# NON-ABELIAN FACTORS FOR ACTIONS OF $\mathbb{Z}$ AND OTHER NON-$C^{*}$-SIMPLE
GROUPS.

Tattwamasi Amrutam  Ben Gurion University of the Negev. Department of
Mathematics. Be’er Sheva, 8410501, Israel<EMAIL_ADDRESS>,  Eli
Glasner  Tel-Aviv University. Department of Mathematics. Tel-Aviv, Israel.
<EMAIL_ADDRESS>and  Yair Glasner  Ben Gurion University of the Negev.
Department of Mathematics. Be’er Sheva, 8410501, Israel.
<EMAIL_ADDRESS>
(Date: January, 2024)

###### Abstract.

Let $\Gamma$ be a countable group and $(X,\Gamma)$ a compact topological
dynamical system. We study the question of the existence of an intermediate
$C^{*}$-subalgebra $\mathcal{A}$

$C^{*}_{r}(\Gamma)<\mathcal{A}<C(X)\rtimes_{r}\Gamma,$

which is not of the form $\mathcal{A}=C(Y)\rtimes_{r}\Gamma$, corresponding to
a factor map $(X,\Gamma)\to(Y,\Gamma)$. Here $C^{*}_{r}(\Gamma)$ is the
reduced $C^{*}$-algebra of $\Gamma$ and $C(X)\rtimes_{r}\Gamma$ is the reduced
$C^{*}$-crossed-product of $(X,\Gamma)$. Our main results are: (1) For
$\Gamma$ which is not $C^{*}$-simple, when $(X,\Gamma)$ admits a
$\Gamma$-invariant probability measure, then such a sub-algebra always exists.
(2) For $\Gamma=\mathbb{Z}$ and $(X,\Gamma)$ an irrational rotation of the
circle $X=\mathbb{R}/\mathbb{Z}$, we give a full description of all these non-
crossed-product subalgebras.

###### Key words and phrases:

$C^{*}$-crossed products, intermediate subalgebras, irrational rotation
crossed product, $C^{*}$-simple groups

###### 2020 Mathematics Subject Classification:

Primary 37A55, 37B05; Secondary 46L55

This research was supported by grants from the Israel Science Foundation: ISF
1175/18 for the first author, ISF 1194/19 for the second, and ISF 2919/19 for
the third. The first named author was also supported by research funding from
the European Research Council (ERC) under the European Union’s Seventh
Framework Program (FP7-2007-2013) (Grant agreement No. 101078193)

###### Contents

  1. 1 Introduction
  2. 2 Some notations and preliminaries
    1. 2.1 The crossed product $C(X)\rtimes_{r}\mathbb{Z}$
    2. 2.2 Two circles
    3. 2.3 Generalized Fourier coefficients
    4. 2.4 A basic lemma
    5. 2.5 Ideal functions
    6. 2.6 The algebra $\mathcal{A}_{\mathfrak{c}}$
    7. 2.7 Basic Subalgebras
    8. 2.8 Ian Putnam’s $Y$-orbit breaking subalgebras
    9. 2.9 Ideals in intermediate subalgebras
  3. 3 The main theorem
    1. 3.1 Extraction of generalized Fourier coefficients.
    2. 3.2 Proof of Theorems 1.2 and 1.4
  4. 4 Structural results about subalgebras
    1. 4.1 The closed join formula
    2. 4.2 Residual algebras
    3. 4.3 Small algebras
    4. 4.4 About the simplicity of intermediate algebras.
  5. 5 Crossed products of non $C^{*}$-simple groups
    1. 5.1 Crossed products
    2. 5.2 Intermediate $C^{*}$-algebra associated to an ideal
    3. 5.3 Intermediate algebras for i.c.c. group actions.
    4. 5.4 Intermediate algebras for non-i.c.c. group actions.
    5. 5.5 An intermediate $C^{*}$-algebra associated to an ideal-II

##  1\. Introduction

Let $\Gamma$ be a countable group and $(X,\Gamma)$ a compact topological
dynamical system. We form the reduced $C^{*}$-crossed product
$C(X)\rtimes_{r}\Gamma$.

A factor map $\pi:(X,\Gamma)\to(Y,\Gamma)$ of $\Gamma$-systems gives rise to
an intermediate subalgebra
$C^{*}_{r}(\Gamma)<C(Y)\rtimes_{r}\Gamma<C(X)\rtimes_{r}\Gamma$. Now a
$C^{*}$-subalgebra $\mathcal{A}$ of the crossed product which contains
$C^{*}_{r}(\Gamma)$ is invariant under the group of automorphisms induced by
$\Gamma$ on $C(X)\rtimes_{r}\Gamma$ hence, by analogy, we can think of
intermediate subalgebras $\\{\mathcal{A}\ |\
C^{*}_{r}(\Gamma)<\mathcal{A}<C(X)\rtimes_{r}\Gamma\\},$ as non-commutative
factors of the dynamical system $(X,\Gamma)$. In [AK20], it was shown that
every intermediate $C^{*}$-algebra $\mathcal{A}$ of the form
$C_{r}^{*}(\Gamma)\subset\mathcal{A}\subset C(X)\rtimes_{r}\Gamma$ is simple,
provided that $\Gamma$ is $C^{*}$-simple and $X$ is a minimal $\Gamma$-space.
In a few cases (e.g., [Amr19]), it is possible to show that all the
intermediate algebras come from dynamical factors. Other results prove a
relative statement. Suzuki [Suz20], for example, shows that when $(\Gamma,Y)$
is a free action, all the intermediate algebras of the form
$C(Y)\rtimes_{r}\Gamma<\mathcal{A}<C(X)\rtimes_{r}\Gamma$ come from
intermediate dynamical factors $X\rightarrow Z\rightarrow Y$. In this work, we
study the existence of non-commutative factors when the group $\Gamma$ is not
$C^{*}$-simple.

Our main results concern the case when the group $\Gamma$ is the group of
integers and the dynamical system is an irrational circle rotation.

Let $\alpha\in 2\pi\left({\mathbb{R}}\setminus{\mathbb{Q}}\right)$,
$X=[-\pi,\pi]/(-\pi\sim\pi)$ and let $T:X\rightarrow X$ be the irrational
rotation given by $T(x)=x+\alpha\pmod{2\pi}$. The dynamical system $(X,T)$ is
minimal and uniquely ergodic with Lebesgue measure $\mu$, given by

$\mu(f)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx\qquad\qquad f\in C(X),$

as the unique invariant probability measure on $X$. We construct the crossed
product $\mathcal{C}=\mathcal{C}_{\alpha}=C(X)\rtimes_{r}{\mathbb{Z}}$. Note
that $C(X)\rtimes_{r}{\mathbb{Z}}$ is the same as the full crossed product
$C(X)\rtimes{\mathbb{Z}}$ since $\mathbb{Z}$ is amenable (See [BO08, Theorem
4.2.6]). Let $\pi_{n}:X\rightarrow X$ denote the $n$-fold covering maps
$\pi_{n}(x)=nx\pmod{2\pi}$, and $\pi_{n}^{*}:C(X)\to C(X)$ the dual map, given
by $f\mapsto f\circ\pi_{n}$. These maps give rise to the $T$-invariant
subalgebras $C^{n}(X)=\pi_{n}^{*}(C(X))$ and consequently to intermediate sub-
crossed products $\mathcal{C}^{n}=C^{n}(X)\rtimes_{r}{\mathbb{Z}}$. By
Suzuki’s result [Suz20] (also see [Ryo21]) every intermediate algebra of the
form $\mathcal{C}^{n}<\mathcal{A}<\mathcal{C}$ is of the form
$\mathcal{C}^{q}$ for some $q$ dividing $n$.

The situation in the absolute case turns out to be more complex. We show that
there are unaccountably many intermediate subalgebras
$C^{*}_{r}({\mathbb{Z}})<\mathcal{A}<\mathcal{C}$ which are not isomorphic to
such crossed products. Still, the situation is nice enough, and our main
theorem gives a complete classification of all the intermediate subalgebras of
$\mathcal{C}$. This classification, in turn, yields structural information
about these subalgebras. To state our results, we first introduce some
terminology.

Let $\operatorname{\mathcal{I}}$ denote the collection of all closed two-sided
ideals in $C^{*}_{r}({\mathbb{Z}})$. We denote by

$\mathfrak{A}=\\{\mathcal{A}\ |\
C^{*}_{r}({\mathbb{Z}})<\mathcal{A}<\mathcal{C}\\},\qquad\mathfrak{D}=\\{\mathfrak{c}:{\mathbb{Z}}\rightarrow\operatorname{\mathcal{I}}\
|\ \mathfrak{c}(0)=C^{*}_{r}({\mathbb{Z}})\\},$

the collections of all intermediate subalgebras, and all
$\operatorname{\mathcal{I}}$ valued functions on ${\mathbb{Z}}$ such that
$\mathfrak{c}(0)=C^{*}_{r}({\mathbb{Z}})$, respectively. We will refer to
elements of $\mathfrak{D}$ as ideal functions. The support of an ideal
function is defined as

$\operatorname{Supp}(\mathfrak{c})=\\{n\in{\mathbb{Z}}\ |\
\mathfrak{c}(n)\neq(0)\\}.$ (1)

For an intermediate algebra $\mathcal{A}\in\mathfrak{A}$ and for an ideal
function $\mathfrak{c}\in\mathfrak{D}$ we set

$\mathfrak{I}_{\mathcal{A}}=\\{I\ |\
I\lhd\mathcal{A}\\}\qquad\mathfrak{D}_{\mathfrak{c}}=\\{\mathfrak{j}:{\mathbb{Z}}\rightarrow\mathcal{I}\
|\ \mathfrak{j}(n)\leq\mathfrak{c}(n),\forall n\in{\mathbb{Z}}\\}.$

$\mathfrak{I}_{\mathcal{A}}$ is the collection of all closed two-sided ideals
in $\mathcal{A}$. Note that such an ideal is automatically ∗-invariant. In the
sequel, when we say ‘ideal,’ we mean a closed, two-sided one.

We refer to the elements of $\mathfrak{D}_{\mathfrak{c}}$ as
$\mathfrak{c}$-ideal functions. Note that
$\mathfrak{D}_{\mathfrak{c}}\not\subset\mathfrak{D}$ because we do not require
$\mathfrak{j}(0)=C^{*}_{r}({\mathbb{Z}})$ for
$\mathfrak{j}\in\mathfrak{D}_{\mathfrak{c}}$. The support of a
$\mathfrak{c}$-ideal function is defined by an equation analogous to Equation
(1). Since by definition $\mathfrak{j}(n)\leq\mathfrak{c}(n),\forall
n\in{\mathbb{Z}},$ we have
$\operatorname{Supp}(\mathfrak{j})\subset\operatorname{Supp}(\mathfrak{c}),\forall\mathfrak{j}\in\mathfrak{D}_{\mathfrak{c}}$.

All of these collections admit a natural lattice structure. The partial order
on $\mathfrak{A}$ is given by inclusion. Similarly, on $\mathfrak{D}$ we write
$\mathfrak{c}_{1}\preceq\mathfrak{c}_{2}$ when
$\mathfrak{c}_{1}(n)\leq\mathfrak{c}_{2}(n),\ \forall n\in{\mathbb{Z}}$. The
lattice operations are given by:

$\mathcal{A}_{1}\wedge\mathcal{A}_{2}=\mathcal{A}_{1}\cap\mathcal{A}_{2}$, |  $\mathcal{A}_{1}\vee\mathcal{A}_{2}=\overline{\langle\mathcal{A}_{1},\mathcal{A}_{2}\rangle}$, | $\forall\mathcal{A}_{1},\mathcal{A}_{2}\in\mathfrak{A}$
---|---|---
$(\mathfrak{c}_{1}\wedge\mathfrak{c}_{2})(n)=\mathfrak{c}_{1}(n)\cap\mathfrak{c}_{2}(n)$, |  $(\mathfrak{c}_{1}\overline{\vee}\mathfrak{c}_{2})(n)=\overline{\mathfrak{c}_{1}(n)+\mathfrak{c}_{2}(n)}$, | $\forall\mathfrak{c}_{1},\mathfrak{c}_{2}\in\mathfrak{D}$

Given $\mathcal{A}\in\mathfrak{A}$ and $\mathfrak{c}\in\mathfrak{D}$ the
lattice structure on $\mathfrak{I}_{\mathcal{A}}$ and on
$\mathfrak{D}_{\mathfrak{c}}$ is defined similarly.

Via the Fourier transform $C^{*}_{r}({\mathbb{Z}})\cong C(S^{1})$, and hence
$\operatorname{\mathcal{I}}$ can be identified with the collection of closed
subsets of the circle $\operatorname{Cl}(S^{1})$. Under this identification,
with every $\mathfrak{c}\in\mathfrak{D}$, we associate a set function
$\ddot{\mathfrak{c}}:{\mathbb{Z}}\rightarrow\operatorname{Cl}(S^{1})$, such
that $\ddot{\mathfrak{c}}(0)=\emptyset$. By $\ddot{\mathfrak{D}}$, we denote
the collection of all such functions. Similarly for
$\mathfrak{c}\in\mathfrak{D}$ we define the collection
$\ddot{\mathfrak{D}}_{\mathfrak{c}}$.

It is crucial to distinguish between the given dynamical system $(X,T)$ and
the circle $S^{1}=\hat{{\mathbb{Z}}}$ introduced in the above paragraph (see
Section 2.2 below). Still, perhaps surprisingly, we will often use the
irrational rotation given by $\tau(t)=t+\alpha\pmod{2\pi}$. Thus, $\tau$
represents the irrational rotation on $S^{1}$ by the same angle $\alpha$
appearing in the given irrational rotation $T:X\rightarrow X$.

######  Definition 1.1.

An ideal function $\mathfrak{c}\in\mathfrak{D}$ is called closed if the
following properties are satisfied:

  * $\mathtt{Cif}1$

$\ddot{\mathfrak{c}}(-n)=\tau^{n}\ddot{\mathfrak{c}}(n),\ \forall
n\in{\mathbb{Z}}$,

  * $\mathtt{Cif}2$

$\ddot{\mathfrak{c}}(m+n)\subset\tau^{-m}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m)$
for every $m,n\in\operatorname{Supp}(\mathfrak{c})$.

By $\mathfrak{C}\subset\mathfrak{D}$, we denote the collection of closed ideal
functions.

We define functions $\Phi:\mathfrak{D}\rightarrow\mathfrak{A}$ and
$\Psi:\mathfrak{A}\rightarrow\mathfrak{D}$ by:

$\Phi(\mathfrak{c})=\overline{\langle e^{inx}\eta\ |\
n\in{\mathbb{Z}},\eta\in\mathfrak{c}(n)\rangle},\quad\Psi(\mathcal{A})(n)=\\{\eta\in
C^{*}_{r}({\mathbb{Z}})\ |\ e^{inx}\eta\in\mathcal{A}\\}.$

Note that here $e^{inx}$ is a function in $C(X)$. See Section 2 for more
details concerning these notations.

The meet and join operations that we defined above translate to a standard
union and intersection:

$(\ddot{\mathfrak{c}}_{1}\wedge\ddot{\mathfrak{c}}_{2})(n)=\ddot{\mathfrak{c}}_{1}(n)\cup\ddot{\mathfrak{c}}_{2}(n),\quad(\ddot{\mathfrak{c}}_{1}\overline{\vee}\ddot{\mathfrak{c}}_{2})(n)=\ddot{\mathfrak{c}}_{1}(n)\cap\ddot{\mathfrak{c}}_{2}(n),\quad\forall\ddot{\mathfrak{c}}_{1},\ddot{\mathfrak{c}}_{2}\in\ddot{\mathfrak{D}}.$

It is straightforward to verify that $\ddot{\mathfrak{C}}$, and hence
$\mathfrak{C}$, are closed under the meet operation. On the other hand, the
join of two closed ideal functions is not generally closed (see Example 2.14).
We are, therefore, led to define the closed join operation:

$\mathfrak{c}_{1}\vee\mathfrak{c}_{2}:=\Psi\circ\Phi(\mathfrak{c}_{1}\overline{\vee}\mathfrak{c}_{2}).$

In Proposition 4.1, we will establish an explicit (albeit long) formula for
the closed join. The meet and the closed join now give $\mathfrak{C}$ a
structure of a lattice, which is not a sublattice of $\mathfrak{D}$. This is
the lattice structure alluded to in item (4) of our main theorem below.

######  Theorem 1.2.

(Main theorem) Let $\Phi:\mathfrak{D}\rightarrow\mathfrak{A}$ and
$\Psi:\mathfrak{A}\rightarrow\mathfrak{D}$ be as above, then:

  1. (1)

The maps $\Phi,\Psi$ form a (monotone) Galois connection between the two
lattices $\mathfrak{D},\mathfrak{A}$. Namely, for
$\mathfrak{c}\in\mathfrak{D},\mathcal{A}\in\mathfrak{A}$ we have
$\Phi(\mathfrak{c})<\mathcal{A}$ if and only if
$\mathfrak{c}<\Psi(\mathcal{A})$.

  2. (2)

The connection is perfect on the algebra side. Namely
$\mathcal{A}=\Phi\circ\Psi(\mathcal{A})$ for every
$\mathcal{A}\in\mathfrak{A}$.

  3. (3)

$\mathfrak{c}=\Psi\circ\Phi(\mathfrak{c})$ if and only if it is closed in the
sense of Definition 1.1.

  4. (4)

$\Phi\upharpoonright_{\mathfrak{C}}:\mathfrak{C}\rightarrow\mathfrak{A}$ is an
isomorphism of lattices, with $\Psi$ as its inverse.

It is worthwhile noting what is easy and what requires new ideas in the above
theorem. Once all the definitions are in place, the fact that $\Phi,\Psi$ form
a Galois connection is clear. We invite the readers to verify that before
proceeding. From this, it follows, by standard properties of Galois
connections, that $\Phi,\Psi$ form an isomorphism between the sublattices of
closed elements on both sides. The deeper part lies in showing that this
Galois connection does not degenerate in the sense that closed elements are
ubiquitous on both sides. In particular, the connection is perfect in the
sense that all elements are closed on the algebra side. Indeed, it is a
priory, not clear at all why a given intermediate algebra
$\mathcal{A}\in\mathfrak{A}$ should contain any element of the form
$e^{inx}\eta\ (n\not=0,\ 0\neq\eta\in C^{*}_{r}({\mathbb{Z}}))$. For any given
algebra $\mathcal{A}\in\mathfrak{A}$, with an associated ideal function
$\mathfrak{c}=\Psi(\mathcal{A})\in\mathfrak{C}$, we can state a similar
classification for all closed two-sided ideals of $\mathcal{A}$.

######  Definition 1.3.

Given $\mathfrak{c}\in\mathfrak{C}$ a $\mathfrak{c}$-ideal function
$\mathfrak{j}\in\mathfrak{D}_{\mathfrak{c}}$ will be called closed it
satisfies the following two conditions:

  * $\mathtt{Cif}1(\mathfrak{c})$

$\ddot{\mathfrak{j}}(-n)=\tau^{n}\ddot{\mathfrak{j}}(n),\ \forall
n\in{\mathbb{Z}}$,

  * $\mathtt{Cif}2(\mathfrak{c})$

$\ddot{\mathfrak{j}}(m+n)\subset\tau^{-m}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{j}}(m)$
for every
$n\in\operatorname{Supp}(\mathfrak{c}),m\in\operatorname{Supp}(\mathfrak{j})$.

