in I\subset
J$, and hence

$\lim_{w^{*},\,\alpha}a_{\alpha}-a_{\alpha}u=b-bu\in J$

by weak*-closedness. Hence $b=(b-bu)+bu\in J+A$. As $b$ was arbitrary we have
$J+A=B$.

Observe that as left $B$-modules we have

(4.1) $\frac{B}{J}=\frac{J+A}{J}\cong\frac{A}{J\cap A}=\frac{A}{I}.$

Since $A/I$ is a simple left $A$-module, it is also a simple left $B$-module,
and hence $B/J$ is simple by the isomorphism. It follows that $J$ is a maximal
left ideal.

(ii) We shall suppose that $I$ is not weak*-dense in $B$, and show that
$J:=\overline{I}^{w^{*}}$ must be maximal. Indeed, in this case we cannot have
$J\supset A$, and hence $J\cap A$ is a proper left ideal of $A$ containing
$I$, which is equal to $I$ by maximality. Hence the result follows from (i). ∎

######  Corollary 4.2.

Let $G$ be a locally compact group, and let $\omega$ be a weight on $G$. Let
$\pi\in\widehat{G}$ and let $\xi\in S_{\mathcal{H}_{\pi}}$, and suppose that
$\mathcal{J}_{\pi,\xi,\omega}$ is weak*-closed. If
$\mathcal{I}_{\pi,\xi,\omega}$ is a maximal modular left ideal of
$L^{1}(G,\omega)$, then $\mathcal{J}_{\pi,\xi}$ is a maximal left ideal of
$M(G)$.

###### Proof.

Since $\mathcal{I}_{\pi,\xi,\omega}=\mathcal{J}_{\pi,\xi,\omega}\cap
L^{1}(G,\omega)$, this follows from Lemma 4.1(i). ∎

Our next lemma complements Lemma 4.1, and tells us how weak*-closed left
ideals in $M(A)$ give us maximal left ideals in $A$ itself.

######  Lemma 4.3.

Let $A$ be a Banach algebra with a bounded approximate identity, such that
$M(A)$ is a dual Banach algebra. Let $J$ be a maximal left ideal of $M(A)$
which is weak*-closed. Then $J\cap A$ is a maximal modular left ideal in $A$.

###### Proof.

Set $I=J\cap A$, and note that $A$ is weak*-dense in $M(A)$ by [21, Lemma
5.2]. Since $J$ is proper and weak*-closed it does not contain $A$, so that
$I$ is proper. Also we have $J+A=M(A)$ by maximality. Hence we may take $u\in
A$ satisfying $1-u\in J$, and for all $a\in A$ it then follows that $a-au\in
I$, so that $u$ is a modular right identity for $I$. As such $I$ is a proper
modular left ideal of $A$. Note that we cannot have $I=\\{0\\}$, since this
would mean that $J$ annihilates $A$, contradicting the fact that it contains
non-zero multipliers.

Observe that the isomorphism (4.1) again holds, this time with $B=M(A)$. Since
$M(A)/J$ is a simple left $M(A)$-module, so is $A/I$. To see that $A/I$ is in
fact a simple left $A$-module, consider a maximal modular left ideal $K$ of
$A$ containing $I$, and note that $K$ is also a left ideal in $M(A)$ since $A$
has an approximate identity. This forces $K=I$ by $M(A)$-simplicity of $A/I$.
Hence $I$ is maximal modular, completing the proof. ∎

######  Corollary 4.4.

Let $G$ be a locally compact Hermitian group, and let $J$ be a weak*-closed
maximal left ideal of $M(G)$. Then there exists $\pi\in\widehat{G}$ and
$\xi\in S_{\mathcal{H}_{\pi}}$ such that $J=\mathcal{J}_{\pi,\xi}$.

###### Proof.

By Lemma 4.3 $J\cap L^{1}(G)$ is a maximal modular left ideal in $L^{1}(G)$,
and hence has the form $\mathcal{I}_{\pi,\xi}$, for some $\pi\in\widehat{G}$
and $\xi\in S_{\mathcal{H}_{\pi}}$ by Palmer’s Theorem (Theorem 2.3). Let
$(e_{\alpha})\subset L^{1}(G)$ be a net that weak*-converges to $\delta_{e}$.
Then for all $\mu\in\mathcal{J}_{\pi,\xi}$ we have $(e_{\alpha}*\mu)\subset
L^{1}(G)\cap\mathcal{J}_{\pi,\xi}$, and also $\lim_{\alpha}e_{\alpha}*\mu=\mu$
in the weak*-topology. Therefore
$\mathcal{J}_{\pi,\xi}\subset\overline{L^{1}(G)\cap\mathcal{J}_{\pi,\xi}}^{w^{*}}$.
The same argument shows that $J\subset\overline{L^{1}(G)\cap J}^{w^{*}},$ from
which it follows that $J=\overline{L^{1}(G)\cap J}^{w^{*}}$ because $J$ is
weak*-closed. Hence

$\mathcal{J}_{\pi,\xi}\subset\overline{L^{1}(G)\cap\mathcal{J}_{\pi,\xi}}^{w^{*}}=\overline{\mathcal{I}_{\pi,\xi}}^{w^{*}}=\overline{L^{1}(G)\cap
J}^{w^{*}}=J.$

By Lemma 4.1(ii) $\mathcal{J}_{\pi,\xi}$ is a maximal left ideal in $M(G)$,
and so $J=\mathcal{J}_{\pi,\xi}$, as required. ∎

The next lemma has consequences both for the weighted and unweighted case.
Given a Hilbert space $\mathcal{H}$, we shall denote the conjugate Hilbert
space by $\overline{\mathcal{H}}$. Also, for a dual Banach space $E$ with
predual $X$, we shall write $\langle x,\lambda\rangle$ as $\langle
x,\lambda\rangle_{(X,\,E)}$, for $x\in X$ and $\lambda\in E$, when we wish to
clarify the exact dual pairing.

######  Lemma 4.5.

Let $G$ be a locally compact group, and let $\omega$ be a weight on $G$. Let
$(\pi,\mathcal{H}_{\pi})$ be a representation of $G$, and suppose either that
$\pi$ vanishes at infinity, or that $1/\omega\in C_{0}(G)$. Then for each
$\xi\in\mathcal{H}_{\pi}$ the map

$\theta_{\xi}\colon
M(G,\omega)\rightarrow\mathcal{H}_{\pi},\qquad\mu\mapsto\pi(\mu)\xi$

is weak*-weak* continuous, with preadjoint given by

$\varphi_{\xi}\colon\overline{\mathcal{H}}_{\pi}\rightarrow
C_{0}(G,1/\omega),\qquad\eta\mapsto\overline{\xi*_{\pi}\eta}.$

###### Proof.

If $\pi$ vanishes at infinity then the map $\varphi_{\xi}$ is well-defined
since in this case $\xi*_{\pi}\eta\in C_{0}(G)\subset C_{0}(G,1/\omega)$ for
all $\eta\in\overline{\mathcal{H}}_{\pi}$. Similarly, for general $\pi$ the
condition $1/\omega\in C_{0}(G)$ ensures that $\varphi_{\xi}$ is well-defined,
since $\xi*_{\pi}\eta$ is always bounded. In either case $\varphi_{\xi}$ is a
bounded linear map with $\|\varphi_{\xi}\|\leq\|\xi\|$. Moreover, for every
$\eta\in\overline{\mathcal{H}}_{\pi}$, and every $\mu\in M(G)$, we have

$\displaystyle\langle\eta,\theta_{\xi}(\mu)\rangle_{(\overline{\mathcal{H}}_{\pi},\,\mathcal{H}_{\pi})}$
$\displaystyle=\langle\eta,\pi(\mu)\xi\rangle=\overline{\langle\pi(\mu)\xi,\eta\rangle}=\int_{G}\overline{\langle\pi(t)\xi,\eta\rangle}{\rm\,d}\mu(t)$
$\displaystyle=\langle\overline{\xi*_{\pi}\eta},\mu\rangle_{(C_{0}(G,1/\omega),\,M(G,\omega))}=\langle\eta,\varphi_{\xi}^{\prime}(\mu)\rangle_{(\overline{\mathcal{H}}_{\pi},\,\mathcal{H}_{\pi})}.$

Hence $\theta_{\xi}=\varphi_{\xi}^{\prime}$. ∎

######  Corollary 4.6.

Let $G$ be a locally compact group, let $(\pi,\mathcal{H}_{\pi})$ be a
representation of $G$ which vanishes at infinity, and let
$\xi\in\mathcal{H}_{\pi}$.

  1. (i)

The left ideal of $M(G)$ given by $\mathcal{J}_{\pi,\xi}$ is weak*-closed.

  2. (ii)

If $\mathcal{I}_{\pi,\xi}$ is a maximal modular left ideal of $L^{1}(G)$, then
$\mathcal{J}_{\pi,\xi}$ is a weak*-closed maximal left ideal of $M(G)$.

###### Proof.

In the notation of Lemma 4.5, $\mathcal{J}_{\pi,\xi}=\ker\theta_{\xi}$, and
part (i) now follows from that lemma. Part (ii) then follows from Corollary
4.2. ∎

We record the following folklore result for the convenience of the reader.

######  Lemma 4.7 (folklore).

Let $\mathcal{M}$ be a commutative von Neumann algebra which admits an ergodic
action by automorphisms by a finite group $G$. Then $\mathcal{M}$ is finite
dimensional.

###### Proof.

Assume towards a contradiction that $\mathcal{M}$ is infinite dimensional, and
let $u_{1},u_{2},\ldots$ be an infinite linearly-independent subset of
$\mathcal{M}$. Let $\mathcal{N}$ be the von Neumann algebra generated by
$\\{gu_{i}:g\in G,i\in\mathbb{N}\\}$. Then $G$ also acts ergodically on
$\mathcal{N}$, which is separable, and as such may be identified with
$L^{\infty}(X)$, for $X$ one of the following spaces with the obvious measure:
$[0,1],\mathbb{N},[0,1]\cup\mathbb{N}$, or
$\\{1,\ldots,n\\},[0,1]\cup\\{1,\ldots,n\\}$ for some $n\in\mathbb{N}$. The
action of $G$ on $\mathcal{N}$ corresponds to an ergodic action of $G$ on the
underlying measure space $X$. However, there is no such action of a finite
group on any of the infinite sets listed above, and hence
$\mathcal{N}=L^{\infty}\\{1,\ldots,n\\}$ for some $n\in\mathbb{N}$. In
particular $\mathcal{N}$ is finite-dimensional, contradicting the choice of
$\\{u_{1},u_{2},\ldots\\}$. ∎

The following proposition is the remaining key ingredient in the proof of our
main classification theorem (Theorem 4.11). It will also be an important tool
in proving Theorem 1.3(iii).

######  Proposition 4.8.

Let $G$ be a locally compact group, and let $\pi\in\widehat{G}$. Suppose that
$G$ has a finite-index, closed normal subgroup $N$ such that
$\pi^{-1}(Z(\pi(N)))$ is not compact. Then for every $\xi\in
S_{\mathcal{H}_{\pi}}$ the left ideal $\mathcal{J}_{\pi,\xi}$ is weak*-dense
in $M(G)$.

###### Proof.

Suppose towards a contradiction that there exists $\xi\in
S_{\mathcal{H}_{\pi}}$ such that $\mathcal{J}_{\pi,\xi}$ is weak*-closed.
Consider $\mathcal{M}$, the von Neumann algebra generated by $\pi(Z(N))$. The
action of $G$ on $Z(N)$ by conjugation induces an action of $G/N$ on
$\mathcal{M}$ by automorphisms, and the fixed point set for this action
consists of operators which commute with $\pi(G)$, and hence consists of
scalars. In other words, $G/N$ acts ergodically on $\mathcal{M}$, forcing
$\mathcal{M}$ to be finite dimensional by Lemma 4.7.

Let $H=\pi^{-1}(Z(\pi(N)))$, which is a closed subgroup of $G$. Set
$\mathcal{K}=\mathcal{M}\xi$, which is a finite-dimensional linear subspace of
$\mathcal{H}_{\pi}$, and define $\rho\colon
H\rightarrow\mathcal{B}(\mathcal{K})$ to be

$\rho(z)\eta=\pi(z)\eta\quad(z\in H,\ \eta\in\mathcal{K}).$

Since $\rho$ is a finite-dimensional representation of $H$, the left ideal
$\mathcal{J}_{\rho,\xi}$ of $M(H)$ has finite codimension, and hence must be
weak*-dense by Corollary 3.3. Moreover, we have $\mathcal{J}_{\pi,\xi}\cap
M(H)=\mathcal{J}_{\rho,\xi}$.

Let $(\mu_{\alpha})\subset\mathcal{J}_{\rho,\xi}$ be a net converging in the
weak*-topology on $M(H)$ to $\delta_{e}$. Given $f\in C_{0}(G)$, we have

$\int_{G}f{\rm\,d}\mu_{\alpha}=\int_{H}f{\rm\,d}\mu_{\alpha}\rightarrow f(e),$

since each $\mu_{\alpha}$ is supported on $H$, and $f|_{H}\in C_{0}(H)$. Hence
$\mu_{\alpha}$ converges to $\delta_{e}$ with respect to the weak*-topology
$M(G)$ as well. We have
$\delta_{e}\in\overline{\mathcal{J}_{\pi,\xi}}^{w^{*}}$, which is a left ideal
in $M(G)$, and hence $\overline{\mathcal{J}_{\pi,\xi}}^{w^{*}}=M(G)$, as
required. ∎

######  Corollary 4.9.

Let $G$ be a locally compact group.

  1. (i)

If $\pi\in\widehat{G}$ has the property that $\operatorname{pker}\pi$ is not
compact, then $\mathcal{J}_{\pi,\xi}$ is weak*-dense in $M(G)$ for every
$\xi\in S_{\mathcal{H}_{\pi}}$.

  2. (ii)

If $G$ is Hermitian and has a finite index closed normal subgroup $N$ such
that $Z(N)$ is non-compact, then $M(G)$ has no weak*-closed, maximal left
ideals. ∎

The next lemma gives a converse to Corollary 4.6 for a particular class of
groups.

######  Lemma 4.10.

Let $G$ be a locally compact group for which every irreducible representation
vanishes at infinity modulo its projective kernel. Let $\pi\in\widehat{G}$,
and let $\xi\in S_{\mathcal{H}_{\pi}}$ be such that $\mathcal{J}_{\pi,\xi}$ is
weak*-closed. Then $\pi$ vanishes at infinity.

###### Proof.

By Lemma 4.9(i) $\operatorname{pker}\pi$ must be compact, since otherwise
$\mathcal{J}_{\pi,\xi}$ would be weak*-dense. Since by hypothesis $\pi$
vanishes at infinity modulo $\operatorname{pker}\pi$, it follows that $\pi$
itself vanishes at infinity. ∎

We can now prove our main classification theorem.

######  Theorem 4.11.

Let $G$ be a Hermitian locally compact group with the property that every
$\pi\in\widehat{G}$ vanishes at infinity modulo its projective kernel. Then
the weak*-closed maximal left ideals of $M(G)$ are given by
$\mathcal{J}_{\pi,\xi}$, where $\pi\in\widehat{G}$ vanishes at infinity, and
$\xi$ is a unit vector in $\mathcal{H}_{\pi}$ such that
$\mathcal{I}_{\pi,\xi}$ is a maximal modular left ideal in $L^{1}(G)$.