For $\mathfrak{c}\in\mathfrak{D}$ and $\mathcal{A}\in\mathfrak{A}$ we define:
$\Phi_{\mathfrak{c}}:\mathfrak{D}_{\mathfrak{c}}\rightarrow\mathfrak{I}_{\Phi(\mathfrak{c})}$
and
$\Psi:\mathfrak{I}_{\mathcal{A}}\rightarrow\mathfrak{D}_{\Psi(\mathcal{A})}$
by:

$\Phi_{\mathfrak{c}}(\mathfrak{j})=\overline{\langle e^{inx}\eta\ |\
n\in{\mathbb{Z}},\eta\in\mathfrak{j}(n)\rangle},\quad\Psi_{\mathcal{A}}(I)(n)=\\{\eta\in
C^{*}_{r}({\mathbb{Z}})\ |\ e^{inx}\eta\in I\\}.$

######  Theorem 1.4.

Fix $\mathfrak{c}\in\mathfrak{D}$ and set $\mathcal{A}=\Phi(\mathfrak{c})$.

  1. (1)

The maps $\Phi_{\mathfrak{c}},\Psi_{\mathcal{A}}$ form a (monotone) Galois
connection between the two lattices
$\mathfrak{D}_{\mathfrak{c}},\mathfrak{I}_{\mathcal{A}}$.

  2. (2)

The connection is perfect at $\mathfrak{I}_{\mathcal{A}}$.

  3. (3)

$\mathfrak{j}=\Psi_{\mathcal{A}}\circ\Phi_{\mathfrak{c}}(\mathfrak{j})$ if and
only if it is closed in the sense of Definition 1.3.

  4. (4)

$\Phi_{\mathfrak{c}}\upharpoonright_{\mathfrak{C}}:\mathfrak{C}_{\mathfrak{c}}\rightarrow\mathfrak{I}_{\mathcal{A}}$
is an isomorphism of lattices, with $\Psi_{\mathcal{A}}$ as its inverse.

As in the main theorem, the lattice structure on $\mathfrak{C}_{\mathfrak{c}}$
uses the closed join operation, which is defined as in Proposition 4.1. Let us
turn to some applications of this structure theory.

######  Definition 1.5.

We say that an intermediate algebra $\mathcal{A}\in\mathfrak{A}$ is residual
if $\ddot{\Psi(\mathcal{A})}(n)\in\operatorname{Cl}(S^{1})$ is nowhere dense
for every $n\in\text{Supp}\left(\Psi(\mathcal{A})\right)$. An ideal
$J\lhd\mathcal{A}$ in a residual algebra is called a residual ideal if
$\ddot{\Psi_{\mathcal{A}}(J)}(n)$ is nowhere dense for every
$n\in\text{Supp}\left(\ddot{\Psi_{\mathcal{A}}(J)}\right)$. We exclude the
trivial algebra and the trivial ideal. These will not be considered residual
even though they qualify formally because there is nothing nontrivial in their
support.

Here are some properties of these algebras following the classification (see
Sub-section 4.2).

######  Theorem 1.6.

cor:residual Let $\mathcal{A},\\{\mathcal{A}_{i}\\}_{i\in I}\in\mathfrak{A}$
be residual algebras and $(0)\neq J\lhd\mathcal{A}$ a closed two sided
nontrivial ideal. Let $\mathfrak{c},\\{\mathfrak{c}_{i}\\}_{i\in
I},\mathfrak{j}$ be the associated ideal functions. Set

$\Omega=\bigcup_{m,n\in\operatorname{Supp}(\mathfrak{c})}\tau^{-m}\ddot{\mathfrak{c}}(n)$

Then

  1. (1)

Every intermediate algebra containing a residual one is also residual.

  2. (2)

$\mathcal{A}_{1}\wedge\mathcal{A}_{2}$ is residual.

  3. (3)

$\operatorname{Supp}(\mathfrak{c})=\\{n\in{\mathbb{Z}}\ |\
\mathfrak{c}(n)\neq(0)\\}$ is a subgroup of ${\mathbb{Z}}$.

  4. (4)

$\operatorname{Supp}(\mathfrak{c})=\operatorname{Supp}(\mathfrak{j})$.

  5. (5)

$\ddot{\mathfrak{j}}(k)\subset\Omega$ for every $k\in\mathbb{Z}$ and in
particular every non trivial ideal $J\lhd\mathcal{A}$ is residual.

  6. (6)

$\mathcal{A}$ is center free.

Property (6) holds more generally, even if we assume that
$\operatorname{Supp}(\mathfrak{c})(n)$ is nowhere dense for one $0\neq
n\in\operatorname{Supp}(\mathfrak{c})$.

Residual algebras should be thought of as large intermediate subalgebras. At
the other extreme stand the algebras highlighted in the following definition:

######  Definition 1.7.

An intermediate algebra $\mathcal{A}\in\mathcal{A}$ with associated ideal
function $\mathfrak{c}=\Psi(\mathcal{A})$ is called small if
$\operatorname{Supp}(\mathfrak{c})$ is finite.

Here are some properties of small intermediate subalgebras.

######  Proposition 1.8.

cor:small Let $\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}\in\mathfrak{A}$ be
small subalgebras $\mathfrak{c}_{1},\mathfrak{c}_{2},\mathfrak{c}$ the
corresponding ideal functions, then

  1. (1)

$\mathfrak{c}(n)$ has a nonempty interior for every $0\neq n$. In particular,
a small subalgebra is never residual.

  2. (2)

A basic subalgebra $\mathcal{A}_{q,P}$ (see Example 2.11 and Subsection 2.7)
is small if and only if it is not residual.

  3. (3)

$\mathcal{A}$ is finite dimensional as a module over
$C^{*}_{r}({\mathbb{Z}})$.

  4. (4)

Many small algebras admit a non-trivial center.

  5. (5)

There are small algebras $\mathcal{A}_{1},\mathcal{A}_{2}\in\mathfrak{A}$ such
that $\mathcal{A}_{1}\vee\mathcal{A}_{2}$ is not small.

In Subsection 4.4, we use our machinery to study the simplicity of
intermediate algebras. We give examples of some algebras that are simple and
others that are not. A partial characterization for simplicity is given in
Propositions 4.6 and 4.13.

######  Remark 1.9.

The universal property of $\mathcal{C}$: Contrary to what the notation
$\mathcal{C}=C(X)\rtimes_{r}{\mathbb{Z}}$ seems to imply there is a symmetry
between the two factors, $C(X)$ and $C^{*}(\mathbb{Z})$, of the crossed
product. This is realized by an involutive automorphism
$\iota:\mathcal{C}\rightarrow\mathcal{C}$ with the property that $\iota
C(X)=C^{*}_{r}({\mathbb{Z}})$. When $\mathcal{C}=\mathcal{C}_{\alpha}$ is
viewed as a crossed product over the algebra $C^{*}_{r}({\mathbb{Z}})$ it is
isomorphic to $\mathcal{C}_{-\alpha}$.

It follows that all our results concerning the intermediate subalgebras
$C^{*}_{r}({\mathbb{Z}})<\mathcal{A}<\mathcal{C}$, are also valid for the
analogous intermediate subalgebras $C(X)<\mathcal{A}<\mathcal{C}$, when they
are reformulated with respect to the rotation $R_{-\alpha}$. See Remark 2.1
below.

To sum up, we have:

######  Theorem 1.10.

notcrossedprodZ There are uncountably many intermediate subalgebras
$C^{*}_{r}({\mathbb{Z}})<\mathcal{A}<\mathcal{C}$ which are not of the form
$\mathcal{A}=C(Y)\rtimes_{r}{\mathbb{Z}}$, with $X\to Y$ a dynamical factor of
the system $(X,R_{\alpha})$. They are indexed by the system of ideal functions
$\mathfrak{C}$ as described in Theorem 1.2.

The fact that the crossed product $C(X)\rtimes_{r}\mathbb{Z}$ admits
intermediate sub-algebras, which are not themselves crossed products, are not
at all special for $\mathbb{Z}$ dynamical systems. Let $(X,\Gamma)$ be a
dynamical system that admits an invariant probability measure of full support.
For such a dynamical system $(X,\Gamma)$, in Section 5, we establish that to
every non-trivial ideal $I<C^{*}_{r}(\Gamma)$, corresponds an intermediate
subalgebra $\mathcal{B}_{I}$,

$C^{*}_{r}(\Gamma)\subset\mathcal{B}_{I}\subset C(X)\rtimes_{r}\Gamma,$

such that $\mathcal{B}_{I}\cap C(X)=\mathbb{C}$. Such a non-trivial ideal
exists if and only if $\Gamma$ is not $C^{*}$-simple. Moreover, this
construction is also valid for general $\Gamma$-$C^{*}$-algebras $\mathcal{A}$
as long as $\mathcal{A}$ admits a faithful $\Gamma$-invariant state. However,
to conclude that $\mathcal{B}_{I}$ is not a crossed product
$C^{*}$-subalgebra, we need additional assumptions on the group $\Gamma$, or
the $C^{*}$-algebra $\mathcal{A}$. More precisely, we have:

######  Theorem 1.11.

notcrossedprod Let $\Gamma$ be a discrete group that is not $C^{*}$-simple and
$\mathcal{A}\neq{\mathbb{C}}$ a $\Gamma$-$C^{*}$-algebra admitting a faithful
$\Gamma$-invariant state. Assume that at least one of the following conditions
holds:

  * •

$\mathcal{A}=C(X)$ with $\left\lvert X\right\rvert\geq 3$.

  * •

$\Gamma_{f}\neq\nicefrac{{{\mathbb{Z}}}}{{2{\mathbb{Z}}}}$, where $\Gamma_{f}$
denotes the FC-center of $\Gamma$.

Then, there is an intermediate $C^{*}$-algebra
$C_{r}^{*}(\Gamma)\subset\mathcal{D}\subset\mathcal{A}\rtimes_{r}\Gamma$ which
is not of the form $\mathcal{D}=\mathcal{B}\rtimes_{r}\Gamma$ for any
$\Gamma$-invariant subalgebra $\mathcal{B}\subset\mathcal{A}$.

Note that $\Gamma_{f}=\langle e\rangle$ exactly when $\Gamma$ is an i.c.c.
group. In this case, we show (see i.c.c-intermediate) that every non-trivial
ideal $I<C_{r}^{*}(\Gamma)$ corresponds to an intermediate sub-algebra
$\mathcal{B}_{I}$ which is not a crossed product $C^{*}$-algebra.

Contrary to the spirit of the above theorem, Example
LABEL:ex:necessaryassumption shows that for the action of the (non
$C^{*}$-simple) group $\Gamma=\mathbb{Z}/2\mathbb{Z}$ on $X=\\{p_{1},p_{2}\\}$
there is no intermediate subalgebra $\mathcal{B}$ of the form
$C_{r}^{*}(\mathbb{Z}/2\mathbb{Z})\subset\mathcal{B}\subset
C(X)\rtimes_{r}\mathbb{Z}/2\mathbb{Z}$ at all. This example is responsible for
the assumption in the second bulleted alternative in theorem
LABEL:notcrossedprod.

Using similar methods, we also show that, for a $\Gamma$-$C^{*}$-algebra
$\mathcal{A}$, every non-trivial two-sided closed $\Gamma$-invariant ideal
$I\leq\mathcal{A}$ corresponds to an intermediate object
$\mathcal{A}\subset\mathcal{B}\subset\mathcal{A}\rtimes_{r}\Gamma$ such that
$\mathcal{B}$ is not a crossed product of the form
$\mathcal{A}\rtimes_{r}\Lambda$ for any normal subgroup
$\Lambda\triangleleft\Gamma$ (see notcomingfromasubgroup).

We attempt to generalize our classification in the subsequent two works in
progress. First, to the general setting of $C(X)\rtimes_{r}\Gamma$ with $X$ is
a compact Abelian group and $\Gamma$ a countable dense subgroup of $X$. In the
second, we treat analogous questions regarding intermediate von Neumann
algebras $\mathcal{N}$ of the form
$L(\Gamma)\subset\mathcal{N}\subset\mathcal{M}\rtimes\Gamma$, where
$\mathcal{M}$ is a von Neumann algebra. In particular, we study the case where
$\mathcal{M}$ is the commutative $L^{\infty}(X,\mu)$ equipped with the
automorphism corresponding to the irrational rotation $R_{\alpha}$.

When we showed a draft of our work to Ilan Hirshberg, he pointed out that our
classification results are closely related and, in fact, partly overlap
several existing works in the literature. More specifically, the works of Ruy
Exel (see [Exe94]), where the author shows the existence and investigates some
of the properties of our ideal functions in a much more general setup. Then,
his extensive book on “Fell Bundles” [Exe17], and in particular, the results
in Chapter 23 of this book. The second source of examples arises as a result
of the dual nature of $\mathcal{C}$. Our construction of ideal functions turns
out to be a generalization (within $\mathcal{C}$) of the well-known Putnam’s
“$Y$-orbit breaking” subalgebras, as described, for example, in Phillips’
contribution to the book [GKPT18] (see Subsection 2.8). Nonetheless, our
approach and methods seem new, and so does our complete classification of
intermediate subalgebras.

Acknowledgements: We thank Ilan Hirshberg, Hanfeng Li, Mehrdad Kalantar, Sven
Raum, and Yongle Jiang for their helpful remarks, discussions, and
corrections. Finally, we thank the anonymous reviewer for carefully reading
our manuscript and for many insightful comments and suggestions. In
particular, we thank the referee for pointing out two serious mistakes: (i) in
the formula for the closed ideal function of the join of two intermediate
algebras, and (ii) in our original proof of Proposition LABEL:prop:Q which is
now replaced by a proof kindly suggested by the referee.

##  2\. Some notations and preliminaries

###  2.1. The crossed product $C(X)\rtimes_{r}\mathbb{Z}$

Let $(X,\mu,T)$ be a metric minimal free uniquely ergodic cascade (i.e. a
$\mathbb{Z}$-action, $n\mapsto T^{n}$, where $T$ is a homeomorphism of $X$).
Let $\mathcal{C}=C(X)\rtimes_{r}\mathbb{Z}$ be the corresponding reduced
crossed product $C^{*}$-algebra and, as usual, we denote by
$C_{r}^{*}(\mathbb{Z})$ the $C^{*}$-algebra of $\mathbb{Z}$. Via the Fourier
transform $\mathcal{F}:\ell^{2}(\mathbb{Z})\to L^{2}(S^{1},\frac{dt}{2\pi})$,
$(a_{n})\mapsto\sum_{n\in\mathbb{Z}}a_{n}e^{int}$ (with
$S^{1}=\mathbb{R}/2\pi\mathbb{Z}$), we can view $C_{r}^{*}(\mathbb{Z})$ as the
commutative algebra $C(S^{1})$, where for $\varphi\in C(S^{1})$ the
corresponding operator in $C^{*}(\mathbb{Z})$ is given by multiplication
$v\mapsto\varphi v,\ v\in L^{2}(S^{1})$. Under this correspondence, we
identify $\lambda_{1}$ (which corresponds to the shift operator on
$\ell^{2}(\mathbb{Z})$) with the function $e^{it}\in C(S^{1})$, and
$\lambda_{n}$ with $e^{nit}$, for every $n\in{\mathbb{Z}}$. We will denote the
opposite isomorphism $C(S^{1})\rightarrow C^{*}_{r}({\mathbb{Z}})$, by
$\varphi\mapsto\hat{\varphi}$. For trigonometric polynomials (and other nice
functions), this is explicitly given by regular Fourier series
$\hat{\varphi}:=\sum_{n\in{\mathbb{Z}}}\hat{\varphi}(n)\lambda_{n}$ with
$\hat{\varphi}(n)=\frac{1}{2\pi}\int_{S^{1}}\varphi(t)e^{-int}dt$, and then
extended by continuity to the whole of $C(S^{1})$.

We then identify $\mathcal{C}$ with the $C^{*}$-subalgebra of
$\mathbb{B}(L^{2}(X,\mu))$ generated by the (unitary) Koopman operator
$U_{T}$, which we also denote as $\lambda_{1}$, and the commutative
$C^{*}$-algebra of multiplication operators $M_{f}:v\mapsto fv,\ v\in
L^{2}(X,\mu),\ f\in C(X)$. We then let $\lambda_{n}=U_{T}^{n}$, so in
particular $\lambda_{0}=I$ and $\lambda_{-1}=U_{T}^{-1}=\lambda_{1}^{*}$. A
general element $a$ of the group ring over $C(X)$ (which is dense in
$\mathcal{C}$) has the form

$a=\sum_{n=-N}^{N}M_{f_{n}}U_{T}^{n}$

which, when no confusion can arise, we also write as

$a=\sum_{n=-N}^{N}f_{n}\lambda_{n}.$

With these notations the group ring that generates $C_{r}^{*}(\mathbb{Z})$
consists of elements of the form

$a=\sum_{n=-N}^{N}a_{n}\lambda_{n},$

with $a_{n}\in{\mathbb{C}}$.

Recall that the $T$-action on $C(X)$ is implemented by conjugation by
$\lambda_{1}$:

$\lambda_{1}M_{f}\lambda_{1}^{*}=M_{f\circ T},$

or simply

$\lambda_{1}f\lambda_{1}^{*}=f\circ T.$

The reduced crossed product $C(X)\rtimes_{r}\mathbb{Z}$ comes equipped with a
$\mathbb{Z}$-equivariant canonical conditional expectation
$\mathbb{E}:C(X)\rtimes_{r}\mathbb{Z}\to C(X)$ defined on the algebraic
crossed product by

$\mathbb{E}(a)=\mathbb{E}\left(\sum_{n=-N}^{N}f_{n}\lambda_{n}\right)=f_{0}.$

It follows from [BO08, Proposition 4.1.9] that $\mathbb{E}$ extends to a
faithful conditional expectation from $C(X)\rtimes_{r}\Gamma$ onto $C(X)$.

For a general element $a\in\mathcal{C}=C(X)\rtimes_{r}\mathbb{Z}$, its Fourier
coefficient of order $n$ is the function

$\hat{a}(n)=\mathbb{E}(a\lambda^{*}_{n})\in C(X).$

###  2.2. Two circles

From here until the end of Section 4, unless we say otherwise, we take
$(X,\mu,T)$ to be the circle $X$, equipped with normalized Lebesgue measure
$\mu$, and with $T=R_{\alpha}$ an irrational rotation. Thus the set of
characters $\\{e^{inx}|\ n\in\mathbb{Z}\\}$ forms a basis for $L^{2}(X,\mu)$
and their linear combinations form a dense $*$-subalgebra of $C(X)$.

When $(X,T)$ is an irrational rotation on the circle, it is important not to
confuse the two circles. We will always refer to the Gelfand dual of
$C^{*}_{r}({\mathbb{Z}})$ as $S^{1}$ and denote functions
$\varphi(t),\psi(t)\in C(S^{1})$ by the lowercase Greek letters, with a
variable $t$. The phase space of the dynamical system will always be denoted
by $X$, and functions on $X$ will be denoted by Roman letters, $f(x),g(x)\in
C(X)$ with a variable $x$. The action of ${\mathbb{Z}}$ on $X$ is by the
rotation $T:X\rightarrow X$, $Tx=x+\alpha\pmod{2\pi}$. We will also consider
the rotation, by the same angle $\alpha$, on the other circle and denote it by
$\tau:S^{1}\rightarrow S^{1}$, $\tau t=t+\alpha\pmod{2\pi}$.

The rotation $\tau$ yields an action on $C(S^{1})$ by
$\varphi\mapsto\varphi\circ\tau$. Via the standard identification of
$C^{*}_{r}({\mathbb{Z}})\cong C(S^{1})$ we obtain an action
$\tau:C^{*}_{r}({\mathbb{Z}})\rightarrow C^{*}_{r}({\mathbb{Z}})$, which is
explicitly given in terms of Fourier coefficients by
$\widehat{\tau\varphi}(n)=\hat{\varphi}(n)e^{in\alpha}$. We denote the
normalized Lebesgue measure on $S^{1}$ by $\nu$. It is the unique
$\tau$-invariant Borel probability measure on $S^{1}$. Note that
$\mathbb{E}:\mathcal{C}\to C(X)$, our canonical conditional expectation, is
given by

$\mathbb{E}\left(\sum_{-N}^{N}f_{n}\lambda_{n}\right)=\mathbb{E}\left(\sum_{-N}^{N}f_{n}\widehat{e^{int}}\right)=\sum_{-N}^{N}f_{n}\int_{S^{1}}e^{int}\;d\nu(t)=f_{0}.$

The $\tau$-action naturally extends to an action on the space of closed ideals
$\operatorname{\mathcal{I}}$ in $C^{*}_{r}({\mathbb{Z}})$, identified with the
closed subsets of the circle $\operatorname{Cl}(S^{1})$. By abuse of notation,
we will denote all of these by the same letter $\tau$. The following
discussion yields the involutive automorphism alluded to in Remark 1.9

######  Remark 2.1.

There exists an involutive automorphism
$\iota:\mathcal{C}\rightarrow\mathcal{C}$ with $\iota
C(X)=C^{*}_{r}({\mathbb{Z}})$. As is shown e.g. in [Dav96] the $C^{*}$-algebra
$\mathcal{C}=\mathcal{C}_{\alpha},\ \alpha\in[0,1)$ irrational, is isomorphic
to the universal $C^{*}$-algebra $C^{*}(U,V)$, with $U,V$ two unitaries
satisfying the relation

$UV=e^{2\pi i\alpha}VU.$ (2)

It is also shown that
$K_{0}(\mathcal{C}_{\alpha})=\mathbb{Z}+\mathbb{Z}\alpha$ and it follows that
two such algebras $\mathcal{C}_{\alpha}$ and $\mathcal{C}_{\beta}$ are
isomorphic if and only if $\beta=\pm\alpha$. In our setup we have
$U=M_{e^{ix}}\in C(X)$ and $V=\lambda_{1}\in C^{*}(\mathbb{Z})\cong C(S^{1})$.
If we take $U=\lambda_{1}=M_{e^{it}}\in C(S^{1})$ and $V=M_{e^{ix}}\in C(X)$
we obtain exactly Equation (2) by Lemma 2.5 below (with $\tau^{-1}$ replacing
$T$).

###  2.3. Generalized Fourier coefficients

In addition to the standard Fourier coefficients of functions $\varphi\in
C(S^{1})$ defined above, we use $C^{*}_{r}({\mathbb{Z}})$-valued Fourier
expansion on $X$.

Using $\mu$, the Lebesgue probability measure on $X$, we define a
$\mathbb{Z}$-equivariant conditional expectation
$\mathbb{E}_{\mu}:\mathcal{C}\to C^{*}_{r}({\mathbb{Z}})$, by setting
$\mathbb{E}_{\mu}(f\lambda_{n})=\mu(f)\lambda_{n}$ and extending to the whole
of $\mathcal{C}$ using linearity and continuity (see [BO08, Exercise 4.1.4]).
Be careful not to confuse this with the standard conditional expectation
$\mathbb{E}:\mathcal{C}\to C(X)$.

######  Definition 2.2.

Let $a\in\mathcal{C}$ we define its $C^{*}_{r}({\mathbb{Z}})$ valued Fourier
coefficients by the following formula

$\overbracket{a}(n)=\mathbb{E}_{\mu}\left(e^{-inx}a\right).$

For finite sums of the form

$a=\sum_{k=-N}^{N}e^{inx}\hat{\varphi}_{n},\quad\varphi_{n}\in C(S^{1}).$ (3)

we just have $\overbracket{a}(n)=\hat{\varphi}_{n}$. We will refer to such
finite sums as generalized trigonometric polynomials. For a function $f\in
C(X)$,

$\overbracket{f}(n)=\int_{X}e^{-inx}f(x)\frac{dx}{2\pi},$

is the standard $n^{th}$ Fourier coefficient. Note that our notation clearly
distinguishes this from Fourier coefficients $\hat{\varphi}(n)$ for a function
$\varphi\in C(S^{1})$.

Given $a\in\mathcal{C}$ we will denote by

$Q_{n}(a)=\sum_{k=-n}^{n}e^{ikx}\overbracket{a}(k)=\sum_{k=-n}^{n}e^{ikx}\hat{\varphi}_{k}.$

Such generalized Fourier polynomial approximations play a central role in our
arguments. Note that these differ from the more standard way of approximating
elements of $\mathcal{C}$ by expressions of $\sum f_{k}\lambda_{k},\ f_{k}\in
C(X)$.

Heuristically, one can think of general elements of $\mathcal{C}$ as infinite
Fourier expansions of the form $\sum_{n}e^{inx}\overbracket{a}(n)$. Of course,
such infinite series often fail to converge in the norm. We will overcome this
by approximating elements of the algebra using the following (also see [ST09,
Proposition 3.3]):

######  Theorem 2.3.

(The Fejér approximation theorem) The following two sequences, of generalized
Cesàro means of an element $a\in\mathcal{C}$

$\displaystyle\hat{\sigma}_{n}(a)$ $\displaystyle=$
$\displaystyle\sum_{j=-n}^{n}\left(1-\frac{\left\lvert
j\right\rvert}{n+1}\right)\hat{a}(j)\lambda_{j}$
$\displaystyle{\overbracket{\sigma}}_{n}(a)$ $\displaystyle=$
$\displaystyle\sum_{j=-n}^{n}\left(1-\frac{\left\lvert
j\right\rvert}{n+1}\right)e^{ijx}\overbracket{a}(j).$

Both converge to $a$ in the norm.

######  Note 2.4.

For any $a\in\mathcal{C}$ and fixed $q\in{\mathbb{N}}$ we have that

$\lim_{n\rightarrow\infty}\left\lVert
Q_{q}({\overbracket{\sigma}}_{n}(a))-Q_{q}(a)\right\rVert=0.$

###  2.4. A basic lemma

######  Lemma 2.5.

Let $m,n\in{\mathbb{Z}}$ and $\hat{\varphi},\hat{\psi}\in
C^{*}_{r}({\mathbb{Z}})$ then:

  1. (1)

$\hat{\varphi}e^{inx}=e^{inx}\widehat{\varphi\circ\tau^{n}},\quad\forall
n\in{\mathbb{Z}},\hat{\varphi}\in C^{*}_{r}({\mathbb{Z}})$,

  2. (2)

$\left(e^{inx}\hat{\varphi}\right)^{*}=e^{-inx}\widehat{\overline{\varphi}\circ\tau^{-n}}$,

  3. (3)

$e^{inx}\hat{\varphi}e^{imx}\hat{\psi}=e^{i(n+m)x}\left[\widehat{(\varphi\circ\tau^{m})\psi}\right]$.

###### Proof.

The first property directly implies the other two. Also, it is enough to prove
the first property when $\varphi$ is a finite sum of the form
$\hat{\varphi}=\sum_{k=-K}^{K}a_{k}\lambda_{k}$ because such functions are
uniformly dense in $C(S^{1})$. In this case, we obtain a direct computation:

$\displaystyle\sum_{k}a_{k}\lambda_{k}e^{inx}$ $\displaystyle=$
$\displaystyle\sum_{k}a_{k}\lambda_{k}e^{inx}\lambda_{-k}\lambda_{k}=\sum_{k}a_{k}e^{in(x+k\alpha)}\lambda_{k}$
$\displaystyle=$ $\displaystyle
e^{inx}\sum_{k}a_{k}e^{ikn\alpha}\lambda_{k}=e^{inx}\widehat{\varphi\circ\tau^{n}}$

Where the last equality is a standard property of Fourier coefficients. ∎

###  2.5. Ideal functions

Given an intermediate algebra $\mathcal{A}\in\mathfrak{A}$, with any function
$f\in C(X)$ we associate a closed ideal of $C^{*}_{r}({\mathbb{Z}})$

$\displaystyle I_{\mathcal{A}}(f)$ $\displaystyle=$ $\displaystyle\\{\eta\in
C^{*}_{r}({\mathbb{Z}})\ |\
f\eta\in\mathcal{A}\\}\in\operatorname{\mathcal{I}}.$

Recall that $\mathcal{I}$ was defined in the introduction as the set of all
closed ideals in $C^{*}_{r}({\mathbb{Z}})$.

######  Definition 2.6.

Given an intermediate algebra
$C^{*}_{r}({\mathbb{Z}})<\mathcal{A}<\mathcal{C}$ we define the ideal function
$\Psi(\mathcal{A})=\mathfrak{c}_{\mathcal{A}}:{\mathbb{Z}}\rightarrow\operatorname{\mathcal{I}}$
associated with $\mathcal{A}$ by

$\mathfrak{c}_{\mathcal{A}}(n)=I_{\mathcal{A}}(e^{inx}).$

Note that $\mathfrak{c}_{\mathcal{A}}(0)=C^{*}_{r}({\mathbb{Z}})$ since by
assumption $C^{*}_{r}({\mathbb{Z}})<\mathcal{A}$, so that
$\mathfrak{c}_{\mathcal{A}}\in\mathfrak{D}$. The support of
$\mathfrak{c}_{\mathcal{A}}$ is the set

$\operatorname{Supp}(\mathfrak{c}_{\mathcal{A}})=\\{n\in\mathbb{Z}\ |\
\mathfrak{c}_{\mathcal{A}}(n)\neq(0)\\}=\\{n\in\mathbb{Z}\ |\
\ddot{\mathfrak{c}}(n)\not=S^{1}\\}.$

and we say that $\mathfrak{c}_{\mathcal{A}}$ is trivial if
$\operatorname{Supp}(\mathfrak{c}_{\mathcal{A}})=\\{0\\}$.

It is unclear that $\mathfrak{c}_{\mathcal{A}}$ should be nontrivial for a
given $C^{*}_{r}({\mathbb{Z}})\neq\mathcal{A}\in\mathfrak{A}$. Still, this
turns out to be true, and our main Theorem 1.2 completely classifies
intermediate algebras in terms of their ideal functions. With this in mind, we
study the basic properties of an ideal function that comes from an algebra.
The basic Lemma 2.5 is our main tool.

######  Proposition 2.7.

Let $\mathcal{A}\in\mathfrak{A}$ and $\mathfrak{c}=\Psi(\mathcal{A})$. The
following properties are satisfied:

  * $\mathtt{Cif}1$

$\ddot{\mathfrak{c}}(-n)=\tau^{n}\ddot{\mathfrak{c}}(n),\ \forall
n\in{\mathbb{Z}}$,

  * $\mathtt{Cif}2$

$\ddot{\mathfrak{c}}(m+n)\subset\tau^{-m}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m)$
for every $m,n\in\operatorname{Supp}(\mathfrak{c})$.

###### Proof.

Let $\varphi,\psi\in C(S^{1})$ be functions with
$\mathcal{Z}(\varphi):=\\{t\in S^{1}\ |\
\varphi(t)=0\\}=\ddot{\mathfrak{c}}(n)$ and
$\mathcal{Z}(\psi)=\ddot{\mathfrak{c}}(m)$. Namely $\hat{\varphi},\hat{\psi}$
are generators of $\mathfrak{c}(n),\mathfrak{c}(m)$ respectively. Now
$\mathcal{Z}((\varphi\circ\tau^{m})\psi)=\tau^{-m}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m)$
and $\mathtt{Cif}2$ follows directly from equation (3) of Lemma 2.5. Similarly
applying Equation (2) of the same lemma, to both $\pm n$, yields property
$\mathtt{Cif}1$. ∎

######  Definition 2.8.

closedidealfunction An ideal function $\mathfrak{c}\in\mathfrak{D}$ is called
closed if it satisfies properties $\mathtt{Cif}1,\mathtt{Cif}2$. We denote the
collection of closed ideal functions by $\mathfrak{C}\subset\mathfrak{D}$. As
noted in the introduction, $\mathfrak{C}\subset\mathfrak{D}$ is not a
sublattice, as it does not respect the join operation. We will elaborate on
this later.

######  Note 2.9.

Condition $\mathtt{Cif}2$ can be rephrased more symmetrically:

$\ddot{\mathfrak{c}}(m+n)\subset\left(\tau^{-m}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m)\right)\cap\left(\tau^{-n}\ddot{\mathfrak{c}}(m)\cup\ddot{\mathfrak{c}}(n)\right),\
\forall m,n\in\operatorname{Supp}(\mathfrak{c})$

######  Note 2.10.

Condition $\mathtt{Cif}1$ implies that every closed ideal function is
completely determined by its values on ${\mathbb{N}}$. Thus, every function
$\mathfrak{c}:{\mathbb{N}}\rightarrow\operatorname{\mathcal{I}}$ subject to
the following three conditions admits a unique extension to a closed ideal
function defined on all of ${\mathbb{Z}}$.

  * $\mathtt{Cif}2a$

$\ddot{\mathfrak{c}}(m+n)\subset\tau^{-m}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m),\quad\forall
m,n\in{\mathbb{N}}\cap\operatorname{Supp}(\mathfrak{c})$

  * $\mathtt{Cif}2b$

$\ddot{\mathfrak{c}}(m-n)\subset\tau^{n-m}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m),\quad\forall
1\leq n<m,\quad m,n\in\operatorname{Supp}(\mathfrak{c})$

  * $\mathtt{Cif}2c$

$\ddot{\mathfrak{c}}(-n+m)\subset\tau^{n}(\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m)),\quad\forall
1\leq n<m,\quad m,n\in\operatorname{Supp}(\mathfrak{c})$

###### Proof.

We extend $\ddot{\mathfrak{c}}$ to ${\mathbb{Z}}$ by setting
$\ddot{\mathfrak{c}}(0)=\emptyset,\ddot{\mathfrak{c}}(-n)=\tau^{n}\ddot{\mathfrak{c}}(n),\forall
n\in{\mathbb{N}}$. With these definitions, condition $\mathtt{Cif}2$ holds
when $m=n$. Hence, without loss of generality, we assume that $m>n$. This
leaves us with the task of verifying $\mathtt{Cif}2$ in six different cases,
three of which are the given equations. Let us verify that, assuming
$\mathtt{Cif}2a$, $\mathtt{Cif}2b$, $\mathtt{Cif}2c$ the other three hold as
well:

$\displaystyle\ddot{\mathfrak{c}}(n-m)$ $\displaystyle=$
$\displaystyle\tau^{m-n}\ddot{\mathfrak{c}}(m-n)\subset\tau^{m-n}\left(\tau^{-m}\tau^{n}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m)\right)=\ddot{\mathfrak{c}}(n)\cup\tau^{-n}\ddot{\mathfrak{c}}(-m)$
$\displaystyle\ddot{\mathfrak{c}}(-m+n)$ $\displaystyle=$
$\displaystyle\tau^{m-n}\ddot{\mathfrak{c}}(-n+m)\subset\tau^{m-n}\tau^{n}\left(\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m)\right)=\tau^{m}\left(\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m)\right)$
$\displaystyle\ddot{\mathfrak{c}}(-n-m)$ $\displaystyle=$
$\displaystyle\tau^{m+n}\ddot{\mathfrak{c}}(m+n)\subset\tau^{m+n}\left(\tau^{-m}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(m)\right)=\ddot{\mathfrak{c}}(-n)\cup\tau^{n}\ddot{\mathfrak{c}}(-m)$

Which is exactly what we had to show. ∎

######  Example 2.11.

Given $q\in{\mathbb{N}}$ and a closed subset $P\in\operatorname{Cl}(S^{1})$.
We define a closed ideal function $\mathfrak{b}_{q,P}\in\mathfrak{C}$ by

$\ddot{\mathfrak{b}}_{q,P}(m)=\left\\{\begin{array}[]{ll}\tau^{q(1-n)}P\cup\ldots\cup\tau^{-q}P\cup
P&m=nq,\ n>0\\\ \emptyset&m=0\\\
\tau^{q}P\cup\tau^{2q}P\cup\ldots\cup\tau^{nq}P&m=-nq,\ n>0\\\
S^{1}&q\not{\mid}m\end{array}\right.$

As a matter of convention, we also define
$\mathfrak{b}_{0,P}(m)=C^{*}_{r}({\mathbb{Z}})$ for $m=0$ and $(0)$ otherwise.
We refer to this as the basic ideal function with the data $(q,P)$. It is
easiest to verify this is an abstract ideal function using the criterion given
in Note 2.10. In fact, once you verify the case of $q=1$, the general case
follows by considering the subalgebra
$\mathcal{A}<\mathcal{C}^{q}<\mathcal{C}$. If $J\lhd C^{*}_{r}({\mathbb{Z}})$
is the closed ideal associated with the closed subset
$P\in\operatorname{Cl}(S^{1})$, we also denote
$\mathfrak{b}_{q,J}=\mathfrak{b}_{q,P}$.

Directly from the definition of a closed ideal function, it follows that
$\mathfrak{b}_{q,J}$ is minimal among all closed ideal functions with
$J<\mathfrak{c}(q)$. Alternatively, this can be verified using Note 2.10.

In particular for every $\mathfrak{c}\in\mathfrak{C}$ and every
$n\in{\mathbb{N}}$ we have
$\mathfrak{b}_{n,\mathfrak{c}(n)}\preceq\mathfrak{c}$. Taking the join over
successive values of $n$ yields better and better approximations for
$\mathfrak{c}$ from below: Thus, we define:

######  Definition 2.12.

For $\mathfrak{c}\in\mathfrak{C}$ we define the $n^{\mathrm{th}}$
approximation of $\mathfrak{c}$ from below by

$\mathfrak{c}^{n}=\bigvee_{\begin{subarray}{c}m\in\operatorname{Supp}{\mathfrak{c}}\\\
0\leq m\leq
n\end{subarray}}\mathfrak{b}_{m,\mathfrak{c}(m)}=\bigvee_{\begin{subarray}{c}m\in\operatorname{Crit}{\mathfrak{c}}\\\
0\leq m\leq n\end{subarray}}\mathfrak{b}_{m,\mathfrak{c}(m)},$

where $\text{Crit}(c)$ is defined below shortly. When
$\mathfrak{c}(m+1)=\mathfrak{c}^{m}(m+1)$, then
$\mathfrak{b}_{m,\mathfrak{c}(m)}$ can be omitted from the above join without
changing anything. We thus define the critical points for $\mathfrak{c}$ by

$\operatorname{Crit}({\mathfrak{c}})=\\{m\in\operatorname{Supp}(\mathfrak{c})\
|m\geq 0,\ \mathfrak{c}^{m}(m+1)\lneqq\mathfrak{c}(m+1)\\}.$

This explains the notation in the rightmost expression of the above
definition.

######  Theorem 2.13.

(The canonical decomposition theorem) For every $\mathfrak{c}\in\mathfrak{C}$
we have

$\mathfrak{c}=\bigvee_{n\in\operatorname{Crit}(\mathfrak{c})}\mathfrak{b}_{n,\mathfrak{c}(n)}=\bigvee_{n\in\operatorname{Crit}(\mathfrak{c})}\mathfrak{c}^{n}.$

We refer to this join as the Canonical generation of $\mathfrak{c}$.

###### Proof.

The right-hand side is dominated by the left-hand side. Conversely, for every
$n$, we have $\mathfrak{c}(n)\leq\mathfrak{b}_{n,c(n)}(n)$, which accounts for
the converse inclusion. ∎

######  Example 2.14.

Using basic ideal functions, it is easy to demonstrate how the join of two
closed ideal functions need not be closed. Indeed

$(\ddot{bc}_{1,p}\overline{\vee}\ddot{bc}_{1,\tau
p})(n)=\left\\{\begin{array}[]{ll}\emptyset&\left\lvert n\right\rvert\leq 1\\\
\\{p,\tau^{-1}p,\ldots,\tau^{-n+2}p\\}&n\geq 2\\\
\\{\tau^{2}p,\tau^{3}p,\ldots,\tau^{n}p\\}&n\leq-2\end{array}\right.$

which is not a closed ideal function. In fact one easily checks that the
condition $\ddot{\mathfrak{c}}(1)=\emptyset$ immediately forces
$\ddot{\mathfrak{c}}(n)=\emptyset$ for any closed ideal function.