###### Proof.

Let $\pi\in\widehat{G}$ vanish at infinity, and let $\xi\in L^{1}(G)$ be such
that $\mathcal{I}_{\pi,\xi}$ is a maximal modular left ideal in $L^{1}(G)$.
Then $\mathcal{J}_{\pi,\xi}$ is a weak*-closed maximal left ideal by Corollary
4.6. Conversely, if we take a weak*-closed maximal left ideal $J$ of $M(G)$,
then Corollary 4.4 implies that $J=\mathcal{J}_{\pi,\xi}$ for some
$\pi\in\widehat{G}$ and $\xi\in S_{\mathcal{H}_{\pi}}$. Now $\pi$ must vanish
at infinity by Lemma 4.10. ∎

Remark. By the remarks on page 221 of [2] every irreducible representation of
a connected nilpotent Lie group vanishes at infinity modulo its kernel, and as
such the previous theorem applies. By [9] it also applies to Hermitian,
connected, real or complex algebraic groups. The theorem can also be seen to
apply to the Fell groups $\mathbb{Q}_{p}\rtimes\mathcal{O}_{p}\
(p\in\mathbb{N}\text{ prime}),$ and $\mathbb{R}^{n}\rtimes SO(n)\
(n\in\mathbb{N},\ n\geq 2)$, since the representations of these groups are
classified (see for instance [10, Examples 4.42-4.43]).

We conclude this section with a classification result for certain weighted
measure algebras. In the case of a compact group $G$ there is a bijective
correspondence between the maximal modular left ideals of $L^{1}(G)$ and the
weak*-closed maximal left ideals of $M(G)$ [21, Theorem 6.2]. The next
proposition shows that, for certain weights on certain non-compact groups $G$
we get an analogous correspondence for $L^{1}(G,\omega)$ and $M(G,\omega)$. In
particular, we get a description of the weak*-closed maximal left ideals of
$M(G,\omega)$ which does not depend on the asymptotic properties of the
representations of $G$.

######  Theorem 4.12.

Let $G$ be a compactly-generated, locally compact group of polynomial growth.
Let $\omega$ be a weight on $G$ such that $1/\omega\in C_{0}(G)$,
$\omega(t^{-1})=\omega(t)$, and
$\lim_{n\rightarrow\infty}\omega(t^{n})^{1/n}=1\ (t\in G)$. Then every maximal
modular left ideal of $L^{1}(G,\omega)$ has the form
$\mathcal{I}_{\pi,\xi,\omega}$ for some $\pi\in\widehat{G}$, and $\xi\in
S_{\mathcal{H}_{\pi}}$. Moreover, the weak*-closed maximal left ideals of
$M(G,\omega)$ are exactly given by $\mathcal{J}_{\pi,\xi,\omega}$, where
$\pi\in\widehat{G}$, and $\xi\in S_{\mathcal{H}_{\pi}}$ is such that
$\mathcal{I}_{\pi,\xi,\omega}$ is maximal modular in $L^{1}(G,\omega)$.

###### Proof.

Let $\gamma$ denote the maximal C*-norm of $L^{1}(G)$, and $\gamma_{\omega}$
denote that of $L^{1}(G,\omega)$. Also let $r$ and $r_{\omega}$ denote the
spectral radius in $L^{1}(G)$ and $L^{1}(G,\omega)$ respectively. By [7,
Theorem 1.3] both $L^{1}(G)$ and $L^{1}(G,\omega)$ are Hermitian so that for
$f\in L^{1}(G,\omega)$ we have

$\gamma_{\omega}(f)=r_{\omega}(f^{*}*f)^{1/2}=r(f^{*}*f)^{1/2}=\gamma(f),$

where we have used condition (iv) of [7, Theorem 1.3] to get the second
equality. Therefore the universal C*-algebra of $L^{1}(G,\omega)$ is
$C^{*}(G)$. It follows that the pure states on $L^{1}(G,\omega)$ all have the
form $f\mapsto\langle\pi(f)\xi,\xi\rangle\ (f\in L^{1}(G,\omega))$, for some
$\pi\in\widehat{G}$ and $\xi\in S_{\mathcal{H}_{\pi}}$, and hence the maximal
modular left ideals have the required form by Palmer’s Theorem [14].

Now let $J$ be a weak*-closed maximal left ideal of $M(G,\omega)$. Then by
Lemma 4.3 $J\cap L^{1}(G,\omega)$ is a maximal modular left ideal of
$L^{1}(G,\omega)$, and hence has the form $\mathcal{I}_{\pi,\xi,\omega}$, for
some $\pi\in\widehat{G}$, and $\xi\in S_{\mathcal{H}_{\pi}}$, by the previous
paragraph. We may now proceed exactly as in Corollary 4.4 to see that
$J=\mathcal{J}_{\pi,\xi,\omega}$.

Finally consider $\pi\in\widehat{G}$, and $\xi\in S_{\mathcal{H}_{\pi}}$ is
such that $\mathcal{I}_{\pi,\xi,\omega}$ is maximal modular in
$L^{1}(G,\omega)$. By Lemma 4.5 each ideal $\mathcal{J}_{\pi,\xi,\omega}$ is
weak*-closed, and hence maximal by Lemma 4.1(i). ∎

Remark. There are many examples of weights satisfying the hypothesis of the
previous theorem. For instance, given $n\in\mathbb{N}$ and $\alpha>0$, define
a weight $\omega$ on $\mathbb{R}^{n}$ by $\omega(x)=(1+\|x\|)^{\alpha}$.

##  5\. Barnes’ Theorem

The purpose of this section is to prove an analogue of Barnes’ Theorem 2.2 for
the setting of certain Hermitian locally compact groups, and representations
vanishing at infinity. We shall first give a reformulation of Barnes’ Theorem
(Proposition 5.2) that permits a direct comparison with our new result. We
begin with a short lemma.

######  Lemma 5.1.

Let $A$ be a semisimple faithful Banach algebra. Then any minimal left ideal
of $M(A)$ which is contained in $A$ is also minimal in $A$.

###### Proof.

Let $J$ be such a minimal left ideal of $M(A)$ contained in $A$. If $J$ is not
minimal in $A$ then it contains a smaller non-zero left ideal $I$. Since $A$
is semisimple, there must exist $x\in I$ for which $x^{2}\neq 0$, and then in
particular $Ax\neq\\{0\\}$. Since $Ax$ is a left ideal of $M(A)$, we must have
$J=Ax$ by minimality. But $x\in I$ so that $J\subset I$, which is a
contradiction. ∎

The following proposition is essentially a rephrasing of Barnes’ Theorem in
terms of the maximal left ideals of $M(G)$.

######  Proposition 5.2.

Let $G$ be a unimodular locally compact group, and let $\pi\in\widehat{G}$.
There exists $\xi\in\mathcal{H}_{\pi}$ such that $\mathcal{J}_{\pi,\xi}$ is a
maximal left ideal generated by a projection if and only if $\pi$ is
integrable.

###### Proof.

Suppose that $\pi$ is integrable. Then $\pi$ is determined by some minimal
projection $p$ in $L^{1}(G)$ as in [1]. It is clear from the construction of
representations determined by a minimal projection that $\pi(p)$ is a rank-one
projection, onto a vector that we shall call $\xi$. Then
$\delta_{e}-p\in\mathcal{J}_{\pi,\xi}$. Moreover, for every
$\mu\in\mathcal{J}_{\pi,\xi}$ we have
$\pi(\mu*(\delta_{e}-p))\xi=0=\pi(\mu)\xi$ and
$\pi(\mu*(\delta_{e}-p))\eta=\pi(\mu)\eta$ for $\eta\in\\{\xi\\}^{\perp}$, so
that $\pi(\mu*(\delta_{e}-p))=\pi(\mu)$.

By [1, Proposition 1] $\sigma(p)=0$ for all irreducible representations
$\sigma\neq\pi$. Hence
$\sigma(\delta_{e}-p)=\operatorname{id}_{\mathcal{H}_{\sigma}}$ so that
$\sigma(\mu)=\sigma(\mu*(\delta_{e}-p))$ for all
$\mu\in\mathcal{J}_{\pi,\xi}$. Since irreducible representations separate the
points of $M(G)$, this forces $\mu=\mu*(\delta_{e}-p)$ for
$\mu\in\mathcal{J}_{\pi,\xi}$, and hence
$\mathcal{J}_{\pi,\xi}=M(G)*(\delta_{e}-p)$.