######  Example 2.15.

For a fixed $q\in{\mathbb{N}}$ note that

$\mathfrak{b}_{q,C^{*}_{r}({\mathbb{Z}})}(m)=\left\\{\begin{array}[]{ll}C^{*}_{r}({\mathbb{Z}})&m=nq,\\\
(0)&q\not{\mid}m\end{array}\right.$

is exactly the ideal function coming from the “dynamical” intermediate algebra
$\mathcal{C}^{q}=C(Y)\rtimes_{r}C^{*}_{r}({\mathbb{Z}})$. Namely the one
coming from the $q$-fold factor $X\stackrel{{\scriptstyle\times
q}}{{\longrightarrow}}Y$ of the circle. Note that by Suzuki’s theorem [Suz20]
mentioned at the beginning of the introduction, any larger ideal function is
of the form $\mathfrak{b}_{r,C^{*}_{r}({\mathbb{Z}})}$ for some $r|q$.

###  2.6. The algebra $\mathcal{A}_{\mathfrak{c}}$

We now turn to define a functor in the other direction
$\Psi:\mathfrak{D}\rightarrow\mathfrak{A}$.

######  Definition 2.16.

For $\mathfrak{c}\in\mathfrak{D}$ we define

$\mathcal{A}_{\mathfrak{c}}^{\prime}=\left\langle e^{inx}\eta_{n}\ |\
n\in{\mathbb{N}},\eta_{n}\in\mathfrak{c}(n)\right\rangle,$

As the abstract $*$-algebra, generated by these elements, and set
$\Psi(\mathfrak{c})=\mathcal{A}_{\mathfrak{c}}=\overline{\mathcal{A}_{\mathfrak{c}}^{\prime}}$
to be its closure.

For $\mathfrak{c}\in\mathfrak{C}$, the algebra
$\mathcal{A}^{\prime}_{\mathfrak{c}}$ assumes a much more explicit form. In
fact it follows directly from the conditions $\mathtt{Cif}1,\mathtt{Cif}2$
that

$\mathcal{A}_{\mathfrak{c}}^{\prime}=\left\\{\sum_{n=-N}^{N}e^{inx}\eta_{n}\
|\ N\in{\mathbb{N}},\eta_{n}\in\mathfrak{c}(n)\right\\}.$ (4)

namely, the collection of all such generalized trigonometric polynomials is
closed under $*$ and multiplication. For this reason, it is much easier to
work with closed ideal functions $\mathfrak{c}\in\mathfrak{C}$. Moreover,
limiting our attention only to closed ideal functions does not limit the
generality as our main theorem asserts in particular that
$\Psi(\mathfrak{C})=\Psi(\mathfrak{D})$.

######  Proposition 2.17.

For $\mathfrak{c}\in\mathfrak{C}$,
$\mathcal{A}_{\mathfrak{c}}=\Psi(\mathfrak{c})$ is an intermediate
$C^{*}$-subalgebra
$C^{*}_{r}({\mathbb{Z}})<\mathcal{A}_{\mathfrak{c}}<\mathcal{C}$. Moreover,
the following conditions are equivalent:

  1. (1)

$\mathfrak{c}(q)=C^{*}_{r}({\mathbb{Z}})$ for some $0\neq q\in{\mathbb{Z}}$,

  2. (2)

$\mathcal{A}_{\mathfrak{c}}=\mathcal{C}^{q}$ comes from a dynamical factor
$(X,T)\stackrel{{\scriptstyle\times q}}{{\longrightarrow}}(X,T^{q})$,

  3. (3)

$\mathcal{A}_{\mathfrak{c}}=\mathcal{B}\rtimes_{r}{\mathbb{Z}}$ for some
subalgebra $\mathcal{B}$ of $C(X)$,

  4. (4)

$\mathcal{A}_{\mathfrak{c}}\cap C(X)\neq{\mathbb{C}}$,

###### Proof.

$\mathcal{A}_{\mathfrak{c}}=\Psi(\mathfrak{c})$ is by definition a closed
$*$-algebra and since $\mathfrak{c}(0)=C^{*}_{r}({\mathbb{Z}})$, we have
$C^{*}_{r}({\mathbb{Z}})<\mathcal{A}_{\mathfrak{c}}$.

The implications (2) $\Rightarrow$ (3) $\Rightarrow$ (4) are all obvious.

To prove (1) $\Rightarrow$ (2) note that if
$\mathfrak{c}(q)=C^{*}_{r}({\mathbb{Z}})$ for some $q\neq 0$ then
$\mathfrak{b}_{q,C^{*}_{r}({\mathbb{Z}})}\preceq\mathfrak{c}$. By Suzuki’s
theorem (see Example 2.15), we know that $\mathfrak{c}$ is necessarily an
intermediate dynamical algebra of the form $\mathcal{C}^{r}$ for some $r|q$.
If $q$ were the minimum positive value such that
$\mathfrak{c}(q)=C^{*}_{r}({\mathbb{Z}})$, then $q=r$.

To show that (4) $\Rightarrow$ (1), let $f\in\mathcal{A}_{\mathfrak{c}}\cap
C(X)$ be a non-constant function, with a non-trivial Fourier coefficient
$\mu(fe^{-iqx})\neq 0$ for some $0\neq q\in\mathbb{Z}$. Since
$f\in\mathcal{A}_{\mathfrak{c}}$, we can find an element
$\sum_{n=-N}^{N}e^{inx}\eta_{n}\in\mathcal{A}_{\mathfrak{c}}^{\prime}$ such
that

$\left\|f-\sum_{n=-N}^{N}e^{inx}\eta_{n}\right\|<\epsilon.$

Where the precise value of $\epsilon$ will be determined later. By possibly
adding zero coefficients, we may assume that $q\in\\{-N,-N+1,\ldots,N-1,N\\}$.
Multiplying the expression inside the norm, from the left by $e^{-iqx}$ and
applying the conditional expectation $\mathbb{E}_{\mu}$, we obtain:

$\left\|\mu(fe^{-iqx})-\eta_{q}\right\|=\left\|\mu(fe^{-iqx})-\mathbb{E}_{\mu}\left(\sum_{n=-N}^{N}e^{i(n-q)x}\eta_{n}\right)\right\|<\epsilon$

Where we used the fact that $\mathbb{E}_{\mu}\left(\sum_{n\neq
q}e^{i(n-q)x}\eta_{n}\right)=0.$ Dividing out by $\mu(fe^{-iqx})$ we obtain

$\left\|1-\frac{\eta_{q}}{\mu(fe^{-iqx})}\right\|<\frac{\epsilon}{\left\lvert\mu(fe^{-iqx})\right\rvert}$

But if $\epsilon<\left\lvert\mu(fe^{-iqx})\right\rvert$ this latter expression
guarantees that $\eta_{q}$ is an invertible (and in particular non-zero)
element in the algebra. And since by definition $\eta_{q}\in\mathfrak{c}(q)$
we conclude that $\mathfrak{c}(q)=C^{*}_{r}({\mathbb{Z}})$ proving (1). ∎

###  2.7. Basic Subalgebras

We denote the algebra associated with a basic ideal function
$\mathfrak{b}_{q,P}$ by $\mathcal{A}_{q,P}$. It is easy to find a generator
for this subalgebra.

######  Proposition 2.18.

If $\varphi\in C(S^{1})$ is any function with $\mathcal{Z}\varphi:=\\{t\in
S^{1}\ |\ \varphi(t)=0\\}=P$, namely any generator for the ideal of functions
vanishing on $P$, then $\mathcal{A}_{q,P}=\overline{\langle
C^{*}_{r}({\mathbb{Z}}),e^{iqx}\hat{\varphi}\rangle}$, the smallest closed
subalgebra of $\mathcal{C}$ containing the element $a=e^{iqx}\hat{\varphi}$
and $C^{*}_{r}({\mathbb{Z}})$.

###### Proof.

Set $\mathcal{B}=\overline{\langle
C^{*}_{r}({\mathbb{Z}}),e^{iqx}\hat{\varphi}\rangle}$. By definition
$C^{*}_{r}({\mathbb{Z}}),e^{iqx}\hat{\varphi}$ are both contained in
$\mathcal{A}_{q,P}$ and hence $\mathcal{B}<\mathcal{A}_{q,P}$. Conversely for
any $m\in{\mathbb{N}}$ applying successively Equation $(3)$ from Lemma 2.5 $m$
times we obtain

$a^{m}=e^{imqx}\left[(\varphi\circ\tau^{(m-1)q})(\varphi\circ\tau^{(m-2)q})\ldots\varphi\right]^{\wedge}.$

If we denote the function inside the square brackets by $\varphi(m)$ we see
immediately that
$\mathcal{Z}\varphi(m)=P\cup\tau^{-q}P\cup\ldots\cup\tau^{-q(m-1)}P={\ddot{{\mathfrak{b}}}}_{q,P}(m)$.
And consequently
$e^{imqx}\widehat{\varphi(m)}C^{*}_{r}({\mathbb{Z}})=e^{imqx}\mathfrak{b}_{q,P}(mq)\subset\mathcal{B}$.
Similarly, using Equation $(2)$ from the same lemma, we obtain a similar
inclusion for the negative values of $m$. Hence,
$\mathcal{A}_{q,P}^{\prime}\subset\mathcal{B}$, and we get the desired
inclusion by passing to the closure. ∎

###  2.8. Ian Putnam’s $Y$-orbit breaking subalgebras

Following [GKPT18], we recall the following definition:

######  Definition 2.19.

Let $X$ be a compact metric space and $T:X\to X$ be a homeomorphism. Consider
the transformation group $C^{*}$-algebra
$\mathcal{C}=C(X)\rtimes_{r}{\mathbb{Z}}$. For a closed subset $Y\subset X$,
we define the $C^{*}$-subalgebra $\mathcal{C}_{Y}$ to be the
$C^{*}$-subalgebra of $\mathcal{C}$ generated by $C(X)$ and
$\lambda_{1}C_{0}(X\setminus Y)$. We say that $\mathcal{C}_{Y}$ is the
$Y$-orbit breaking subalgebra of $\mathcal{C}$.

Applying the duality involution interchanging the unitary operators
$U=M_{e^{it}}$ with $V=\lambda_{1}$ and $T=R_{\alpha}$ with
$\tau=R_{-\alpha}$, as in of Remark 2.1 above, we can interpret Putnam’s
$Y$-orbit breaking subalgebras as follows:

######  Definition 2.20.

Let $X$ be a compact metric space and $T:X\to X$ be a homeomorphism. Consider
the transformation group $C^{*}$-algebra
$\mathcal{C}=C(X)\rtimes_{r}{\mathbb{Z}}$. For a closed subset $P\subset
S^{1}$, we define the $C^{*}$-subalgebra $\mathcal{C}_{P}$ to be the
$C^{*}$-subalgebra of $\mathcal{C}$ generated by $C(S^{1})$ and
$e^{ix}C_{0}(S^{1}\setminus P)$.

Now comparing this definition with our definition of the basic subalgebra
$\mathcal{A}_{1,P}$, we conclude that $\mathcal{C}_{P}=\mathcal{A}_{1,P}$, so
that in the particular case of the irrational crossed product $C^{*}$-algebra,
the basic algebra $\mathcal{A}_{1,P}$ and Putnam’s $Y$-orbit breaking
subalgebra is the same object.

###  2.9. Ideals in intermediate subalgebras

Let $\mathcal{A}_{\mathfrak{c}}=\Psi(\mathfrak{c})\in\mathfrak{A}$, and
$J\lhd\mathcal{A}_{\mathfrak{c}}$ be a closed two-sided ideal. Just like we
did in the case of algebras, we can associate with $J$ an ideal valued
function
$\Phi_{\mathfrak{c}}(J)=\mathfrak{j}:{\mathbb{Z}}\rightarrow\mathcal{I}$ by
setting

$\mathfrak{j}(n)=\\{\eta\in\mathfrak{c}(n)\ |\ e^{inx}\eta\in J\\}.$

A calculation analogous to the one carried out in Lemma 2.5 and Proposition
2.7 yields the following properties for this function.

######  Definition 2.21.

Given $\mathfrak{c}\in\mathfrak{C}$ a function
$\mathfrak{j}:{\mathbb{Z}}\rightarrow\mathcal{I}$ will be called a
$\mathfrak{c}$-ideal function if
$\ddot{\mathfrak{j}}(n)\supset\ddot{\mathfrak{c}}(n),\forall
n\in{\mathbb{Z}}$. The support of $\mathfrak{j}$ is the set

$\operatorname{Supp}(\mathfrak{j})=\\{n\in\mathbb{Z}\ |\
\mathfrak{j}(n)\neq(0)\\}=\\{n\in\mathbb{Z}\ |\
\ddot{\mathfrak{j}}(n)\not=S^{1}\\}.$

A $\mathfrak{c}$-ideal function will be called closed if in addition it
satisfies the following two conditions:

  * $\mathtt{Cif}1(\mathfrak{c})$

$\ddot{\mathfrak{j}}(-n)=\tau^{n}\ddot{\mathfrak{j}}(n),\ \forall
n\in{\mathbb{Z}}$,

  * $\mathtt{Cif}2(\mathfrak{c})$

$\ddot{\mathfrak{j}}(m+n)\subset\tau^{-m}\ddot{\mathfrak{j}}(n)\cup\ddot{\mathfrak{c}}(m)$
for every
$m\in\operatorname{Supp}(\mathfrak{c}),n\in\operatorname{Supp}(\mathfrak{j})$.

######  Remark 2.22.

Note that the second condition above is stated in a highly non-symmetric form.
A longer but more explicit form, in the case where
$m,n\in\operatorname{Supp}(\mathfrak{j})$ would read:

$\displaystyle\ddot{\mathfrak{j}}(m+n)$
$\displaystyle\subset\left(\tau^{-m}\ddot{\mathfrak{j}}(n)\cup\ddot{\mathfrak{c}}(m)\right)\cap\left(\tau^{-n}\ddot{\mathfrak{j}}(m)\cup\ddot{\mathfrak{c}}(n)\right)\cap\left(\tau^{-m}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{j}}(m)\right)\cap\left(\tau^{-n}\ddot{\mathfrak{c}}(m)\cup\ddot{\mathfrak{j}}(n)\right)$

######  Remark 2.23.

Similar to the case of ideal functions, $\mathfrak{c}$-ideal functions are
completely determined by their values on
${\mathbb{N}}_{0}={\mathbb{N}}\cup\\{0\\}$. For a function
$\ddot{\mathfrak{j}}:{\mathbb{N}}_{0}\rightarrow\operatorname{Cl}(S^{1})$
satisfying $\ddot{\mathfrak{j}}(n)\supset\ddot{\mathfrak{c}}(n),\forall
n\in{\mathbb{N}}$ it is enough to verify the following conditions for
$m,n\in{\mathbb{N}}_{0}$ with
$m\in\operatorname{Supp}(\mathfrak{c}),n\in\operatorname{Supp}(\mathfrak{j})$
in order to guarantee that the unique extension to ${\mathbb{Z}}$ forms a
legal closed $\mathfrak{c}$-ideal function:

$\displaystyle\ddot{\mathfrak{j}}(m+n)$ $\displaystyle\subset$
$\displaystyle\left[\tau^{-m}\ddot{\mathfrak{j}}(n)\cup\ddot{\mathfrak{c}}(m)\right]\cap\left[\tau^{-n}\ddot{\mathfrak{c}}(m)\cup\ddot{\mathfrak{j}}(n)\right],$
$\displaystyle\ddot{\mathfrak{j}}(m-n)$ $\displaystyle\subset$
$\displaystyle\left[\tau^{n-m}\ddot{\mathfrak{j}}(n)\cup\ddot{\mathfrak{c}}(m)\right]\cap\left[\tau^{n}\left(\ddot{\mathfrak{c}}(m)\cup\ddot{\mathfrak{j}}(n)\right)\right],\forall
0\leq n\leq m,$ $\displaystyle\ddot{\mathfrak{j}}(n-m)$ $\displaystyle\subset$
$\displaystyle\left[\tau^{m-n}\ddot{\mathfrak{c}}(m)\cup\ddot{\mathfrak{j}}(n)\right]\cap\left[\tau^{m}\left(\ddot{\mathfrak{c}}(m)\cup\ddot{\mathfrak{j}}(n)\right)\right],\forall
0\leq m\leq n,$

###### Proof.

We extend the function $\mathfrak{j}$ to ${\mathbb{Z}}$ using
$\mathtt{Cif}1(\mathfrak{c})$. We must verify condition
$\mathtt{Cif}2(\mathfrak{c})$ in the three missing cases. In all cases we
assume $m,n\in{\mathbb{N}}_{0}$ with
$m\in\operatorname{Supp}(\mathfrak{c}),n\in\operatorname{Supp}(\mathfrak{j})$.

$\displaystyle\ddot{\mathfrak{j}}(-m-n)$
$\displaystyle=\tau^{m+n}\ddot{\mathfrak{j}}(m+n)\subset\tau^{m+n}\left(\tau^{-n}\ddot{\mathfrak{c}}(m)\cup\ddot{\mathfrak{j}}(n)\right)=\tau^{m}\ddot{\mathfrak{j}}(-n)\cup\ddot{\mathfrak{c}}(-m)$
$\displaystyle\mathfrak{j}(-m+n)$
$\displaystyle=\tau^{m-n}\mathfrak{j}(m-n)\subset\tau^{m-n}\tau^{n}(\ddot{\mathfrak{c}}(m)\cup\ddot{\mathfrak{j}}(n)=\tau^{m}(\ddot{\mathfrak{c}}(m)\cup\ddot{\mathfrak{j}}(n),$
$\displaystyle 0\leq n\leq m$ $\displaystyle\ddot{\mathfrak{j}}(m-n)$
$\displaystyle=\tau^{n-m}\mathfrak{j}(n-m)\subset\tau^{n-m}(\tau^{m-n}\ddot{\mathfrak{c}}(m)\cup\ddot{\mathfrak{j}}(n))=\tau^{-m}\mathfrak{j}(-n)\cup\ddot{\mathfrak{c}}(m),$
$\displaystyle 0\leq m\leq n$

As required. ∎

Conversely, with any $\mathfrak{c}$-ideal function, we can associate an ideal.

######  Definition 2.24.

Given any $\mathfrak{c}$-ideal function $\mathfrak{j}$ as above, we define

$J_{\mathfrak{j}}=\Psi_{\mathfrak{c}}(\mathfrak{j})=\overline{\left\langle\left.e^{inx}\eta_{n}\right|n\in{\mathbb{Z}},\
\eta_{n}\in\mathfrak{j}(n)\right\rangle}\lhd\Psi(\mathfrak{c}).$

######  Proposition 2.25 (Properties of $\mathfrak{c}$-ideal functions).

Let $J=\Psi_{\mathfrak{c}}(\mathfrak{j})$ for some closed $\mathfrak{c}$-ideal
function as above. Then

  * •

$\mathfrak{j}(0)=\mathbb{E}_{\mu}(J)=J\cap C^{*}({\mathbb{Z}})$.

  * •

$J=\mathcal{A}_{\mathfrak{c}}$ if and only if
$\mathfrak{j}(0)=C^{*}_{r}({\mathbb{Z}})$, if and only if
$\ddot{\mathfrak{j}}(0)=\emptyset$,

  * •

$J=(0)$ if and only if $J\cap C^{*}({\mathbb{Z}})=(0)$, if and only if
$\ddot{\mathfrak{j}}(0)=S^{1}$.

  * •

$\ddot{\mathfrak{j}}(0)\subset\ddot{\mathfrak{j}}(n),\forall
n\in{\mathbb{Z}}$.

In particular $J$ is nontrivial in the sense that $(0)\lneqq
J\lneqq\mathcal{A}$, if and only if
$\emptyset\subsetneq\ddot{\mathfrak{j}}(0)\subsetneq S^{1}$.

###### Proof.

The first three items are clear; for the last item, it is enough to consider
$n\in\operatorname{Supp}(\mathfrak{j})$. Using $\mathtt{Cif}2(\mathfrak{c})$
we obtain for such $n$

$\ddot{\mathfrak{j}}(0)=\ddot{\mathfrak{j}}(n+(-n))\subset\tau^{-n}\tau^{n}\ddot{\mathfrak{j}}(n)\cup\ddot{\mathfrak{c}}(n)=\ddot{\mathfrak{j}}(n)\cup\ddot{\mathfrak{c}}(n)=\ddot{\mathfrak{j}}(n).$

∎

##  3\. The main theorem

###  3.1. Extraction of generalized Fourier coefficients.

Let $V_{n}=\operatorname{Span}\\{e^{imx}\ |\ \left\lvert m\right\rvert\leq
n\\}<C(X)$ be the space of trigonometric polynomials of degree at most $n$.
With respect to the standard $L^{2}$-metric, let $Q_{n}:C(X)\rightarrow V_{n}$
be the orthogonal projection and $V_{n}^{\perp}=\ker(Q_{n})$. The projection
$Q_{n}$ is explicitly given by the formula
$Q_{n}f=\sum_{m=-n}^{n}\overbracket{f}(m)e^{imx}$. It is easy to verify that
$Q_{n}$ is a $T$-equivariant map.

It turns out that $Q_{n}$ can be extended to the whole $\mathcal{C}$. By abuse
of notation, we denote the extended operator by the same name.

######  Proposition 3.1.

extensionofQ The map $Q_{n}$ extends naturally to a bounded linear map,
$Q_{n}:\mathcal{C}\rightarrow\mathcal{C}$, that restricts to the identity on
$C^{*}_{r}({\mathbb{Z}})$.

###### Proof.

Clearly $Q_{n}\upharpoonright_{C(X)}:C(X)\to C(X)$ is a bounded linear
operator. Using [Pis03, Proposition 1.10], we see that $Q_{n}$ is completely
bounded. Now, it follows from [RSW89, Theorem 3.5] that $Q_{n}$ extends
uniquely to a completely bounded map, denoted again by $Q_{n}$ from
$\mathcal{C}\to\mathcal{C}$ such that for every $N$

$Q_{n}\left(\sum_{i=-N}^{N}f_{i}\lambda_{i}\right)\mapsto\sum_{i=-N}^{N}Q_{n}(f_{i})\lambda_{i}.$

∎

######  Definition 3.2.

For every element $a\in\mathcal{C}$, let $q(a)$ be the smallest
$q\in\mathbb{N}$ such that $Q_{q(a)}(a)\not\in C^{*}_{r}({\mathbb{Z}})$, with
the convention that $q(a)=\infty$ whenever $a\in C^{*}_{r}({\mathbb{Z}})$ and
$Q_{\infty}=\operatorname{id}$.

Explicitly for non-constant $f\in C(X)$ we have $f\in V_{q(f)-1}^{\perp}$ and

$f=Q_{q(f)}(f)+g=e^{-iq(f)x}\overbracket{f}(-q(f))+\overbracket{f}(0)+e^{iq(f)x}\overbracket{f}(q(f))+g$
(5)

with $g\in V_{q(f)}^{\perp}$. A similar decomposition holds for every
$a\in\mathcal{C}$. Using Proposition 3.1 above, Equation (5) can be applied to
each Fourier coefficient $f_{i}=\mathbb{E}(a\lambda_{i}^{*})$ of $a$
separately and $q(a)$ is just the minimum over all $q(f_{i})$. The following
two propositions are crucial to our proof.

######  Proposition 3.3.

prop:Q Let $\mathcal{A}\in\mathfrak{A}$. Then for every
$a\in\mathcal{A}\setminus C_{r}^{*}(\mathbb{Z})$ with $q=q(a)$, we have:

$Q_{q(a)}(a)=e^{-iqx}\overbracket{a}(-q)+\overbracket{a}(0)+e^{iqx}\overbracket{a}(q)\in\mathcal{A}.$

If in addition $a\in J\lhd\mathcal{A}$, with $J$ a closed two-sided ideal,
then $Q_{q(a)}(a)\in J$.

######  Lemma 3.4.

operatorwithnormone Let $N\subset{\mathbb{N}}$ be a finite set with each $r\in
N$ satisfying $r>q$. Then, for every $\epsilon>0$ there exist
$n\in{\mathbb{N}}$ and numbers $\\{k_{i}\in{\mathbb{Z}}\ |\ 1\leq i\leq n\\}$
and $\\{\beta_{i}\in{\mathbb{C}}\ |\ 1\leq i\leq n\\}$, with
$\sum_{i=1}^{n}\left\lvert\beta_{i}\right\rvert=1$ such that the operator

$L_{N,\epsilon}(a)=\sum_{i=1}^{n}\beta_{i}\lambda_{k_{i}}a\lambda_{k_{i}}^{*},$

has norm at most one and satisfies
$L_{N,\epsilon}(e^{iqx})\approx_{\epsilon}e^{iqx}$ and
$L_{N,\epsilon}(e^{irx})\approx_{\epsilon}0$ for every $r\in N$. Moreover, the
operator $L_{N,\epsilon}$ can also be made to satisfy
$L_{N,\epsilon}(e^{-iqx})\approx_{\epsilon}e^{-iqx}$, or
$L_{N,\epsilon}(e^{-iqx})\approx_{\epsilon}0$.

###### Proof.

It is enough to prove the claim for $N=\\{r\\}$. Indeed, if this were the
case, then given any finite subset $N=\\{r_{1},r_{2},\ldots,r_{n}\\}$, a
standard induction argument shows that $L_{N,\epsilon}$ can be taken to be
$L_{r_{n},\frac{\epsilon}{n}}\circ
L_{r_{n-1},\frac{\epsilon}{n}}\circ\dots\circ L_{r_{1},\frac{\epsilon}{n}}$.
We now prove the claim for $N=\\{r\\}$. Consider the subgroup $H_{1}\leq
S^{1}$ of $r$-th roots of unity. Under the map $x\mapsto x^{q}$, the image of
$H_{1}$ forms a smaller non-trivial subgroup $H_{2}\leq S^{1}$ since $q<r$.
Given any root of unity $w_{j}\in H_{2}$, we can choose some $k_{j}$ with the
property that $e^{ik_{j}\alpha}$ is approximately an $r$-th root of unity such
that $(e^{ik_{j}\alpha})^{q}\approx_{\epsilon}w_{j}$.
Note that

$\lambda_{k}e^{iqx}\lambda_{k}^{*}=(e^{ik\alpha})^{q}e^{iqx}\text{ and
}\lambda_{k}e^{irx}\lambda_{k}^{*}=(e^{ik\alpha})^{r}e^{irx}$

Therefore, we see that

$\lambda_{k_{j}}e^{iqx}\lambda_{k_{j}}^{*}\approx_{\epsilon}w_{j}e^{iqx}$
$\lambda_{k_{j}}e^{irx}\lambda_{k_{j}}^{*}\approx_{\epsilon}e^{irx}$

Multiplying by the scalar $\overline{w_{j}}$ on both sides, we see that

$\overline{w_{j}}\lambda_{k_{j}}e^{iqx}\lambda_{k_{j}}^{*}\approx_{\epsilon}e^{iqx}$
$\overline{w_{j}}\lambda_{k_{j}}e^{irx}\lambda_{k_{j}}^{*}\approx_{\epsilon}\overline{w_{j}}e^{irx}$

It is now easy to see that $L_{r,\epsilon}$ given by

$L_{r,\epsilon}(a)=\frac{1}{|H_{2}|}\sum_{w_{j}\in
H_{2}}\overline{w_{j}}\lambda_{k_{j}}a\lambda_{k_{j}}^{*}$

is the required map. This finishes the proof. ∎

Note that $L$ is not a positive map on the crossed product, but it is still a
completely bounded map with (CB) norm at most $1$.

###### Proof of prop:Q.

Given any $a\in\mathcal{A}$, we may assume without loss of generality that the
Fourier coefficient at zero is zero, and choose to write it as

$a=e^{-iqx}a_{-q}+e^{iqx}a_{q}+g+h,$

where $g$ is a finitely supported element with Fourier coefficients only
supported on $|r|>q$, and $h$ is some element with $\|h\|<\epsilon$. Then
choosing the operator $L$ as in operatorwithnormone, we either have

$L(a)=L(e^{-iqx}a_{-q}+e^{-iqx}a_{q})+L(g)+L(h)\approx
e^{-iqx}a_{-q}+e^{iqx}a_{q}+0+0,$

or, we have

$L(a)=L(e^{-iqx}a_{-q}+e^{-iqx}a_{q})+L(g)+L(h)\approx 0+e^{iqx}a_{q}+0+0.$

Since $L(a)\in\mathcal{A}$, at least one of $e^{-iqx}a_{-q}+e^{iqx}a_{q}$ or
$e^{iqx}a_{q}$ lies in $\mathcal{A}$, since at least one of the above
statements is true for a sequence of epsilons converging to zero. By
performing a similar argument with $a^{*}$ instead of $a$, we can also
conclude that at least one of $e^{-iqx}a_{-q}+e^{iqx}a_{q}$ or
$e^{-iqx}a_{-q}$ lies in $\mathcal{A}$. In any case, regardless of which
combination of statements is true, we may one way or another obtain that
$e^{-iqx}a_{-q}+e^{iqx}a_{q}\in\mathcal{A}$.
A similar argument works for an ideal $J\triangleleft\mathcal{A}$ since $J$ is
invariant under the conjugation action of $\mathbb{Z}$. ∎

We go one step further and extract individual generalized Fourier
coefficients.

######  Proposition 3.5.

Let $\mathcal{A}\in\mathfrak{A}$ and $a\in\mathcal{A}$. Then for every
$q\in{\mathbb{Z}}$ we have $e^{iqx}\overbracket{a}(q)\in\mathcal{A}$. If in
addition $a\in J\lhd\mathcal{A}$, with $J$ a closed two-sided ideal, then
$Q_{q}(a)\in J$.

###### Proof.

First, we claim that it is enough to prove the proposition for the special
case $q=\pm q(a)$. Indeed set $q_{0}=q(a)$, Proposition LABEL:prop:Q implies
that $a_{1}:=a-Q_{q_{0}}(a)\in\mathcal{A}$. Moreover, for
$q_{1}=q(a_{1})=q_{0}+1$ we have
$\overbracket{a_{1}}(q_{1})=\overbracket{a}(q_{1})$. Continuing inductively in
this form, we can extract all Fourier coefficients of $a$. Thus, we will
assume that $q=q(a)$. Since $C^{*}_{r}({\mathbb{Z}})<\mathcal{A}$ by
assumption we may assume that $\overbracket{a}(0)=0$. By Proposition
LABEL:prop:Q we can replace $a$ by $Q_{q}(a)$ and thus assume from now on that

$a=e^{iqx}\overbracket{a}(q)+e^{-iqx}\overbracket{a}(-q).$

Let us define the derivative of $a$ to be the element

$a^{\prime}=iqe^{iqx}\overbracket{a}(q)-iqe^{-iqx}\overbracket{a}(-q).$

We claim that $a^{\prime}\in\mathcal{A}$. Indeed, since $\alpha$ is assumed to
be irrational, we can find a sequence such that
$\lim_{j\rightarrow\infty}\overline{n_{j}\alpha}=0$ where
$\overline{n_{j}\alpha}$ denotes the representative of
$n_{j}\alpha+{\mathbb{Z}}$ in the interval $(-1/2,1/2]$. Now

$\displaystyle a^{\prime}$
$\displaystyle=\lim_{j\rightarrow\infty}\left(\frac{e^{iq(x+\overline{n_{j}\alpha})}-e^{iqx}}{\overline{n_{j}\alpha}}\overbracket{a}(q)+\frac{e^{-iq(x+\overline{n_{j}\alpha})}-e^{-iqx}}{\overline{n_{j}\alpha}}\overbracket{a}(-q)\right)$
$\displaystyle=\lim_{j\rightarrow\infty}\frac{\lambda_{n_{j}}a\lambda_{-n_{j}}-a}{\overline{n_{j}\alpha}}$

This convergence is uniform in $x$ because the exponential functions are twice
continuously differentiable. Thus $a^{\prime}\in\mathcal{A}$ as claimed.
Finally

$e^{iqx}\overbracket{a}(q)=\frac{a}{2}+\frac{a^{\prime}}{2iq}\in\mathcal{A},\qquad\qquad
e^{-iqx}\overbracket{a}(-q)=\frac{a}{2}-\frac{a^{\prime}}{2iq}\in\mathcal{A}.$

Which completes the proof of the proposition. Again, the same proof works in
the case $a\in J\lhd\mathcal{A}$. ∎

###  3.2. Proof of Theorems 1.2 and 1.4

Assume that $\Phi(\mathfrak{c})<\mathcal{A}$ for
$\mathfrak{c}\in\mathfrak{D},\mathcal{A}\in\mathfrak{A}$. By definition this
means that $e^{inx}\varphi\in\mathcal{A}$ for every
$n\in{\mathbb{Z}},\varphi\in\mathfrak{c}(n)$; which in turn shows that
$\varphi\in\Psi(\mathcal{A})(n)$. Thus $\mathfrak{c}<\Psi(\mathcal{A})$.
Conversely suppose $\mathfrak{c}<\Psi(\mathcal{A})$ and assume that
$\varphi\in\mathfrak{c}(n)<\Psi(\mathcal{A})(n)$ for some $n\in{\mathbb{Z}}$.
By definition of $\Psi(\mathcal{A})$ this means that
$e^{inx}\varphi\in\mathcal{A}$. But by definition
$\Phi(\mathfrak{c})=\overline{\langle e^{inx}\varphi\ |\
\varphi\in\mathfrak{c}(n)\rangle}$ and the desired inclusion
$\Phi(\mathfrak{c})<\mathcal{A}$ follows. This verifies (1) from the main
theorem, namely that $\Phi$ and $\Psi$ form a Galois connection. Unlike the
one in Galois theory, this is a monotone (i.e., order-preserving) Galois
connection. However, one naturally obtains an order reversing Galois
connection upon composing with the natural identification
$\mathfrak{D}\cong\ddot{\mathfrak{D}}$.

The images $\Psi(\mathfrak{A})\subset\mathfrak{D}$ and
$\Phi(\mathfrak{D})\subset\mathfrak{A}$ are now called closed ideal functions
and closed intermediate algebras, respectively. Statement (4) of the main
theorem now follows from standard results about Galois connections, which we
recall here:

######  Proposition 3.6.

For a monotone Galois connection as above, the following hold:

  1. (i)

$\mathfrak{c}\preceq\Psi\Phi\mathfrak{c},\ \forall\mathfrak{c}\in\mathfrak{D}$
and $\mathcal{A}\geq\Phi\Psi\mathcal{A},\forall\mathcal{A}\in\mathfrak{A}$.

  2. (ii)

$\Phi(\mathfrak{c}_{1})\leq\Phi(\mathfrak{c}_{2})$ whenever
$\mathfrak{c}_{1}\preceq\mathfrak{c}_{2}$ and similarly
$\Psi(\mathcal{A}_{1})\preceq\Psi(\mathcal{A}_{2})$ whenever
$\mathcal{A}_{1}\leq\mathcal{A}_{2}$.

  3. (iii)

$\Phi\Psi\Phi\mathfrak{c}=\Phi\mathfrak{c},\
\forall\mathfrak{c}\in\mathfrak{D}$ and
$\Psi\Phi\Psi\mathcal{A}=\Psi\mathcal{A},\ \forall\mathcal{A}\in\mathfrak{A}$.

  4. (iv)

$\Phi\upharpoonright_{\Psi\mathfrak{A}}:\Psi\mathfrak{A}\rightarrow\Phi\mathfrak{D}$
is an isomorphism of lattices between the closed elements on both sides. And
$\Psi$ is its inverse.

###### Proof.

Applying the Galois connection property to
$\Phi(\mathfrak{c})\leq\Phi(\mathfrak{c})$ and to
$\Psi\mathcal{A}\preceq\Psi\mathcal{A}$ yields i. Assume that
$\mathfrak{c}_{1}\preceq\mathfrak{c}_{2}$. By i we have
$\mathfrak{c}_{1}\preceq\mathfrak{c}_{2}\preceq\Psi\Phi\mathfrak{c}_{2}$, and
the defining property of the Galois connection yields
$\Phi(\mathfrak{c}_{1})\leq\Phi(\mathfrak{c}_{2})$. The dual assertion in ii
is proved similarly. Now applying i to $\Phi(\mathfrak{c})\in\mathfrak{A}$,
immediately implies that $\Phi\mathfrak{c}\geq\Phi\Psi\Phi\mathfrak{c},\
\forall\mathfrak{c}\in\mathfrak{D}$. Similarly i also implies that
$\mathfrak{c}\preceq\Psi\Phi\mathfrak{c}$, and by ii
$\Phi\mathfrak{c}\leq\Phi\Psi\Phi\mathfrak{c}$. Equality
$\Phi\mathfrak{c}=\Phi\Psi\Phi\mathfrak{c},\
\forall\mathfrak{c}\in\mathfrak{D}$ follows which is one clause in iii. The
dual clause is proved identically. The last property iv is a direct
consequence of iii. ∎

The property ii above is referred to as the monotonicity property for obvious
reasons. In our case, it is evident from the definitions of $\Phi,\Psi$, but
as we saw above, it also follows formally from the defining property of a
monotone Galois connection. We now turn to the less formal statements of the
main theorem.

Let $\mathcal{A}\in\mathfrak{A}$ be an intermediate algebra and denote
$\mathfrak{c}=\Psi\mathcal{A}$. By the above Proposition,
$\mathcal{A}\geq\Phi\mathfrak{c}$. We must establish the opposite inclusion to
prove (2). Thus let $a\in\mathcal{A}$, by Proposition 3.5 we know that
$e^{inx}\overbracket{a}(n)\in\mathcal{A},\ \forall n\in{\mathbb{Z}}$, which by
definition means that $\overbracket{a}(n)\in\mathfrak{c}(n),\ \forall
n\in{\mathbb{Z}}$. By definition of $\Phi$ this means that
$e^{inx}\overbracket{a}(n)\in\Phi(\mathfrak{c}),\ \forall n\in{\mathbb{Z}}$.
Thus $Q_{n}(a)=\sum_{k=-n}^{n}e^{ikx}\overbracket{a}(k)\in\Phi(\mathfrak{c})$
and by Fejér’s Theorem 2.3 we deduce that

$a=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}Q_{n}(a)\in\Phi(\mathfrak{c}),$

showing that the Galois connection is perfect on the algebra side, and
finishing the proof of (2).

On the other side of the Galois connection, Proposition 3.6 implies
$\mathfrak{c}\leq\Psi\Phi\mathfrak{c}$ for every
$\mathfrak{c}\in\mathfrak{D}$. To prove (3), we must show that the ideal
functions satisfying the opposite inclusion are exactly these in
$\mathfrak{C}$. From Proposition 2.7, we already know that the conditions
$\mathtt{Cif}1,\mathtt{Cif}2$, defining the class $\mathfrak{C}$, are
necessary for an ideal function to be closed. It remains to show that these
conditions are also sufficient.

Let $\mathfrak{c}\in\mathfrak{C}$, set $\mathcal{A}=\Phi\mathfrak{c}$ and
assume, towards a contradiction, that
$\eta\in\Psi\mathcal{A}(n)\setminus\mathfrak{c}(n)$ for some $n$. We may
assume that $n\in{\mathbb{N}}$ is the minimal such index. By definition
$\eta\in\Psi(\mathcal{A})(n)$ means that $e^{inx}\eta\in\mathcal{A}$.