Now suppose instead that there exists $\xi\in S_{\mathcal{H}_{\pi}}$ such that
$\mathcal{J}_{\pi,\xi}$ is maximal and that $\mathcal{J}_{\pi,\xi}=M(G)*q$ for
some projection $q\in M(G)$. Then by routine algebra $M(G)*p$ is a minimal
left ideal in $M(G)$, where $p:=\delta_{e}-q$. Note that $L^{1}(G)*p\neq 0$,
and hence $L^{1}(G)*p=M(G)*p$ by minimality, whence $p\in L^{1}(G)$. It
follows from Lemma 5.1 that $L^{1}(G)*p$ is a minimal left ideal in
$L^{1}(G)$.

We may now apply Barnes’ Theorem 2.2 to conclude that $p$ determines an
irreducible integrable representation $\sigma\in\widehat{G}$. Now note that,
since $\delta_{e}-p=q\in\mathcal{J}_{\pi,\xi}$, we have
$\pi(p)\xi=\operatorname{id}_{\mathcal{H}_{\pi}}\xi-\pi(q)\xi=\xi\neq 0$, so
that by [1, Proposition 1] $\pi=\sigma$. ∎

Although Palmer’s Theorem tells us that, for a Hermitian locally compact group
$G$, every maximal modular left ideal of $L^{1}(G)$ has the form
$\mathcal{I}_{\pi,\xi}$, for some $\pi\in\widehat{G}$, and some $\xi\in
S_{\mathcal{H}_{\pi}}$, it seems very difficult to determine, for any non-
Moore group $G$, for which $\pi\in\widehat{G}$ and $\xi\in
S_{\mathcal{H}_{\pi}}$ the left ideal $\mathcal{I}_{\pi,\xi}$ is maximal
modular. (On the other hand, in the C*-algebraic setting all such left ideals
are maximal modular). Indeed, the only general result that we have been able
to establish is the following modest proposition.

######  Proposition 5.3.

Let $G$ be a Hermitian CCR locally compact group. For every
$\pi\in\widehat{G}$ there exists $\xi\in S_{\mathcal{H}_{\pi}}$ such that
$\mathcal{I}_{\pi,\xi}$ is a maximal modular left ideal.

###### Proof.

Fix $\pi\in\widehat{G}$, set $\mathcal{A}_{\pi}=\pi(L^{1}(G))$, which is a
Banach *-algebra with the quotient topology. We may suppose that
$\dim\mathcal{A}_{\pi}>1$, since otherwise the result is trivial. We shall
show that every maximal modular left ideal of $\mathcal{A}_{\pi}$ lifts to a
maximal modular left ideal of $L^{1}(G)$ of the form $\mathcal{I}_{\pi,\xi}$.
Since $\mathcal{A}_{\pi}$ is semisimple and not equal to $\mathbb{C}$, it has
at least one maximal modular left ideal, and this will imply the result.

Let $I$ be a maximal modular left ideal of $\mathcal{A}_{\pi}$. Since
$L^{1}(G)$ is Hermitian, so is $\mathcal{A}_{\pi}$ by [11], and therefore it
follows from Palmer’s Theorem [14] that $I$ is of the form
$\\{a\in\mathcal{A}_{\pi}:\sigma(a)\zeta=0\\},$ for some irreducible
*-representation
$\sigma\colon\mathcal{A}_{\pi}\rightarrow\mathcal{B}(\mathcal{H}_{\sigma}),$
and some $\zeta\in S_{\mathcal{H}_{\sigma}}.$ Defining $J=\\{f\in
L^{1}(G):\pi(f)\in I\\}$ gives a maximal modular left ideal of $L^{1}(G)$.

The *-representation $\sigma$ extends via $\pi$ to $L^{1}(G)$, and hence to an
irreducible *-representation of $C^{*}(G)$, which we shall denote by
$\widetilde{\sigma}$. With this notation we have $J=\\{f\in
L^{1}(G):\widetilde{\sigma}(f)\zeta=0\\}.$ Note that, since
$\widetilde{\sigma}$ factors through $\pi$, we have
$\ker\widetilde{\sigma}\supset\ker\pi$ in $C^{*}(G)$. In terms of the Fell
topology this says that $\widetilde{\sigma}\in\overline{\\{\pi\\}}$, which
forces $\widetilde{\sigma}$ and $\pi$ to be equal as elements of
$\widehat{G}$, since $\widehat{G}$ is $T_{1}$ by [6, Theorem (7.7)] (as $G$ is
CCR). It follows that $J$ can be rewritten as $J=\\{f\in
L^{1}(G):\pi(f)\xi=0\\}$, for some $\xi\in S_{\mathcal{H}_{\pi}}$. ∎

Next we present our analogue of Barnes’ Theorem for irreducible
representations vanishing at infinity.

######  Theorem 5.4.

Let $G$ be a Hermitian CCR locally compact group for which every irreducible
representation vanishes at infinity modulo its projective kernel, and let
$\pi\in\widehat{G}$. Then there exists $\xi\in S_{\mathcal{H}_{\pi}}$ for
which $\mathcal{J}_{\pi,\xi}$ is a weak*-closed maximal left ideal of $M(G)$
if and only if $\pi$ vanishes at infinity.

###### Proof.

First suppose that $\pi$ vanishes at infinity. By Proposition 5.3 there exists
$\xi\in S_{\mathcal{H}_{\pi}}$ such that $\mathcal{I}_{\pi,\xi}$ is a maximal
modular left ideal of $L^{1}(G)$. By Theorem 4.11 $\mathcal{J}_{\pi,\xi}$ is
maximal and weak*-closed.

Now suppose instead that there exists $\xi\in S_{\mathcal{H}_{\pi}}$ such that
$\mathcal{J}_{\pi,\xi}$ is a weak*-closed maximal left ideal of $M(G)$. Then
Theorem 4.11 implies that $\pi$ vanishes at infinity. ∎

Remark. Examples of groups satisfying the conditions of the previous theorem
include connected nilpotent Lie groups and $\mathbb{R}^{n}\rtimes SO(n)\
(n\in\mathbb{N},\ n\geq 2)$.

##  6\. Examples

In this section we use the theory that we have developed to give some examples
of measure algebras for which the weak*-closed maximal left ideals can be
described. Our first example provides a class of groups whose measure algebras
possess an abundance of weak*-closed maximal left ideals, none of which is
generated by a projection. We do not know if they can be finitely-generated.

######  Example 6.1.

Let $G=\mathbb{R}^{n}\rtimes SO(n)\ (n\geq 2)$. Then $G$ is Hermitian by [13,
e.g. (a), pg. 191]. It is known (see e.g. [10, Example 4.42]) that the
irreducible representations of $G$ fall into two kinds: those factoring
through $SO(n)$, and those which are induced from irreducible representations
of $\mathbb{R}^{n}\rtimes SO(n-1)$; the latter are the infinite-dimensional
irreducible representations, and they vanish at infinity by [10, Theorem 7.6].
However, $G$ has no integrable irreducible representations: this follows from
[1, Corollary 2], and the fact that $\widehat{G}$ with the fell topology has
no open points. It follows that $M(G)$ has no maximal left ideals generated by
a projection by Proposition 5.2. As such, Proposition 5.3 and Theorem 4.11
together imply that for each $\pi\in\widehat{G}\setminus\widehat{SO(n)}$ there
exists $\xi\in\mathcal{H}_{\pi}$ such that $\mathcal{J}_{\pi,\xi}$ is a
weak*-closed maximal left ideal, but is not generated by a projection.

The next example contrasts with the previous one.