Since $\mathfrak{c}\in\mathfrak{C}$

$\mathcal{A}_{\mathfrak{c}}^{\prime}=\left\\{\sum_{k=-K}^{K}e^{ikx}\eta_{k}\
|\ K\in{\mathbb{N}},\eta_{k}\in\mathfrak{c}(k)\right\\}.$

is an algebra, closed under the $*$-operation and by definition
$\mathcal{A}=\Phi(\mathfrak{c})$ is its closure in $\mathcal{C}$. Thus for
every $\epsilon>0$ we can find an approximation

$\left\lVert
e^{inx}\eta-\sum_{k=-K}^{K}e^{ikx}\eta_{k}\right\rVert=\left\lVert(\eta-\eta_{k})-\sum_{\begin{subarray}{c}k=-K\\\
k\neq n\end{subarray}}^{K}e^{i(k-n)x}\eta_{k}\right\rVert<\epsilon$

Applying the conditional expectation $\mathbb{E}_{\mu}$ on both sides we
obtain that

$\left\lVert\mathbb{E}_{\mu}(\eta-\eta_{k})-\mathbb{E}_{\mu}\left(\sum_{\begin{subarray}{c}k=-K\\\
k\neq n\end{subarray}}^{K}e^{i(k-n)x}\eta_{k}\right)\right\rVert<\epsilon$

Since $\eta_{k}$ falls in the multiplicative domain of $\mathbb{E}_{\mu}$ and
$\mathbb{E}_{\mu}(e^{i(k-n)x})=0$ for $k\neq n$, we obtain that
$\left\lVert\eta-\eta_{k}\right\rVert<\epsilon$. Since $\epsilon>0$ was
arbitrary and the ideal $\mathfrak{c}(n)\lhd C^{*}_{r}({\mathbb{Z}})$ is
closed, we obtain the desired contradiction. This completes the proof of
Theorem 1.2. ∎

We now turn to the proof of Theorem 1.4. It is quite similar to the proof of
Theorem 1.2, but we include all the details. Fix $\mathfrak{c}\in\mathfrak{C}$
and let $\mathcal{A}=\Phi(\mathfrak{c})$. That
$\Phi_{\mathfrak{c}}:\mathfrak{D}_{\mathfrak{c}}\rightarrow\mathfrak{J}_{\mathcal{A}}$
and
$\Psi_{\mathcal{A}}:\mathfrak{J}_{\mathcal{A}}\rightarrow\mathfrak{D}_{\mathfrak{c}}$
form a monotone Galois connection follows as before. This proves (1), as well
as (4), using Proposition 3.6.

To prove (2), namely that the connection is perfect on the ideal side, we have
to show that $J\leq\Phi_{\mathfrak{c}}(\mathfrak{j})$ for every
$J\in\mathfrak{J}_{\mathcal{A}}$ and $\mathfrak{j}=\Psi_{\mathcal{A}}(J)$.
Indeed, by Proposition 3.5, together with every $a\in J$ we have
$e^{inx}\overbracket{a}(n)\in J$ for every $n\in{\mathbb{Z}}$. Hence by the
definition of $\mathfrak{j}=\Psi_{\mathcal{A}}(J)$ we know that
$\overbracket{a}(n)\in\mathfrak{j}(n),\ \forall n\in{\mathbb{Z}}$. We conclude
using Fejér approximations:

$a=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}Q_{n}(a)\in\Phi_{\mathfrak{a}}(\mathfrak{j}).$

To prove (3) we must show that the $\mathfrak{c}$-ideal functions satisfying
$\Psi_{\mathcal{A}}\Phi_{\mathfrak{c}}\mathfrak{j}=\mathfrak{j}$ are exactly
those satisfying conditions
$\mathtt{Cif}1(\mathfrak{c}),\mathtt{Cif}2(\mathfrak{c})$ of Definition 2.21.
Set $J=\Phi_{\mathfrak{c}}(\mathfrak{j})$. In view of Lemma 2.5, the
conditions $\mathtt{Cif}1(\mathfrak{c})$ and $\mathtt{Cif}2(\mathfrak{c})$
follow directly from the respective requirements that
$(e^{inx}\hat{\varphi})^{*}\in J$ and
$e^{inx}\hat{\varphi}e^{imx}\hat{\psi}\in J$, whenever
$e^{inx}\hat{\varphi}\in J,e^{imx}\hat{\psi}\in\mathcal{A}$. Namely, they
follow from $J$ being $*$-closed right ideal. It remains to show that if
$\mathfrak{j}$ satisfies the conditions
$\mathtt{Cif}1(\mathfrak{c}),\mathtt{Cif}2(\mathfrak{c})$ then
$\Psi_{\mathcal{A}}\Phi_{\mathfrak{c}}(\mathfrak{j})\preceq\mathfrak{j}$,
where by property i of Galois connections the other inclusion is automatic for
every $\mathfrak{j}$. So let us assume by way of contradiction that
$\eta\in\Psi_{\mathcal{A}}J(n)\setminus\mathfrak{j}(n)$ for some
$n\in{\mathbb{N}}$.

By conditions $\mathtt{Cif}1(\mathfrak{c}),\mathtt{Cif}2(\mathfrak{c})$

$J^{\prime}=\left\\{\sum_{k=-K}^{K}e^{ikx}\eta_{k}\ |\
K\in{\mathbb{N}},\eta_{k}\in\mathfrak{j}(k)\right\\}.$

is a $*$-closed right ideal of $\mathcal{A}$. Hence, it is also automatically
a left ideal. $J=\Phi_{\mathfrak{c}}(\mathfrak{j})=\overline{J^{\prime}}$ is
by definition the closure. Thus for every $\epsilon>0$ we can find an
approximation

$\left\lVert
e^{inx}\eta-\sum_{k=-K}^{K}e^{ikx}\eta_{k}\right\rVert=\left\lVert(\eta-\eta_{k})-\sum_{\begin{subarray}{c}k=-K\\\
k\neq n\end{subarray}}^{K}e^{i(k-n)x}\eta_{k}\right\rVert<\epsilon$

Applying the conditional expectation $\mathbb{E}_{\mu}$ on both sides we
obtain that

$\left\lVert\mathbb{E}_{\mu}(\eta-\eta_{k})-\mathbb{E}_{\mu}\left(\sum_{\begin{subarray}{c}k=-K\\\
k\neq n\end{subarray}}^{K}e^{i(k-n)x}\eta_{k}\right)\right\rVert<\epsilon$

Since $\eta_{k}$ falls in the multiplicative domain of $\mathbb{E}_{\mu}$ and
$\mathbb{E}_{\mu}(e^{i(k-n)x})=0$ for $k\neq n$, we obtain that
$\left\lVert\eta-\eta_{k}\right\rVert<\epsilon$. Since $\epsilon>0$ was
arbitrary and the ideal $\mathfrak{j}(n)$ is closed, we obtain the desired
contradiction and conclude the proof of Theorem 1.4. ∎

##  4\. Structural results about subalgebras

###  4.1. The closed join formula

In this section, we finally establish a closed formula for the closed join of
two algebras $\mathcal{A}_{1},\mathcal{A}_{2}\in\mathcal{A}$:

######  Proposition 4.1.

Let $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ be two intermediate algebras with
associated ideal functions $\mathfrak{c}_{1},\mathfrak{c}_{2}\in\mathfrak{C}$,
so that $\mathcal{A}_{i}=\Phi(\mathfrak{c}_{i})$. Then

$\Psi\circ\Phi(\mathfrak{c}_{1}\overline{\vee}\mathfrak{c}_{2})=\Psi(\mathcal{A}_{1}\vee\mathcal{A}_{2}).$

We call this closed ideal function the closed join of $\mathfrak{c}_{1}$ and
$\mathfrak{c}_{2}$. We denote it by $\mathfrak{c}_{1}\vee\mathfrak{c}_{2}$. A
more explicit expression for the closed join
$\mathfrak{d}=\mathfrak{c}_{1}\vee\mathfrak{c}_{2}$ is given by

$\displaystyle\ddot{\mathfrak{d}}(n)=$
$\displaystyle\bigcap\tau^{-\sum_{k=2}^{s}j_{k}}\ddot{\mathfrak{c}}_{\epsilon}(j_{1})\cup\left(\tau^{-\sum_{k=3}^{s}j_{k}}\ddot{\mathfrak{c}}_{\epsilon+1}(j_{2})\right)\cup\ldots\cup\left(\tau^{-j_{s}}\ddot{\mathfrak{c}}_{\epsilon+s-2}(j_{s-1})\right)\cup\ddot{\mathfrak{c}}_{\epsilon+s-1}(j_{s}),$

where it is understood that $\ddot{\mathfrak{c}}_{h}$ is either
$\ddot{\mathfrak{c}}_{1}$ or $\ddot{\mathfrak{c}}_{2}$, depending on the
parity of $h$. The intersection is taken over all possible values $s\geq 2$,
$\epsilon\in\\{1,2\\}$ and $\\{j_{1},j_{2},\ldots,j_{s}\\}\subset{\mathbb{Z}}$
such that $\sum_{k=1}^{s}j_{k}=n$.

###### Proof.

For this let $\mathfrak{c}_{1},\mathfrak{c}_{2}\in\mathfrak{C}$ be two closed
ideal functions with corresponding intermediate algebras
$\mathcal{A}_{i}=\Phi(\mathfrak{c}_{i})$. Since
$\mathfrak{c}_{1}\overline{\vee}\mathfrak{c}_{2}>\mathfrak{c}_{i},\forall
i\in\\{1,2\\}$ we have
$\Phi(\mathfrak{c}_{1}\overline{\vee}\mathfrak{c}_{2})>\mathcal{A}_{i},\forall
i\in\\{1,2\\}$. Hence
$\Phi(\mathfrak{c}_{1}\overline{\vee}\mathfrak{c}_{2})>\mathcal{A}_{1}\vee\mathcal{A}_{2}$
so that
$\Psi\circ\Phi(\mathfrak{c}_{1}\overline{\vee}\mathfrak{c}_{2})>\Psi(\mathcal{A}_{1}\vee\mathcal{A}_{2})$.
Conversely for any $n\in{\mathbb{Z}}$ let $\varphi_{i}\in C(S^{1})$ be a non-
negative function with $\mathcal{Z}\varphi_{i}=\\{t\in S^{1}\ |\
\varphi_{i}(t)=0\\}=\ddot{\mathfrak{c}}_{i}(n)$. By definition of the algebra
$\mathcal{A}_{i}=\Phi(\mathfrak{c}_{i})$ we have
$e^{inx}\hat{\varphi}_{i}\in\mathcal{A}_{i}$. Hence
$e^{inx}(\hat{\varphi}_{1}+\hat{\varphi}_{2})\in\mathcal{A}_{1}\vee\mathcal{A}_{2}$.
But
$\mathcal{Z}(\varphi_{1}+\varphi_{2})=\ddot{\mathfrak{c}}_{1}(n)\cap\ddot{\mathfrak{c}}_{2}(n)=(\ddot{\mathfrak{c}}_{1}\overline{\vee}\ddot{\mathfrak{c}}_{2})(n)$.
Thus,
$\Phi(\mathfrak{c}_{1}\overline{\vee}\mathfrak{c}_{2})<\mathcal{A}_{1}\vee\mathcal{A}_{2}$
and applying $\Psi$ we obtain
$\Psi\circ\Phi(\mathfrak{c}_{1}\overline{\vee}\mathfrak{c}_{2})<\Psi(\mathcal{A}_{1}\vee\mathcal{A}_{2})$.
These two estimates yield the first equality of our proposition.

Now let $\ddot{\mathfrak{d}}(n)$ be the ideal function defined by the right-
hand side formula claimed in Proposition 4.1. That is

$\displaystyle\ddot{\mathfrak{d}}(n)=\bigcap
Q^{(n,s)}_{\epsilon}(j_{1},j_{2},\ldots,j_{s}),\qquad{\text{where}}$ (6)
$\displaystyle Q^{(n,s)}_{\epsilon}(j_{1},j_{2},\ldots,j_{s}):=$
$\displaystyle\qquad\tau^{-\sum_{k=2}^{s}j_{k}}\ddot{\mathfrak{c}}_{\epsilon}(j_{1})\cup\left(\tau^{-\sum_{k=3}^{s}j_{k}}\ddot{\mathfrak{c}}_{\epsilon+1}(j_{2})\right)\cup\ldots\cup\left(\tau^{-j_{s}}\ddot{\mathfrak{c}}_{\epsilon+s-2}(j_{s-1})\right)\cup\ddot{\mathfrak{c}}_{\epsilon+s-1}(j_{s})$

We claim that

$\ddot{\mathfrak{c}}_{1}\overline{\vee}\ddot{\mathfrak{c}}_{2}\supset\ddot{\mathfrak{d}}\supset\ddot{\mathfrak{c}}_{1}\vee\ddot{\mathfrak{c}}_{2}.$

Indeed, clearly
$Q^{(n,s)}_{\epsilon}(n,0,\ldots,0)=\ddot{\mathfrak{c}}_{\epsilon}(n)$ for
every $n\in{\mathbb{Z}}$ and $\epsilon\in\\{1,2\\}$. Since by definition
$\left(\ddot{\mathfrak{c}}_{1}\overline{\vee}\ddot{\mathfrak{c}}_{2}\right)(n)=\ddot{\mathfrak{c}}_{1}(n)\cap\ddot{\mathfrak{c}}_{2}(n)$
the first inclusion follows.

To establish the second inclusion let us fix $s,\epsilon$ and numbers $j_{k}$
such that $\sum_{k=1}^{s}j_{k}=n$. Consider the product

$\displaystyle(e^{ij_{1}x}\hat{\varphi}_{1})\cdot(e^{ij_{2}x}\hat{\varphi}_{2})\ldots(e^{ij_{s}x}\hat{\varphi}_{s})=$
$\displaystyle
e^{inx}\left[(\varphi_{1}\circ\tau^{j_{2}+j_{3}+\ldots+j_{s}})(\varphi_{2}\circ\tau^{j_{3}+\ldots+j_{s}})\cdot\ldots\cdot(\varphi_{s-1}\circ\tau^{j_{s}})\varphi_{s}\right]^{\wedge}=:e^{inx}\hat{\varphi}\in\mathcal{A}_{1}\vee\mathcal{A}_{2}.$

Here, the $\wedge$ sign indicates that we have taken the Fourier transform on
the function inside the square brackets. We name this function $\varphi$. The
equality is obtained by an iterated application of the multiplication formula
in Lemma 2.5(3).

If the functions $\varphi_{k}(t)\in C(S^{1})$ are chosen to be non-negative
with $\mathcal{Z}(\varphi_{k})=\ddot{\mathfrak{c}}_{\epsilon+k-1}(j_{k})$,
this element is inside $\mathcal{A}_{1}\vee\mathcal{A}_{2}$. Since
$\mathcal{Z}(\varphi)=Q^{s}_{\epsilon}(j_{1},j_{2},\ldots,j_{s})$, we obtain
$Q^{s}_{\epsilon}(j_{1},\ldots,j_{s})\supset(\ddot{\mathfrak{c}}_{1}\vee\ddot{\mathfrak{c}}_{2})(n)$.
Taking finite sums over various possible choices of
$s,\epsilon,_{1},\ldots,j_{s}$ and then passing to the limit yields the second
desired containment. It follows from this that

$\Phi(\ddot{\mathfrak{c}}_{1}\overline{\vee}\ddot{\mathfrak{c}}_{2})=\Phi(\ddot{\mathfrak{d}})=\Phi(\ddot{\mathfrak{c}}_{1}\vee\ddot{\mathfrak{c}}_{2}).$

Thus it would be enough to show that $\mathfrak{d}$ is closed in order to show
the desired equality
$\ddot{\mathfrak{d}}=\ddot{\mathfrak{c}}_{1}\vee\ddot{\mathfrak{c}}_{2}$.

Consider first $\mathtt{Cif}1$. We see that

$\displaystyle\tau^{n}Q^{(n,s)}_{\epsilon}(j_{1},j_{2},\ldots,j_{s})=\tau^{j_{1}}\ddot{\mathfrak{c}}_{\epsilon}(j_{1})\cup\ldots\cup\left(\tau^{\sum_{k=1}^{s-1}j_{k}}\ddot{\mathfrak{c}}_{\epsilon+s-1}(j_{s-1})\right)\cup\tau^{n}\ddot{\mathfrak{c}}_{\epsilon+s}(j_{s})$
$\displaystyle=\ddot{\mathfrak{c}}_{\epsilon}(-j_{1})\cup\ldots\cup\left(\tau^{\sum_{k=1}^{s-2}j_{k}}\ddot{\mathfrak{c}}_{\epsilon+s-1}(-j_{s-1})\right)\cup\left(\tau^{\sum_{k=1}^{s-1}j_{k}}\ddot{\mathfrak{c}}_{\epsilon+s}(-j_{s})\right)$
$\displaystyle=Q^{(-n,s)}_{\epsilon+s}(-j_{s},-j_{s-1},\ldots,-j_{1})$

Plugging this into the definition of $\ddot{\mathfrak{d}}$ in Equation (6) we
conclude that $\ddot{\mathfrak{d}}(-n)=\tau^{n}\ddot{\mathfrak{d}}(n)$ which
is exactly $\mathtt{Cif}1$.

Turning to $\mathtt{Cif}2$, consider
$Q^{(m+n,s)}_{\epsilon}(j_{1},\ldots,j_{r},j_{r+1},\ldots,j_{s})$ where
$\sum_{k=1}^{r}j_{k}=n$, $\sum_{k=r+1}^{s}j_{k}=m$. Then we have

$\displaystyle\ddot{\mathfrak{d}}(m+n)$ $\displaystyle\subset
Q^{(m+n,s)}_{\epsilon}(j_{1},j_{2},\ldots,j_{s})$
$\displaystyle=\tau^{-m}\left[\tau^{-\sum_{k=2}^{r}j_{k}}\ddot{\mathfrak{c}}_{\epsilon}(j_{1})\cup\ldots\cup\ddot{\mathfrak{c}}_{\epsilon+r}(j_{r})\right]\cup\left[\tau^{-\sum_{k=r+2}^{s}j_{k}}\ddot{\mathfrak{c}}_{\epsilon+r+1}(j_{r+1})\cup\ldots\cup\ddot{\mathfrak{c}}_{\epsilon+s}(j_{s})\right]$
$\displaystyle=\tau^{-m}Q^{(n,r)}_{\epsilon}(j_{1},\ldots,j_{r})\cup
Q^{(m,s-r)}_{\epsilon+r+1}(j_{r+1},\ldots,j_{s})$

Applying the same calculation to
$Q^{(m+n,s+1)}_{\epsilon}(j_{1},\ldots,j_{r},0,j_{r+1},\ldots,j_{s})$ we
obtain an inclusion
$\ddot{\mathfrak{d}}(m+n)\subset\tau^{-m}Q^{(n,r)}_{\epsilon}(j_{1},\ldots,j_{r})\cup
Q^{(m,s-r)}_{\epsilon+r+1}(j_{r+1},\ldots,j_{s})$ and taking the intersection
over these two we have now

$\ddot{\mathfrak{d}}(m+n)\subset\tau^{-m}Q^{(n,r)}_{\epsilon}(j_{1},\ldots,j_{r})\cup
Q^{(m,s-r)}_{\epsilon^{\prime}}(j_{r+1},\ldots,j_{s}).$

Now the parameters $\epsilon,r,j_{1},\ldots,j_{r}$ appearing in the first part
of this expression and $\epsilon^{\prime},s-r,j_{r+1},\ldots,j_{s}$ appearing
in the second part are completely independent of each other. Taking the
intersection over all of these yields the desired result:

$\ddot{\mathfrak{d}}(m+n)\subset\tau^{-m}\ddot{\mathfrak{d}}(n)\cup\ddot{\mathfrak{d}}(m)$

which is exactly $\mathtt{Cif}2$. This completes our proof. ∎

###  4.2. Residual algebras

The special class of residual intermediate subalgebras was singled out in the
introduction as algebras admitting especially nice properties. Recall that an
ideal function $\mathfrak{c}\in\mathfrak{C}$ (resp. a $\mathfrak{c}$-ideal
function $\mathfrak{j}$) are called residual if $\ddot{\mathfrak{c}}(n)$
(resp. $\ddot{\mathfrak{j}}(n)$) has an empty interior for every $n$ in the
support. We also refer to the corresponding intermediate algebra (resp ideal)
as residual. We denote by $\mathfrak{C}^{r}$ (resp.
$\mathfrak{I}_{\mathfrak{c}}^{r}$) the sub-lattices of residual objects.

######  Note 4.2.

The notation could be confusing. Residual algebras are large, but we are
talking about small sets at the level of closed subsets of the circle. In
fact, $\mathfrak{c}$ is residual exactly when the complement
$S^{1}\setminus\ddot{\mathfrak{c}}(n)$ is residual (in the standard Baire
category sense of the word), for every
$n\in\operatorname{Supp}(\mathfrak{c})$.

###### Proof of cor:residual.

It is clear that the class of residual algebras $\mathfrak{C}^{r}$ is closed
under arbitrary joins. The fact that it is closed under finite intersections
follows from the (finite) Baire category theorem. This establishes Properties
(1) and (2).

Any ideal function’s support $\operatorname{Supp}(\mathfrak{c})$ is symmetric
by property $\mathtt{Cif}1$. In the residual case, it is closed under addition
by $\mathtt{Cif}2$ because the union of two closed nowhere dense sets is still
nowhere dense. This establishes Property (3).

Now let $J\lhd\mathcal{A}\in\mathfrak{A}$ be a nontrivial closed two-sided
ideal with associated $\mathfrak{c}$-ideal function $\mathfrak{j}$ and set
$M:=\ddot{\mathfrak{j}}(0)\subset S^{1}$. When we say that $J$ is nontrivial,
we mean that $(0)\lneqq J\lneqq\mathcal{A}$, which means that
$\emptyset\subsetneq M\subsetneq S^{1}$, as in Proposition (2.25).

As $\ddot{\mathfrak{c}}(n)\subset\ddot{\mathfrak{j}}(n)$ for every
$n\in\mathbb{Z}$, it is clear that
$\operatorname{Supp}(\mathfrak{j})\subset\operatorname{Supp}(\mathfrak{c})$.
The reverse inclusion also holds. Indeed by $\mathtt{Cif}2(\mathfrak{c})$ we
have, for $k\in\operatorname{Supp}(\mathfrak{c})$, that
$\ddot{\mathfrak{j}}(k)=\ddot{\mathfrak{j}}(k+0)\subset\ddot{\mathfrak{c}}(k)\cup\tau^{-k}\ddot{\mathfrak{j}}(0)\subsetneq
S^{1}$, where the proper containment follows since $\ddot{\mathfrak{j}}(0)$ is
a proper closed subset of $S^{1}$ and $\ddot{\mathfrak{c}}(k)$ is closed with
an empty interior. Thus
$\operatorname{Supp}(\mathfrak{c})=\operatorname{Supp}(\mathfrak{j})$,
establishing Property (4).

To prove Property (5) we first argue that $M=\ddot{\mathfrak{j}}(0)$ is
nowhere dense. By (3) and (4)
$\operatorname{Supp}(\mathfrak{c})=\operatorname{Supp}(\mathfrak{j})=r{\mathbb{Z}}$
for some $r\geq 1$ is a subgroup of $\mathbb{Z}$. Recall that by Proposition
(2.25) $M\subset\ddot{\mathfrak{j}}(k),\forall k\in{\mathbb{Z}}$. Now applying
$\mathtt{Cif}2(\mathfrak{c})$ we have for every
$k,n\in\operatorname{Supp}(\mathfrak{c})$

$\displaystyle M$ $\displaystyle\subset$
$\displaystyle\ddot{\mathfrak{j}}(k)=\ddot{\mathfrak{j}}(n+k-n)\subset\tau^{-n-k}\ddot{\mathfrak{j}}(-n)\cup\ddot{\mathfrak{c}}(n+k)=\tau^{-k}\ddot{\mathfrak{j}}(n)\cup\ddot{\mathfrak{c}}(n+k)$
(7) $\displaystyle\subset$
$\displaystyle\tau^{-k}\ddot{\mathfrak{j}}(n)\cup\tau^{-k}\ddot{\mathfrak{c}}(n)\cup\ddot{\mathfrak{c}}(k)=\tau^{-k}\ddot{\mathfrak{j}}(n)\cup\ddot{\mathfrak{c}}(k),$

where the last equality uses, again, the fact that
$\ddot{\mathfrak{j}}(n)\supset\ddot{\mathfrak{c}}(n)$.

Taking intersection over all $n\in\operatorname{Supp}(\mathfrak{c})$ and using
the fact that $\bigcap_{n\in{\mathbb{Z}}}\ddot{\mathfrak{j}}(n)=M$, by
Proposition 2.25, we obtain

$M\subset\ddot{\mathfrak{j}}(k)\subset\tau^{-k}\left(\bigcap_{n\in{\mathbb{Z}}}\ddot{\mathfrak{j}}(n)\right)\cup\ddot{\mathfrak{c}}(k)=\tau^{-k}M\cup\ddot{\mathfrak{c}}(k)$
(8)

Let us denote by
$\Omega=\bigcup_{m,n\in\operatorname{Supp}(\mathfrak{c})}\tau^{m}\ddot{\mathfrak{c}}(n)$.
This is a meager $\tau^{r}$-invariant subset of $S^{1}$. Now, using the last
step, we have

$M\setminus\Omega\subset\tau^{-r}\left(M\setminus\Omega\right).$

Namely $M\setminus\Omega$ is $\tau^{r}$-invariant. It cannot be dense because
$M$ is a proper closed subset of $S^{1}$. Hence, it has to be empty by the
minimality of $\tau^{r}$. Thus $M\subset\Omega$, particularly $M$, is meager.

Now Equation (8) implies that
$\ddot{\mathfrak{j}}(k)\subset\tau^{-k}M\cup\ddot{\mathfrak{c}}(k)\subset\Omega$,
for every
$k\in\operatorname{Supp}(\mathfrak{j})=\operatorname{Supp}(\mathfrak{c})$. In
particular, $\ddot{\mathfrak{j}}(k)$ is closed and nowhere dense for every
$k\in\operatorname{Supp}(\mathfrak{j})$. This completes the proof of Property
(5).

Finally, towards a proof of Property (6), suppose that $a$ is in the center of
$\mathcal{A}$. We first show that $a\in C^{*}_{r}({\mathbb{Z}})$. For every
$n$ we have $a\lambda_{n}=\lambda_{1}(a\lambda_{n})\lambda_{1}^{*}$, hence

$\mathbb{E}(a\lambda_{n})=\mathbb{E}(\lambda_{1}(a\lambda_{n})\lambda_{1}^{*})=\lambda_{1}\mathbb{E}(a\lambda_{n})\lambda_{1}^{*}=\mathbb{E}(a\lambda_{n})\circ
T.$

By ergodicity, each Fourier coefficient $\mathbb{E}(a\lambda_{n})$ is a
constant, and we conclude that $a\in C^{*}_{r}({\mathbb{Z}})$. Thus
$a=\hat{\xi}$ for some $\xi\in C(S^{1})$.

Let $n\in\operatorname{Supp}(\mathfrak{c})$ be such that
$\ddot{\mathfrak{c}}(n)$ is nowhere dense and let $\varphi\in C(S^{1})$ be
such that $\hat{\varphi}$ is a generator of the ideal $\mathfrak{c}(n)$. Or in
other words $\mathcal{Z}\varphi=\\{t\in S^{1}\ |\
\varphi(t)=0\\}=\ddot{\mathfrak{c}}(n)$. We then have, using Lemma 2.5(1)

$e^{inx}\widehat{\varphi\xi}=e^{inx}\hat{\varphi}\hat{\xi}=\hat{\xi}e^{inx}\hat{\varphi}=e^{inx}\widehat{\left[(\xi\circ\tau^{n})\varphi\right]},$

so on the open dense set $S^{1}\setminus\ddot{\mathfrak{c}}(n)$ we get
$\xi\circ\tau^{n}=\xi$. Using the ergodicity of $\tau^{n}$, we conclude that
$\xi$ is a constant. This establishes Property (6) and completes the proof of
Corollary (cor:residual). ∎

######  Note 4.3.

According to Property (2) in the above proof, the intersection of any finite
collection of residual subalgebras is still residual. This does not generalize
to countable intersections, though. Consider, for example, the basic algebras
$\mathcal{A}_{i}:=\mathcal{A}_{1,\\{p_{i}\\}}$ with $\\{p_{i}\ |\
i\in{\mathbb{N}}\\}\subset S^{1}$ a dense countable set. Finite intersections
of these algebras are residual, and in fact, they are still basic:

$\Psi\left(\bigcap_{i=1}^{n}\mathcal{A}_{i}\right)=\mathfrak{b}_{1,\\{p_{1},p_{2},\ldots,p_{n}\\}}.$

But clearly

$\bigcap_{i\in{\mathbb{N}}}\mathcal{A}_{i}=C(X).$

###  4.3. Small algebras

This section is dedicated to studying small subalgebras, whose basic
properties are summarized in cor:small.

###### Proof of cor:small.

Let $\mathcal{A}\in\mathcal{A}$ and $\mathfrak{c}=\Psi(\mathcal{A})$. By our
main structure theorem, we know that, as a $C^{*}_{r}({\mathbb{Z}})$ module

$\mathcal{A}=\overline{\oplus_{n\in\operatorname{Supp}(\mathfrak{c})}\mathfrak{c}(n)}.$

Thus, it is clear that $\operatorname{Supp}(\mathfrak{c})$ is finite if and
only if the algebra is finite-dimensional as a module over
$C^{*}_{r}({\mathbb{Z}})$. These are the small subalgebras and (3) directly
follows.

Since for residual subalgebras $\operatorname{Supp}(\mathfrak{c})$ is a
(nontrivial) subgroup of ${\mathbb{Z}}$, residual subalgebras can never be
small. Now let $\mathfrak{b}_{q,P}$ be a basic ideal function and
$\mathcal{A}_{q,P}$ the associated algebra (see Example 2.11 and Subsection
2.7). If $P$ has an empty interior, then $\mathcal{A}_{q,P}$ is residual and
hence not small. Conversely, if $P$ has a nonempty interior, by the minimality
of $\tau:S^{1}\rightarrow S^{1}$, there is some $N$ such that
$\ddot{\mathfrak{b}}_{q,P}(nq)=P\cup\tau^{-q}P\cup\ldots\cup\tau^{q(1-n)}P=S^{1}$
for every $n\geq N$. By the explicit formula for the basic ideal function in
Example 2.11, this immediately implies that
$\operatorname{Supp}(\mathfrak{b}_{q,P})$ is finite and hence
$\mathcal{A}_{q,P}$ is small. This proves (2)

If $P=\ddot{\mathfrak{c}}(q)$ has an empty interior for some $q\neq 0$, then
$\mathcal{A}$ contains the basic residual algebra $\mathcal{A}_{q,P}$ as a
subalgebra. Hence, $\mathcal{A}$ itself must be residual, contrary to the
assumption that it is small. This proves (1). The following examples will
demonstrate parts (4) and (5). ∎

######  Example 4.4.

Let $J=S^{1}\setminus P=(-\pi/10,\pi/10)$ and suppose that
$\pi/10<\alpha<2\pi/10$. Then any function $\varphi\in C(S^{1})$ such that
$\varphi(t)=\varphi(t+\alpha)$ for $t\in J$ and $\varphi(t)=\varphi(t-\alpha)$
for $t\in\tau J$, is a central element of the algebra $\mathcal{A}_{1,P}$.

###### Proof.

Clearly $P\cup\tau P=S^{1}$ so that every element $a$ of $\mathcal{A}_{1,P}$
has the form

$a=e^{-ix}\hat{\psi}_{-1}+\hat{\psi}_{0}+e^{ix}\hat{\psi}_{1},$

with $\psi_{-1}\upharpoonright_{\tau P}=0=\psi_{1}\upharpoonright_{P}$. In
order for an element $\hat{\varphi}\in C^{*}(\mathbb{Z})$ to be central in
$\mathcal{A}_{1,P}$ it should satisfy the equation
$\hat{\varphi}a=a\hat{\varphi}$ for every $a\in\mathcal{A}_{1,P}$. Writing
this equation more explicitly, we get

$e^{-ix}(\hat{\varphi}\circ\tau^{-1})\hat{\psi}_{-1}+\hat{\varphi}\hat{\psi}_{0}+e^{ix}(\hat{\varphi}\circ\tau)\hat{\psi}_{1}=e^{-ix}\hat{\psi}_{-1}\hat{\varphi}+\hat{\psi}_{0}\hat{\varphi}+e^{ix}\hat{\psi}_{1}\hat{\varphi}.$

From this we deduce that on $J$ we have $\varphi=\varphi\circ\tau^{1}$ and on
$\tau J$ we have $\varphi=\varphi\circ\tau^{-1}$. By our assumption the sets
$J$, $\tau^{-1}J$, $\tau J$ and $\tau^{2}J$ are pairwise disjoint, and it
follows that any $\varphi$ which satisfies these conditions will be a central
element of $\mathcal{A}_{1,P}$. ∎

The intersection of two residual algebras is residual. In contrast, the
following example shows that the join of two small algebras need not be small.

######  Example 4.5.

Consider two subsets $P$,$Q\subset S^{1}$ with non-empty interior such that
$P\cap Q=\emptyset$. Let
$\mathfrak{a}=\mathfrak{b}_{1,P},\mathfrak{b}=\mathfrak{b}_{1,Q}$ be the
corresponding ideal functions. It is easy to see that
$(\ddot{\mathfrak{a}}\vee\ddot{\mathfrak{b}})(1)=\emptyset$ as it is contained
in both $\ddot{\mathfrak{a}}(1)=P$ and $\ddot{\mathfrak{b}}(1)=Q$. In other
words, the join of the corresponding algebra contains $e^{ix}$ and
$C_{r}^{*}(\mathbb{Z})$, and is thus the entire crossed product
$C(S^{1})\rtimes_{r}\mathbb{Z}$.

###  4.4. About the simplicity of intermediate algebras.

It is well known that the algebras
$\mathcal{C}^{q}=C(X_{q})\rtimes_{r}\mathbb{Z}$ are simple (each
$(X,R_{q\alpha})$ being minimal). We would like to determine which of the
intermediate algebras are simple. To approach this question, consider a
general intermediate algebra $\mathcal{A}_{\mathfrak{c}}$. The collection of
closed sets $\\{\ddot{\mathfrak{c}}(n):|n|\geq 1\\}$ either has the finite
intersection property with

$Q=Q(\mathfrak{c}):=\bigcap_{|n|\geq 1}\ddot{\mathfrak{c}}(n)\not=\emptyset,$

or it does not. In the latter case there is an $N\geq 1$ such that
$\bigcap_{1\leq|n|\leq N}\ddot{\mathfrak{c}}(n)=\emptyset$.

######  Proposition 4.6.

Let $\mathfrak{c}$ be an ideal function such that $Q=\bigcap_{|n|\geq
1}\ddot{\mathfrak{c}}(n)\not=\emptyset$, then the algebra
$\mathcal{A}_{\mathfrak{c}}$ is not simple. In particular, for the algebra
$\mathcal{A}=\mathcal{A}_{\mathfrak{b}_{q,P}}$ corresponding to the basic
ideal functions $\mathfrak{b}_{q,P}$, we have $Q=P\cap\tau^{q}P$ so that
$\mathcal{A}$ is not simple whenever this set is not empty.

###### Proof.

$Q\not=\emptyset\Rightarrow\mathcal{A}=\mathcal{A}_{\mathfrak{c}}$ is not
simple.

Let

$I=\\{a\in\mathcal{A}:\overbracket{a}(0)\upharpoonright_{Q}=0\\}.$ (9)

It is easy to check that this is a proper ideal: Given $d\in I$ and
$a\in\mathcal{A}$ we need to show that both
$\overbracket{ad}(0)=\mathbb{E}_{\mu}(ad)$ and
$\overbracket{da}(0)=\mathbb{E}_{\mu}(da)$ correspond to functions that vanish
on $Q$. Let $a^{\prime}=\sum_{|n|\leq N}e^{inx}\hat{\varphi}_{n}$ and
$d^{\prime}=\sum_{|k|\leq N}e^{ikx}\hat{\psi}_{k}$ be good Fejér
approximations to $a$ and $d$ respectively. Applying the conditional
expectation, we get the following:

$\displaystyle\mathbb{E}_{\mu}(ad)$ $\displaystyle\approx\
\mathbb{E}_{\mu}\left((\sum_{|n|\leq N}e^{inx}\hat{\varphi}_{n})(\sum_{|k|\leq
N}e^{ikx}\hat{\psi}_{k})\right)$ $\displaystyle=\sum_{0<|n|\leq
N}(\widehat{\varphi_{n}\circ\tau^{-n}})\hat{\psi}_{-n}+\widehat{\varphi_{0}\cdot\psi_{0}}.$
(10)

Now the first summoned corresponds to a function on $C(S^{1})$ which vanishes
on $Q$, as, for $n\not=0$, we have that $\psi_{-n}$ vanishes on
$\ddot{\mathfrak{c}}(-n)$, $\varphi_{n}\circ\tau^{-n}$ also vanishes on
$\tau^{n}\ddot{\mathfrak{c}}(n)=\ddot{\mathfrak{c}}(-n)$, and
$Q\subset\ddot{\mathfrak{c}}(-n)$. The second summoned corresponds to the
function $\varphi_{0}\psi_{0}$ which belongs to $I$. Since $I$ is closed, we
conclude that indeed $ad\in I$. A similar computation shows that also $da\in
I$.

For the last assertion, we note that for $\mathfrak{b}_{q,P}$ we have
$Q=P\cap\tau^{q}P$. ∎

Is the condition $Q(\mathfrak{c})\not=\emptyset$ also a necessary condition
for $\mathcal{A}_{\mathfrak{c}}$ to be simple ? We only have a partial answer
to this question in Proposition 4.13 below.

######  Lemma 4.7.

If $I<\mathcal{A}_{\mathfrak{c}}$ is a proper ideal, then
$\mathcal{B}=\overline{C^{*}(\mathbb{Z})+I}$ is a $C^{*}$-subalgebra:
$C^{*}(\mathbb{Z})<\mathcal{B}<\mathcal{A}_{\mathfrak{c}}$, and $I$ is a
proper ideal in $\mathcal{B}$.

###### Proof.

Easy to check. ∎

######  Definition 4.8.

genrideal We say that an ideal $I<\mathcal{A}_{\mathfrak{c}}$ is generating
when $\mathcal{A}_{\mathfrak{c}}=\overline{C^{*}(\mathbb{Z})+I}$.

See also [Ped18, Corollary 1.5.8] for the following lemma.

######  Lemma 4.9.

If $I$ is a generating ideal for $\mathcal{A}_{\mathfrak{c}}$ then we actually
have:

$\mathcal{A}_{\mathfrak{c}}=C^{*}(\mathbb{Z})+I.$

###### Proof.

By Theorem 1.4 we have $I=I_{\mathfrak{j}}$ for some ideal function
$\mathfrak{j}$. Let $I_{0}=I\cap C^{*}({\mathbb{Z}})$. It then follows that
$\mathbb{E}_{\mu}(I)\subset I_{0}$. Let $c\in\mathcal{A}_{\mathfrak{c}}$, then
we have

$c=\lim(a_{n}+b_{n}),$

with $a_{n}\in C^{*}(\mathbb{Z})$ and $b_{n}\in I$. Now

$\mathbb{E}_{\mu}(c)=\lim\ \mathbb{E}_{\mu}(a_{n}+b_{n})=\lim\
(a_{n}+\mathbb{E}_{\mu}(b_{n})).$

It follows that $c-\mathbb{E}_{\mu}(c)=\lim(b_{n}-\mathbb{E}_{\mu}(b_{n}))$.
As, for each $n$, both $b_{n}$ and $\mathbb{E}_{\mu}(b_{n})$ are in $I$ and
$I$ is closed, we conclude that $c-\mathbb{E}_{\mu}(c)\in I$, whence $c\in
C^{*}({\mathbb{Z}})+I$. ∎

######  Corollary 4.10.

Given any ideal $I\triangleleft\mathcal{A}_{\mathfrak{c}}$,
$C_{r}^{*}(\mathbb{Z})+I=\overline{C_{r}^{*}(\mathbb{Z})+I}$.