######  Example 6.2.

Let $H$ denote the Heisenberg group

$H=\left\\{\begin{pmatrix}1&x&z\\\ 0&1&y\\\
0&0&1\end{pmatrix}:x,y,z\in\mathbb{R}\right\\},$

and let $N$ be the subgroup with $x=y=0$ and $z\in\mathbb{Z}$. Let $G$ be
$H/N$, which is sometimes called the Weyl-Heisenberg group. This group is a
connected nilpotent Lie group, and as such Theorem 4.11 applies. Identify $G$
with the set $\mathbb{R}\times\mathbb{R}\times\mathbb{T}$, with the
multiplication given by

$(x_{1},y_{1},a_{1})(x_{2},y_{2},a_{2})=(x_{1}+x_{2},y_{1}+y_{2},a_{1}a_{2}{\rm
e}^{ix_{1}y_{2}}).$

The infinite dimensional irreducible representations of $G$ act on
$L^{2}(\mathbb{R})$ and have the form

$[U_{n}(x,y,a)f](s)=a^{n}{\rm e}^{-insy}f(s-x)\qquad(f\in L^{2}(\mathbb{R}))$

for $n\in\mathbb{Z}\setminus\\{0\\}$. Each $U_{n}$ is integrable (for instance
$\xi*_{U_{n}}\xi\in L^{1}(G)$, where $\xi(s)={\rm e}^{-s^{2}}$), and as such
vanishes at infinity, so that Theorem 4.11 tells us that the weak*-closed
maximal left ideals of $M(G)$ have the form $\mathcal{J}_{U_{n},\xi}$, where
$n\in\mathbb{Z}\setminus\\{0\\}$ and $\xi\in S_{\mathcal{H}_{U_{n}}}$. In fact
we can say more: by [22], for every $n\in\mathbb{Z}\setminus\\{0\\}$ there
exists a dense subspace $E\subset L^{2}(\mathbb{R})$ such that, for every
$\xi\in E$, the function $\xi*_{U_{n}}\xi$ can be scaled to give a minimal
projection in $L^{1}(G)$. Denoting this projection by $p$, it follows from the
proof of Proposition 5.2 that $\mathcal{J}_{{U_{n}},\xi}=M(G)*(\delta_{e}-p)$,
and that it is a maximal left ideal. We do not know whether all of the
weak*-closed maximal left ideals are generated by projections.

We have seen in Proposition 3.4 that a group with non-compact centre cannot
have any weak*-closed maximal left ideals in its measure algebra, and
Corollary 4.9(ii) gives an even stronger statement for Hermitian groups. The
next example is of a class of groups, not covered by these previous results,
but which nonetheless has no weak*-closed maximal left ideals in its measure
algebra.

######  Example 6.3.

Consider $G=\mathbb{R}^{2}\rtimes_{A}\mathbb{R}$, where $A\in
M_{2}(\mathbb{R}),$ and $\mathbb{R}$ acts on $\mathbb{R}^{2}$ via

$t\colon x\rightarrow{\rm e}^{tA}x\qquad(t\in\mathbb{R},\
x\in\mathbb{R}^{2}).$

We claim that for these groups $M(G)$ has no weak*-closed maximal left ideals.

In [10, Chapter 4] the Mackey Machine is employed to classify the irreducible
representations of the groups $\mathbb{R}^{2}\rtimes_{A}\mathbb{R}\ (A\in
M_{2}(\mathbb{R}))$ according to the Jordan normal form of $A$. In the case
that $A$ is similar to a matrix of the form
$\big{(}\begin{smallmatrix}0&\lambda\\\ -\lambda&0\end{smallmatrix}\big{)},$
for some $\lambda\in\mathbb{R}\setminus\\{0\\}$, the centre of $G$ contains a
copy of $\mathbb{Z}$, so in this case the claim follows by Proposition 3.4. In
the remaining cases the elements of $\widehat{G}$ either factor through
$\mathbb{R}$, or have the form $U^{\gamma}\colon
G\rightarrow\mathcal{B}(L^{2}(\mathbb{R}))$, where $\gamma\in\mathbb{R}^{2}$,
and $U^{\gamma}$ is given by

(6.1) $[U^{\gamma}(x,t)f](s)={\rm e}^{i\gamma\cdot{\rm
e}^{-sA}x}f(s-t)\qquad(s\in\mathbb{R},\ f\in L^{2}(\mathbb{R}),\
(x,t)\in\mathbb{R}^{2}\rtimes_{A}\mathbb{R}).$

Note that $G$ satisfies the hypothesis of Theorem 4.11: the representations
appearing (6.1) are induced from $\mathbb{R}^{2}$, and vanish at infinity
modulo their kernels by [2]. Of course, in the case of a representation that
factors through the subgroup $\mathbb{R}$, the projective kernel is the entire
group $G$, so that these representations vanish at infinity modulo their
projective kernels for trivial reasons. Finally, $G$ is Hermitian by [12]. We
shall show that none of the irreducible representations vanish at infinity,
and hence, by Theorem 4.11, $M(G)$ has no weak*-closed maximal left ideals.

We only need to consider the infinite-dimensional irreducible representations.
We shall divide into two cases.

Case 1. Suppose that $\\{\gamma\\}^{\perp}$ is invariant for the action of
$\mathbb{R}$ on $\mathbb{R}^{2}$ via $A$. Then, taking $f=\chi_{[0,1]},$ we
see that

$\langle
U^{\gamma}(x,0)f,f\rangle=\int_{-\infty}^{\infty}\chi_{[0,1]}(s){\rm\,d}s=1,$

for all $x\in\\{\gamma\\}^{\perp}$. Hence $f*_{U^{\gamma}}f\notin
C_{0}(\mathbb{R}^{2}\rtimes_{A}\mathbb{R})$.

Case 2. Suppose that $\\{\gamma\\}^{\perp}$ is not invariant for the action of
$\mathbb{R}$ on $\mathbb{R}^{2}$ via $A$. Fix
$x\in\\{\gamma\\}^{\perp}\setminus\\{0\\}$ and $a\in\mathbb{R}$ such that
$\gamma\cdot{\rm e}^{-aA}x\neq 0$. By scaling $x$ we may assume that
$\gamma\cdot{\rm e}^{-aA}x=1$. Define $f=\chi_{[0,a]}$, and
$g(s)=-\gamma\cdot(A{\rm e}^{-sA}x)\chi_{[0,a]}(s)\ (s\in\mathbb{R})$, and
note that

$\frac{\rm d}{{\rm d}s}\gamma\cdot{\rm e}^{-sA}x=-\gamma\cdot(A{\rm
e}^{-sA}x).$

Then for all $k\in\mathbb{N}$ we have

$\displaystyle\langle U^{\gamma}((2k+1)\pi x,0)f,g\rangle$
$\displaystyle=\int_{0}^{a}{\rm e}^{(2k+1)\pi i\gamma\cdot{\rm
e}^{-sA}x}(-(2k+1)\pi\gamma\cdot(A{\rm e}^{-sA}x)){\rm\,d}s$
$\displaystyle=-i[{\rm e}^{(2k+1)\pi i\gamma\cdot{\rm
e}^{-sA}x}]^{a}_{0}=-i({\rm e}^{(2k+1)\pi i}-e^{0})=2i.$

Hence $f*_{U^{\gamma}}g\notin C_{0}(\mathbb{R}^{2}\rtimes_{A}\mathbb{R})$, and
this concludes the argument.

Finally, we note that many of these groups are not covered by Corollary
4.9(ii). Indeed, it is routinely shown that these groups have no proper finite
index subgroups, and have trivial centre whenever the action determined by $A$
is faithful and has no non-zero fixed points. This happens, for instance, for
$A=\big{(}\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\big{)}$ or
$A=\big{(}\begin{smallmatrix}\lambda&1\\\ 0&\lambda\end{smallmatrix}\big{)}$,
for $\lambda\in\mathbb{R}\setminus\\{0\\}$.

##  7\. Finitely-Generated Closed Left Ideals of Measure Algebras

In this section we shall show how the results that we have proved so far imply
Theorem 1.3. We shall then take a more detailed look at the weak*-closed left
ideals of $M(\mathbb{R}^{2}\rtimes SO(2))$ (see Example 6.1 above), and in
particular prove Theorem 1.4.

###### Proof of Theorem 1.3.

(i) Let $G$ be a Moore group and suppose that $M(G)$ has a maximal left ideal
$J$ that is weak*-closed. Moore groups are Hermitian by [16, diagram, pg.
1486]. By Corollary 4.4 there exists $\pi\in\widehat{G}$ and $\xi\in
S_{\mathcal{H}_{\pi}}$ such that $J=\mathcal{J}_{\pi,\xi}$. Since $\pi$ is
finite-dimensional, $\mathcal{J}_{\pi,\xi}$ has finite codimension and hence
$G$ must be compact by Corollary 3.3.

(ii) This is immediate from Proposition 3.4.

(iii) $G$ has a finite-index normal nilpotent subgroup $N$, which is also
finitely-generated. By [17, 5.2.22(ii)] $Z(N)$ is infinite. The result now
follows from Corollary 4.9(ii).

(iv) This is Example 6.3. ∎

We now turn to the proof of Theorem 1.4. In very broad terms the idea is to
show that if $\mathcal{J}_{U,1}$ is generated by finitely many compactly-
supported measures, then measures belonging to the ideal decay unexpectedly
rapidly in a certain sense, and this will give a contradiction.

For the remainder of this section we shall set $G=\mathbb{R}^{2}\rtimes
SO(2)$. For notational convenience we shall sometimes formally identify
$SO(2)$ with $[0,2\pi)$. This allows us to represent an element of $G$ as
$(x,\theta)$, where $x\in\mathbb{R}^{2}$, and $\theta\in[0,2\pi)$.

We shall write $B_{0}(\rho)$ for the ball of radius $\rho>0$ about the origin
in $\mathbb{R}^{2}$. Let $B_{1}\subset G$ be a set of the form
$B_{0}(\rho)\times SO(2)$, and let

$B_{n}=\\{t_{1}\cdots t_{n}:t_{1},\ldots,t_{n}\in B_{1}\\}.$

Then $B_{n}=B_{0}(n\rho)\times SO(2)$. We shall also write
$S_{n}=B_{n}\setminus B_{n-1}$. Let $\chi$ be the character of
$\mathbb{R}^{2}$ given by $\chi(x)={\rm e}^{ix_{1}}\ (x\in\mathbb{R}^{2})$,
and let $U\in\widehat{G}$ be the representation on $L^{2}(SO(2))$ given by

$[U(x,\theta)\xi](\varphi)=\exp(i(x_{1}\cos\theta+x_{2}\sin\theta))\xi(\varphi-\theta),$

for $x\in\mathbb{R}^{2},\ \theta,\varphi\in[0,2\pi),$ and $\xi\in
L^{2}(SO(2)).$ In fact, $U$ is the representation of $G$ induced by $\chi$ and
vanishes at infinity. We shall study $\mathcal{J}_{U,1}$.

######  Lemma 7.1.

Fix $\rho>0$, and let $B_{1}$ and $B_{n}\ (n\in\mathbb{N})$ be defined in
terms of $\rho$ as above. Let

$\sigma_{n}(\mu)=\int_{B_{n}}\langle
U(t)1,1\rangle{\rm\,d}\mu(t)\qquad(n\in\mathbb{N},\ \mu\in\mathcal{J}_{U,1}).$

Let $\nu\in\mathcal{J}_{U,1}$ satisfy ${\rm supp\,}\nu\subset B_{1}$. Then for
every $\mu\in M(G)$ we have

$\sum_{n=1}^{\infty}|\sigma_{n}(\mu*\nu)|<\infty.$

###### Proof.

We calculate

$\displaystyle\sigma_{n}(\mu*\nu)$ $\displaystyle=\int_{B_{n}}\langle
U(t)1,1\rangle{\rm\,d}(\mu*\nu)(t)=\int_{G}\int_{G}\mathbbm{1}_{B_{n}}(st)\langle
U(st)1,1\rangle{\rm\,d}\mu(s){\rm\,d}\nu(t)$
$\displaystyle=\int_{B_{1}}\int_{B_{n}t^{-1}}\langle
U(st)1,1\rangle{\rm\,d}\mu(s){\rm\,d}\nu(t)=I_{1}+I_{2},$

where

$I_{1}=\int_{B_{1}}\int_{B_{n-1}}\langle
U(st)1,1\rangle{\rm\,d}\mu(s){\rm\,d}\nu(t),$

and

$I_{2}=\int_{B_{1}}\int_{B_{n}t^{-1}\setminus B_{n-1}}\langle
U(st)1,1\rangle{\rm\,d}\mu(s){\rm\,d}\nu(t).$

Now,

$I_{1}=\int_{B_{n-1}}\int_{B_{1}}\langle
U(st)1,1\rangle{\rm\,d}\nu(t){\rm\,d}\mu(s)=\int_{B_{n-1}}\langle
U(s)U(\nu)1,1\rangle{\rm\,d}\mu(s)=0,$

because ${\rm supp\,}\nu\subset B_{1}$ and $U(\nu)1=0$. Hence

$\displaystyle|\sigma_{n}(\mu*\nu)|$
$\displaystyle\leq|I_{2}|\leq\int_{B_{1}}\int_{B_{n}t^{-1}\setminus
B_{n-1}}|\langle U(st)1,1\rangle|{\rm\,d}|\mu|(s){\rm\,d}|\nu|(t)$
$\displaystyle\leq\|\nu\||\mu|(B_{n+1}\setminus
B_{n-1})=\|\nu\||\mu|(S_{n+1}\cup S_{n}),$

since $|\langle U(st)1,1\rangle|\leq 1\ (s,t\in G)$ and $B_{n}t^{-1}\setminus
B_{n-1}\subset B_{n+1}\setminus B_{n-1}$. Hence

$\sum_{n=1}^{\infty}|\sigma_{n}(\mu*\nu)|\leq\|\nu\|\sum_{n=1}^{\infty}|\mu|(S_{n+1})+\|\nu\|\sum_{n=1}^{\infty}|\mu|(S_{n})\leq
2\|\nu\|\|\mu\|<\infty,$

as required. ∎

From now on we shall write $\chi^{\varphi}(x)={\rm e}^{i(x_{1}\cos\varphi-
x_{2}\sin\varphi)}\ (x\in\mathbb{R}^{2})$, that is $\chi$ applied to $x$
rotated anticlockwise through and angle of $\varphi$.

######  Lemma 7.2.

For every $x\in\mathbb{R}^{2}$ we have

$[U(x,0)1](\varphi)=\chi^{-\varphi}(x)\qquad\varphi\in[0,2\pi).$

###### Proof.

Direct computation based on the formula for an induced representation. ∎

######  Lemma 7.3.

Let $k\in\mathbb{N}$, and set $\rho_{n}=2\pi kn\ (n\in\mathbb{N})$. Let
$B_{1}=B_{0}(2\pi k)$, so that $B_{n}=B_{0}(\rho_{n})\times SO(2)\
(n\in\mathbb{N})$, and let $\sigma_{n}$ be defined as in Lemma 7.1. Then there
exists $\nu\in\mathcal{J}_{U,1}$ such that
$\sum_{n=1}^{\infty}|\sigma_{n}(\nu)|$ diverges.

###### Proof.

Write $S_{0}(\rho_{n})=\\{x\in\mathbb{R}^{2}:\|x\|=\rho_{n}\\}$. Note that
$\|\mathbbm{1}_{S_{0}(\rho_{n})}\|=2\pi\rho_{n}=4\pi^{2}kn.$ We claim that

(7.1)
$\chi(\mathbbm{1}_{S_{0}(\rho_{n})})=\|\mathbbm{1}_{S_{0}(\rho_{n})}\|J_{0}(\rho_{n})\qquad(n\in\mathbb{N}).$

Indeed,

$\displaystyle\chi(\mathbbm{1}_{S_{0}(\rho_{n})})$
$\displaystyle=\int_{S_{0}(\rho_{n})}{\rm
e}^{ix_{1}}{\rm\,d}x=\int_{S_{0}(\rho_{n})}\cos(x_{1}){\rm\,d}x=4\int_{0}^{\frac{\pi}{2}}\rho_{n}\cos(\rho_{n}\cos(\theta)){\rm\,d}\theta$
$\displaystyle=4\times\frac{\pi\rho_{n}J_{0}(\rho_{n})}{2}=2\pi\rho_{n}J_{0}(\rho_{n}),$

where $J_{0}$ denotes the zeroth order Bessel function of the first kind, and
we have used the identity

$J_{0}(z)=\frac{1}{\pi}\int_{0}^{\pi}\cos(z\sin\tau){\rm\,d}\tau\qquad(z\in\mathbb{C})$

to compute the integral. Using the following formula

$J_{0}(z)=\sqrt{\frac{2}{\pi z}}\left[\cos\left(z-\pi/4\right)+{\rm
e}^{|\operatorname{im}z|}O(|z|^{-1})\right]\qquad(|\arg z|<\pi)$

we see that

(7.2) $J_{0}(\rho_{n})=\frac{1}{\pi\sqrt{2kn}}+O(1/n^{3/2}).$

Define measures $\nu_{n}\in M(\mathbb{R}^{2})$ for $n\in\mathbb{N}$ by

$\nu_{n}=\frac{1}{n^{3/2}\|\mathbbm{1}_{S_{0}(\rho_{n})}\|}\mathbbm{1}_{S_{0}(\rho_{n})}\qquad(n\in\mathbb{N})$

and define

$\nu_{0}=\zeta\delta_{e}-\sum_{n=1}^{\infty}\nu_{n},$

where $\zeta:=\chi\left(\sum_{n=1}^{\infty}\nu_{n}\right)$. Note that
$\nu_{0}$ is well-defined since the $\nu_{n}$’s are disjointly supported, and
$\|\nu_{n}\|=n^{-3/2}$. Since $\nu_{0}$ is rotationally symmetric,
$\chi^{\varphi}(\nu_{0})=\chi(\nu_{0})$ for all $\varphi\in[0,2\pi)$. Finally,
we define $\nu=\nu_{0}\times\delta_{0}\in M(\mathbb{R}^{2}\rtimes SO(2))$.
First we show that $\nu\in\mathcal{J}_{U,1}$. Given $\eta\in L^{2}(SO(2))$ we
have

$\displaystyle\langle U(\nu)1,\eta\rangle$
$\displaystyle=\int_{SO(2)}\int_{SO(2)}\int_{\mathbb{R}^{2}}[U(x,\theta)1](\varphi)\overline{\eta(\varphi)}{\rm\,d}\nu_{0}(x){\rm\,d}\delta_{0}(\theta){\rm\,d}\varphi$
$\displaystyle=\int_{SO(2)}\int_{\mathbb{R}^{2}}[U(x,0)1](\varphi)\overline{\eta(\varphi)}{\rm\,d}\nu_{0}(x){\rm\,d}\varphi$
$\displaystyle=\int_{SO(2)}\int_{\mathbb{R}^{2}}\chi^{-\varphi}(x)\overline{\eta(\varphi)}{\rm\,d}\nu_{0}(x){\rm\,d}\varphi\qquad(\text{by
Lemma \ref{5.4a}})$
$\displaystyle=\left(\int_{SO(2)}\overline{\eta(\varphi)}{\rm\,d}\varphi\right)\left(\int_{\mathbb{R}^{2}}\chi(x){\rm\,d}\nu_{0}(x)\right)\qquad(\text{because
}\nu_{0}\text{ is rotationally symmetric})$
$\displaystyle=\chi(\nu_{0})\int_{SO(2)}\overline{\eta(\varphi)}{\rm\,d}\varphi=0,$

because $\chi(\nu_{0})=0$. Since $\eta$ was arbitrary $U(\nu)1=0$, i.e.
$\nu\in\mathcal{J}_{U,1}$.

Next we show that

(7.3) $\sigma_{n}(\nu)=\sum_{m=n+1}^{\infty}\chi(\nu_{m}).$

Indeed,

$\displaystyle\langle U(\nu_{0}\times\delta_{0}|_{B_{n}})1,1\rangle$
$\displaystyle=\int_{SO(2)}\int_{\mathbb{R}^{2}}\langle
U(x,\theta)1,1\rangle{\rm\,d}(\nu_{0}\times\delta_{0}|_{B_{n}})(x)$
$\displaystyle=\int_{B_{0}(\rho_{n})}\langle
U(x,0)1,1\rangle{\rm\,d}\nu_{0}(x)=\zeta\langle
1,1\rangle-\sum_{m=1}^{\infty}\int_{B_{0}(\rho_{n})}\langle
U(x,0)1,1\rangle{\rm\,d}\nu_{m}(x).$

Since $B_{0}(\rho_{n})\cap{\rm supp\,}\nu_{m}=\emptyset$ for $m>n$, this is
equal to

$\displaystyle\zeta-\sum_{m=1}^{n}\int_{B_{0}(\rho_{n})}$
$\displaystyle\langle
U(x,0)1,1\rangle{\rm\,d}\nu_{m}(x)=\zeta-\sum_{m=1}^{n}\int_{\mathbb{R}^{2}}\int_{SO(2)}\chi^{-\varphi}(x){\rm\,d}\varphi{\rm\,d}\nu_{m}(x)$
$\displaystyle=\zeta-\sum_{m=1}^{n}\int_{SO(2)}\int_{\mathbb{R}^{2}}\chi^{-\varphi}(x){\rm\,d}\nu_{m}(x){\rm\,d}\varphi$
$\displaystyle=\zeta-\sum_{m=1}^{n}\int_{SO(2)}\chi^{-\varphi}(\nu_{m}){\rm\,d}\varphi=\zeta-\sum_{m=1}^{n}\int_{SO(2)}\chi(\nu_{m}){\rm\,d}\varphi$
$\displaystyle=\zeta-\sum_{m=1}^{n}\chi(\nu_{m})=\sum_{m=n+1}^{\infty}\chi(\nu_{m}),$

where we have used Lemma 7.2 in the first line, and the fact that each
$\nu_{m}$ is rotationally symmetric in the penultimate line. This establishes
(7.3). By (7.1) and (7.2) we have

$\displaystyle\chi(\nu_{n})=\frac{1}{n^{3/2}}J_{0}(\rho_{n})=\frac{1}{\pi\sqrt{2k}n^{2}}+O(n^{-3}),$

for all $n\in\mathbb{N}$, so there exists a constant $C>0$ such that

$\left|\chi(\nu_{n})-\frac{1}{\pi\sqrt{2k}n^{2}}\right|\leq\frac{C}{n^{3}}\qquad(n\in\mathbb{N}).$

It follows that for all $n\in\mathbb{N}$ we have

$\sum_{m=n+1}^{\infty}\chi(\nu_{m})\geq\sum_{m=n+1}^{\infty}\frac{1}{\pi\sqrt{2k}m^{2}}-C\sum_{m=n+1}^{\infty}\frac{1}{m^{3}}.$

By an integral estimate

$\sum_{m=n+1}^{\infty}\frac{1}{m^{3}}\leq\frac{1}{(n+1)^{3}}+\frac{1}{2(n+1)^{2}},$

whereas

$\sum_{m=n+1}^{\infty}\frac{1}{m^{2}}\geq\frac{1}{n},$

for $n\in\mathbb{N}$. Combining these estimates with (7.3) gives

$\sum_{n=1}^{N}|\sigma_{n}(\nu)|\geq\frac{1}{\pi\sqrt{2k}}\sum_{n=1}^{N}\frac{1}{n}-C\sum_{n=1}^{N}\frac{1}{(n+1)^{3}}+\frac{1}{2(n+1)^{2}},$

which tends to infinity as $N\rightarrow\infty$. ∎

We can now prove Theorem 1.4.

###### Proof of Theorem 1.4.

Part (i) follows from Example 6.1. We shall prove part (ii). As $U$ vanishes
at infinity, $\mathcal{J}_{U,1}$ is weak*-closed by Corollary 4.6(i). We shall
prove that $\mathcal{J}_{U,1}$ cannot be generated by finitely many compactly
supported elements. Let $G=\mathbb{R}^{2}\rtimes SO(2)$, and suppose that for
some $m\in\mathbb{N}$ there are compactly-supported measures
$\mu_{1},\ldots,\mu_{m}\in\mathcal{J}_{U,1}$ such that

$\mathcal{J}_{U,1}=M(G)*\mu_{1}+\cdots+M(G)*\mu_{m}.$

There exists $k\in\mathbb{N}$ such that ${\rm supp\,}\mu_{i}\subset B_{0}(2\pi
k)\times SO(2)\ (i=1,\ldots,m)$. It follows from Lemma 7.1 with $\rho=2\pi k$
that $\sum_{n=1}^{\infty}|\sigma_{n}(\mu)|<\infty$ for every
$\mu\in\mathcal{J}_{U,1}$. This contradicts Lemma 7.3. ∎

Remark. Continue to write $G=\mathbb{R}^{2}\rtimes SO(2)$, and let $M_{c}(G)$
denote the compactly supported measures in $M(G)$. If we could prove that
$M_{c}(G)\cap\mathcal{J}_{U,1}$, or even
$M(G)*(M_{c}(G)\cap\mathcal{J}_{U,1})$, was norm-dense in $\mathcal{J}_{U,1}$,
then [19, Lemma 2.1], together with Theorem 1.4, would imply that
$\mathcal{J}_{U,1}$ is not finitely-generated, and hence that finite
generation and weak*-closedness are distinct notions for closed left ideals of
measure algebras. However, we have not been able to prove this.

##  8\. Open Questions

We conclude this article with some problems for future study. The main
question we would like to resolve is whether or not Conjecture 1.1 and
Conjecture 1.2 hold for all Hermitian locally compact groups. Another of the
central topics of this article is the connection between finite generation of
a closed left ideal of a measure algebra, and weak*-closedness. This suggests
to us the following natural questions that we have been unable to answer.

######  Question 8.1.

  1. (i)

Does there exist a locally compact group $G$, and a closed left ideal $J$ in
$M(G)$, such that $J$ is weak*-closed, but not finitely-generated?

  2. (ii)

Does there exist a locally compact group $G$, and a closed left ideal $J$ in
$M(G)$, such that $J$ is finitely-generated but not generated by a projection?

It would be particularly interesting to be able to answer these questions for
maximal left ideals.

In [19] it was shown that, if $G$ is a non-discrete locally compact group,
then $L^{1}(G)$ has no finitely-generated maximal modular left ideals. It
would be interesting to know whether the same holds for an infinite discrete
group $G$. We saw in Theorem 1.3 that this does hold when $G$ is finitely-
generated and virtually nilpotent.

######  Question 8.2.

  1. (i)

Does $\ell^{1}(G)$ ever have a finitely-generated maximal left ideal, for an
infinite group $G$?

  2. (ii)

Can $\ell^{1}(G)$ ever have a weak*-closed maximal left ideal, for an infinite
group $G$?

Another class of dual Banach algebras associated with locally compact groups
are the Fourier-Stieltjes algebras $B(G)$. These are commutative algebras, so
the maximal ideals automatically have codimensions 1. Since the predual of
$B(G)$ is given by $C^{*}(G)$ it is clear that, for a discrete group $G$, and
$t_{0}\in G$, the set $\\{u\in B(G):u(t_{0})=0\\}$ is always a weak*-closed
maximal ideal of $B(G)$.

######  Question 8.3.

  1. (i)

Can the Fourier-Stieltjes algebra $B(G)$ ever have a weak*-closed maximal
ideal, for a non-discrete locally compact group $G$?

  2. (ii)

Let $G$ be a discrete group. Is it true that every weak*-closed maximal ideal
of $B(G)$ is of the form $\\{u\in B(G):u(t_{0})=0\\}$ for some $t_{0}\in G$?

It would also be interesting to study the weak*-closed ideals of the reduced
Fourier-Stieltjes algebra $B_{r}(G)$ and the algebra of completely bounded
multipliers of the Fourier algebra $M_{cb}A(G)$. In all of these examples the
absence of a bounded approximate identity in the Fourier algebra $A(G)$ for
non-amenable groups introduces new difficulties not seen in the present work.

### Acknowledgements

Part of this work was undertaken during my postdoctoral position at the
Laboratoire des Mathématiques de Besançon, and I would like to thank Uwe Franz
and Yulia Kuznetsova for their kind hospitality during that period. I would
like to thank Simeng Wang, Keith Taylor, and Michael Leinert for useful email
exchanges. I am indebted to Bence Horváth for his careful reading of the
manuscript.

## References

  * [1] B. Barnes, The role of minimal idempotents in the representation theory of locally compact groups, _Proc. Edinburgh Math. Soc._ 23 (1980), 229–238.
  * [2] L. Baggett and K. F. Taylor, On asymptotic behaviour of induced representations, _Can. J. Math._ 34 (1982), 220–232.
  * [3] H. G. Dales and A. T-M. Lau, _The second duals of Beurling algebras_ , Mem. Amer. Math. Soc., Volume 177, 2005.
  * [4] M. Daws, _Multipliers, self-induced and dual Banach algebras_ , Dissertationes Math., Volume 470, 2010.
  * [5] J. Dixmier, _C*-algebras_ , North Holland Publishing Co., Amsterdam-New York-Oxford (1977). Translated from French by Francis Jellett, North holland Mathematical Library, Vol. 15.
  * [6] G. B. Folland, _A course in abstract harmonic analysis_ , Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1995).
  * [7] G. Fendler, K. Gröchenig, and M. Leinert, Symmetry of weighted $L^{1}$-algebras and the GRS-condition, _Bull. London Math. Soc._ 38 (2006), 625–635.
  * [8] B. Hayati and M. Amini, Connes–amenability of multiplier Banach algebras, _Kyoto J. Math._ 50 (2010), 41–50.
  * [9] R. Howe and C. Moore, Asymptotic properties of unitary representations, _J. Funct. Anal._ 32 (1979), 72–96.
  * [10] E. Kaniuth and K. F. Taylor, _Induced representations of locally compact groups_ , Cambridge University Press, 2013.
  * [11] H. Leptin, Ideal theory in group algebras of locally compact groups, _Inv. Math._ 31 (1976), 259–278.
  * [12] H. Leptin, _Lokal kompakte Gruppen mit symmetrischen Algebren_(German), Symposia Mathematica, Vol. XXII (1977), 267–281.
  * [13] J. Ludwig, A class of symmetric and a class of Weiner group algebras, _J. Funct. Anal._ 31 (1979), 187–194.
  * [14] T. W. Palmer, Hermitian Banach *-algebras, _Bull. Amer. Math. Soc._ 78 (1972), 522–524.
  * [15] T. W. Palmer, _Banach algebras and the general theory of *-algebras, Vol.I_ , volume 49 of _Encyclopedia of Mathematics and its applications_. Cambridge University Press, Cambridge, 1994. Algebras and Banach Algebras.
  * [16] T. W. Palmer, _Banach algebras and the general theory of *-algebras II_ , Encyclopedia of Mathematics, Cambridge University Press, Cambridge, 2001.
  * [17] D. J. S. Robinson, _A course in the theory of groups_ , Springer, 1994.
  * [18] E. Samei and M. Wiersma, Quasi-Hermitian locally compact groups are amenable, _Adv. Math._ 359 (2020), available online.
  * [19] J. T. White, Finitely-generated left ideals in Banach algebras on groups and semigroups, _Studia Math._ 239 (2017), 67–99.
  * [20] J. T. White, Banach algebras on groups and semigroups, PhD thesis, University of Lancaster (2018).
  * [21] J. T. White, Left ideals of Banach algebras and Dual Banach algebras, Proceedings of the 23rd International Conference on Banach Algebras and Applications, De Gruyter Proceedings in Mathematics (De Gruyter 2020), 227-253.
  * [22] A. Valette, Minimal projections, integrable representations and property (T), _Arch. Math. (Basel)_ 45 (1984), 397–406.