###### Proof.

It follows from genrideal that $I$ is a generating ideal for
$\overline{C_{r}^{*}(\mathbb{Z})+I}$. Since
$\overline{C_{r}^{*}(\mathbb{Z})+I}$ is an intermediate algebra of the form
$\mathcal{A}_{\mathfrak{d}}$ for some $\mathfrak{d}\leq\mathfrak{c}$, the
claim follows from Lemma 4.9. ∎

######  Remark 4.11.

We note that the ideal $I<\mathcal{A}_{\mathfrak{c}}$ (defined in Equation (9)
of Proposition 4.6) is of the form $I=I_{\mathfrak{j}}$ with
$\ddot{\mathfrak{j}}(0)=Q$ and $\ddot{\mathfrak{j}}(n)=\ddot{\mathfrak{c}}(n)$
for $n\neq 0$, hence it is generating. Let us denote such an ideal by
$I^{\mathcal{A}}_{Q}$. If $Q_{1}\subset Q$ is a smaller closed subset, then
Formula (4.4) shows that $I^{\mathcal{A}}_{Q_{1}}\lhd\mathcal{A}$ as well. It
is easy to check that if $Q_{1}$ is taken to be a point, we obtain a maximal
ideal (of codimension $1$) inside $\mathcal{A}$.

######  Proposition 4.12.

Let $\mathcal{A}_{\mathfrak{c}}$ be an intermediate algebra with a generating
proper ideal $I<\mathcal{A}_{\mathfrak{c}}$; then
$Q(\mathfrak{c})=\bigcap_{|n|\geq 1}\ddot{\mathfrak{c}}(n)\not=\emptyset$.
Conversely, if $Q(\mathfrak{c})\not=\emptyset$ then
$I=\\{a\in\mathcal{A}:\overbracket{a}(0)\upharpoonright_{Q(\mathfrak{c})}=0\\}$
is a proper generating ideal of $\mathcal{A}_{\mathfrak{c}}$.

###### Proof.

Let $I_{0}=I\cap C^{*}({\mathbb{Z}})$ and let $I=I_{\mathfrak{j}}$ as in
Theorem 1.4. It then follows that $\mathbb{E}_{\mu}(I)\subset I_{0}$.

Now suppose to the contrary that $Q=\bigcap_{|n|\geq
1}\ddot{\mathfrak{c}}(n)=\emptyset$. As we observed above there is an $N\geq
1$ such that $\bigcap_{1\leq|n|\leq N}\ddot{\mathfrak{c}}(n)=\emptyset$.
Denoting $U_{n}=S^{1}\setminus\ddot{\mathfrak{c}}(n)$, we have
$S^{1}=\bigcup_{1\leq|n|\leq N}U_{n}$. Let $\\{\xi_{n}\\}_{1\leq|n|\leq N}$ be
a partition of unity subordinate to the open cover $\\{U_{n}\\}_{1\leq|n|\leq
N}$; i.e. $0\leq\xi_{n}\leq 1$, $\sum_{1\leq|n|\leq N}\xi^{2}_{n}=1$, and
$\operatorname{Supp}\xi_{n}\subset U_{n}$. Let $c=\sum_{1\leq|n|\leq
N}e^{inx}\hat{\xi}_{n}$.

By Lemma 4.9 we have $c=\hat{\varphi}_{0}+d$, with $\hat{\varphi}_{0}\in
C^{*}({\mathbb{Z}})$ and $d\in I$. Now
$0=\mathbb{E}_{\mu}(c)=\mathbb{E}_{\mu}(\hat{\varphi}_{0}+d)=\hat{\varphi}_{0}+\mathbb{E}_{\mu}(d)$,
hence $\hat{\varphi}_{0}=-\mathbb{E}_{\mu}(d)\in I_{0}\subset I$ and therefore
also $c=\hat{\varphi}_{0}+d\in I$. We now conclude that
$\mathbb{E}_{\mu}(c^{*}c)=\sum_{1\leq|n|\leq N}\hat{\xi}_{n}^{2}\in I$. But
$\sum_{1\leq|n|\leq N}\xi_{n}^{2}=1_{S^{1}}$, contradicting our assumption
that $I$ is a proper ideal.

The other direction follows from Proposition 4.6 and Remark 4.11.

∎

######  Proposition 4.13.

Suppose $\mathcal{A}_{\mathfrak{c}}$ is not simple. Let
$I<\mathcal{A}_{\mathfrak{c}}$ be a proper ideal (which we can assume to be
maximal) and let
$\mathcal{B}=C^{*}(\mathbb{Z})+I\subset\mathcal{A}_{\mathfrak{c}}$. Then
$\mathcal{B}$ is a sub-$C^{*}$-algebra, and $I$ is a generating ideal for
$\mathcal{B}$. Moreover, $\mathcal{B}=\mathcal{A}_{\mathfrak{d}}$ for an ideal
function $\mathfrak{d}$ with
$\ddot{\mathfrak{d}}(n)\supset\ddot{\mathfrak{c}}(n)$ for every
$n\in\mathbb{Z}$, and such that
$Q(\mathfrak{d})=\bigcap_{1\leq|n|}\ddot{\mathfrak{d}}(n)\not=\emptyset$.

As was mentioned above, the algebras
$\mathcal{C}^{q}=C(X_{q})\rtimes_{r}\mathbb{Z}$ are simple. We next describe
additional examples of simple intermediate subalgebras. These include the
basic subalgebras corresponding to the ideal functions
$\mathfrak{b}_{q,p}=\mathfrak{b}_{q,\\{p\\}}$, with $0\neq q\in\mathbb{N},p\in
S^{1}$, thus generalizing (in the context of $\mathcal{C}$) [GKPT18,
Proposition 11.3.21] (in view of Subsection 2.8).

######  Proposition 4.14.

residualsimplicity Assume that $\mathcal{A}\in\mathfrak{A}^{r}$ is a residual
intermediate algebra with $\mathfrak{c}=\Psi(\mathcal{A})$ such that
$\mathfrak{c}(q)=\\{p\\}$ is a singleton for some $0\neq q\in{\mathbb{Z}}$.
Then, $\mathcal{A}$ is simple.

###### Proof.

Assume towards a contradiction that $J\lhd\mathcal{A}$ is a non-trivial ideal
and let $\mathfrak{j}$ be the corresponding $\mathfrak{c}$-ideal function.
Since $\mathcal{A}$ is residual, everything from the proof of cor:residual in
sub-Section 4.2 applies. We thus adopt the notation from that proof and, in
particular, set $M=\ddot{\mathfrak{j}}(0)$. As in that proof, the non-
triviality of the ideal implies $M\neq\emptyset$ and $M\neq S^{1}$.

Using Equation (8) from the proof of cor:residual, we conclude that

$\emptyset\subsetneq
M\cap\ddot{\mathfrak{c}}(k)\subsetneq\ddot{\mathfrak{c}}(k),\qquad\forall
k\neq 0.$ (11)

Indeed if $M\cap\ddot{\mathfrak{c}}(k)=\emptyset$ then Equation (8) would read
$M\subset\tau^{-k}M$. By minimality of $\tau$, the closed subset $M\subset
S^{1}$ is either empty or everything, which is not the case since $J$ is a
nontrivial ideal. Similarly if $\ddot{\mathfrak{c}}(k)\subset M$ we can apply
Equation (8) with $-k$ instead of $k$, to obtain

$M\subset\tau^{k}M\cup\ddot{\mathfrak{c}}(-k)=\tau^{k}\left(M\cup\ddot{\mathfrak{c}}(k)\right)=\tau^{k}M,$

leading to the same contradiction. Taking $k=q$, we obtain a contradiction to
the assumption that $\ddot{\mathfrak{c}}(q)=\\{p\\}$ is a singleton. ∎

######  Remark 4.15.

By [GKPT18, Theorem 12.2.5], when a closed subset $P\subset S^{1}$ is such
that $\tau^{n}P\cap P=\emptyset$ for all $n\not=0$, the algebra
$\mathcal{A}_{1,P}$ is centrally large (hence large). It is simple and
infinite dimensional by [GKPT18, Theorem 12.2.6]. Presently, we don’t know how
to prove this result using our methods.

##  5\. Crossed products of non $C^{*}$-simple groups

Many of the methods we used to obtain a complete classification of
intermediate algebras in Theorem 1.2 are pretty specific to the system we
considered there. This section is dedicated to the proof of Theorem
(LABEL:notcrossedprod), where we show that the mere existence of intermediate
algebras that do not come from dynamical factors holds for a much more general
class of crossed products of the form $\mathcal{A}\rtimes_{r}\Gamma$. For
example, this is the case whenever $\mathcal{A}$ admits a faithful
$\Gamma$-invariant state, $\Gamma$ is not $C^{*}$-simple and does not admit a
normal subgroup isomorphic to ${\mathbb{Z}}/2{\mathbb{Z}}$.

###  5.1. Crossed products

We briefly describe the construction of the crossed product for unital
$C^{*}$-algebras in general and refer the reader to [BO08] for more details.

Let $\Gamma$ be a discrete group and $\mathcal{A}$ be an unital
$C^{*}$-algebra. An action of $\Gamma$ on $\mathcal{A}$ is a group
homomorphism $\alpha$ from $\Gamma$ to the group of $*$-automorphisms on
$\mathcal{A}$. A $C^{*}$-algebra equipped with a $\Gamma$-action is called a
$\Gamma$-$C^{*}$-algebra. Suppose that $\mathcal{A}$ is a unital
$\Gamma$-$C^{*}$-algebra. Let $\pi:\mathcal{A}\to\mathbb{B}(\mathcal{H})$ be a
faithful $*$-representation. Let $\ell^{2}(\Gamma,\mathcal{H})$ be the space
of square summable $\mathcal{H}$-valued functions on $\Gamma$, i.e.,

$\ell^{2}(\Gamma,\mathcal{H})=\left\\{\xi:\Gamma\to\mathcal{H}\text{ such that
}\sum_{t\in\Gamma}\|\xi(t)\|_{\mathcal{H}}^{2}<\infty.\right\\}$

There is an action $\Gamma\curvearrowright\ell^{2}(\Gamma,H)$ by left
translation:

$\lambda_{s}\xi(t):=\xi(s^{-1}t),\xi\in\ell^{2}(\Gamma,\mathcal{H}),s,t\in\Gamma$

Let $\sigma$ be the $*$-representation

$\sigma:\mathcal{A}\to\mathbb{B}(\ell^{2}(\Gamma,\mathcal{H}))$

defined by

$\sigma(a)(\xi)(t):=\pi(t^{-1}a)\xi(t),a\in\mathcal{A}$

where $\xi\in\ell^{2}(\Gamma,\mathcal{H})$, $t\in\Gamma$. The reduced crossed
product $C^{*}$-algebra $\mathcal{A}\rtimes_{r}\Gamma$ is the closure in
$\mathbb{B}(\ell^{2}(\Gamma,\mathcal{H}))$ of the subalgebra generated by the
operators $\sigma(a)$ and $\lambda_{s}$. Note that
$\lambda_{s}\sigma(a)\lambda_{s^{-1}}=\sigma(s.a)$ for all $s\in\Gamma$ and
$a\in\mathcal{A}$.

It follows from the construction that $\mathcal{A}\rtimes_{r}\Gamma$ contains
$C_{\lambda}^{*}(\Gamma)$ as a $C^{*}$-sub-algebra. The reduced crossed
product $\mathcal{A}\rtimes_{r}\Gamma$ comes equipped with a
$\Gamma$-equivariant canonical conditional expectation
$\mathbb{E}:\mathcal{A}\rtimes_{r}\Gamma\to\mathcal{A}$ defined by

$\mathbb{E}\left(\sigma(a_{s})\lambda_{s}\right)=\left\\{\begin{array}[]{ll}0&\mbox{if
$s\neq e$}\\\ \sigma(a_{e})&\mbox{otherwise}\end{array}\right\\}$

It follows from [BO08, Proposition 4.1.9] that $\mathbb{E}$ extends to a
faithful conditional expectation from $\mathcal{A}\rtimes_{r}\Gamma$ onto
$\mathcal{A}$. Moreover, for a subgroup $H\leq\Gamma$, there is a faithful
conditional expectation
$\mathbb{E}_{H}:\mathcal{A}\rtimes_{r}\Gamma\to\mathcal{A}\rtimes_{r}H$ (see
[Kho96, Remark 3.2]) defined by

$\mathbb{E}_{H}\left(\sigma(a_{s})\lambda_{s}\right)=\left\\{\begin{array}[]{ll}\sigma(a_{s})\lambda_{s}&\mbox{if
$s\in H$}\\\ 0&\mbox{otherwise}\end{array}\right\\}$

Explicit examples of intermediate algebras not coming from any factor $Z$ of
the form $Y\xrightarrow[]{}Z\xrightarrow[]{}X$, associated with an inclusion
$C(X)\subset C(Y)$ were presented in [Suz20, Proposition 2.6]. Ideals have
also been used to create intermediate algebras for tensor product inclusion
which do not split (see, e.g., [Zac01, Corollary 3.4]).
This section demonstrates that the ideals obstruct the intermediate algebras
being crossed products. Given a unital $\Gamma$-$C^{*}$-algebra, we show that
we can create an algebra that is not a crossed product in a canonical way, as
long as the reduced $C^{*}$-algebra is not simple, under some assumptions on
the unital $C^{*}$-algebra $\mathcal{A}$.

###  5.2. Intermediate $C^{*}$-algebra associated to an ideal

Let $\Gamma$ be a discrete group acting continuously on an unital
$C^{*}$-algebra $\mathcal{A}$ by $*$-automorphisms. Assume that there exists a
normal subgroup $N\triangleleft\Gamma$ such that $C_{r}^{*}(N)$ is not
$\Gamma$-simple, i.e., $C_{r}^{*}(N)$ has a $\Gamma$-invariant closed two
sided ideal. Let $J$ be a two-sided, non-trivial, closed $\Gamma$-invariant
ideal in $C_{r}^{*}(N)$. We observe that if $N\triangleleft\Gamma$ is such
that $C_{r}^{*}(N)$ is not $\Gamma$-simple, then $\Gamma$ is a not a
$C^{*}$-simple group (see [AK20, Theorem 1.1]). Using $J$, we can build a non-
trivial closed two-sided ideal $I$ inside $C_{r}^{*}(\Gamma)$.

######  Lemma 5.1.

biggeridealassociatedtoasmallerone Let $N\triangleleft\Gamma$ be such that
$C_{r}^{*}(N)$ is not $\Gamma$-simple. Let $J$ be a two-sided, non-trivial,
closed $\Gamma$-invariant ideal in $C_{r}^{*}(N)$. Then,

$I=I_{J}=\overline{\text{Span}\left\\{\eta a:\eta\in J,a\in
C_{r}^{*}(\Gamma)\right\\}}$

is a non-trivial closed two sided ideal of $C_{r}^{*}(\Gamma)$ which contains
$J$.

###### Proof.

Clearly, $I$ is closed in $\|.\|$ and also, under addition. By construction,
$I$ is closed under right multiplication. To show that $I$ is closed under the
left multiplication, it is enough to show that $\lambda(s)\eta a\in I$ for any
$\eta\in J$ and $a\in C_{r}^{*}(\Gamma)$. Again writing
$\lambda(s)\eta=\eta_{s}\lambda(s)$ and using the $\Gamma$-invariance of $J$,

$\lambda(s)\eta
a=\left(\lambda(s)\eta\lambda(s)^{*}\right)\lambda(s)a=\eta_{s}\lambda(s)a\in
I.$

This automatically implies that $I$ is a two-sided ideal of
$C_{r}^{*}(\Gamma)$. All that needs to be established now is that $I$ is non-
trivial. Since $J\neq 0$, $I\neq 0$. Let us show that $I\neq
C_{r}^{*}(\Gamma)$. Towards a contradiction, let us assume otherwise. Then,
given $0<\epsilon<1$, we can find $\eta_{1},\eta_{2},\ldots,\eta_{k}\in I$ and
$a_{1},a_{2},\ldots,a_{k}\in C_{r}^{*}(\Gamma)$ such that

$\left\|\lambda(e)-\sum_{i=1}^{k}\eta_{i}a_{i}\right\|<\epsilon.$

Let $\mathbb{E}_{N}:C_{r}^{*}(\Gamma)\to C_{r}^{*}(N)$ denote the canonical
conditional expectation from $C_{r}^{*}(\Gamma)$ onto $C_{r}^{*}(N)$. Note
that $\mathbb{E}_{N}$ sends every element in $N$ to itself and sends each
element of $\Gamma\setminus N$ to zero. Applying $\mathbb{E}_{N}$ to the above
inequality and using the fact that $J$ falls in the multiplicative domain of
$\mathbb{E}_{N}$, we see that

$\left\|\lambda(e)-\sum_{i=1}^{k}\eta_{i}\mathbb{E}_{N}(a_{i})\right\|<\epsilon<1.$

This forces $\sum_{i=1}^{k}\eta_{i}\mathbb{E}_{N}(a_{i})\in J$ to be an
invertible operator. However, $J\triangleleft C_{r}^{*}(N)$ is a non-trivial
ideal. This contradicts our assumption that $I=C_{r}^{*}(\Gamma)$. Therefore,
we are done since $I\neq C_{r}^{*}(\Gamma)$. ∎

Given an ideal $I\triangleleft C_{r}^{*}(\Gamma)$, let us associate the
following operator space $\mathcal{A}_{I}$ to $I$ defined by

$\mathcal{A}_{I}=\text{Span}\left\\{\eta\tilde{a}\eta^{\prime}:\eta,\eta^{\prime}\in
I,\leavevmode\nobreak\ \tilde{a}\in\mathcal{A}\rtimes_{r}\Gamma\right\\}$

Since $I$ is a closed two-sided ideal, $\mathcal{A}_{I}$ is closed under the
$*$-operation. We claim that $\overline{\mathcal{A}_{I}}$ is a
$C^{*}$-algebra. It is enough to show that $\overline{\mathcal{A}_{I}}$ is
closed under multiplication. Let us denote by
$\mathcal{A}_{\rtimes_{r,\text{alg}}}\Gamma$ the collection of finite sums of
the form $\sum_{i=1}^{n}a_{s_{i}}\lambda(s_{i})$, or more formally

$\mathcal{A}\rtimes_{r,\text{alg}}\Gamma=\left\\{\sum_{s\in F}a_{s}\lambda(s)\
|\ F\subset\Gamma,|F|<\infty,\leavevmode\nobreak\
a_{s}\in\mathcal{A}\right\\}$

Also, recall that $\mathcal{A}\rtimes_{r,\text{alg}}\Gamma$ is norm dense
inside $\mathcal{A}\rtimes_{r}\Gamma$. If there exists a $\Gamma$-invariant
state $\varphi$ on $\mathcal{A}$, then the map
$\mathbb{E}_{\varphi}:\mathcal{A}\rtimes_{r,\text{alg}}\Gamma\to
C_{r}^{*}(\Gamma),a\lambda(s)\mapsto\varphi(a)\lambda(s)$ extends to a well-
defined map at the level of $\mathcal{A}\rtimes_{r}\Gamma$ (see e.g., [BO08,
Exercise 4.1.4]).

######  Proposition 5.2.

algebra $\overline{\mathcal{A}_{I}}$ is a $C^{*}$-algebra. Moreover,
$\mathcal{B}=C_{r}^{*}(\Gamma)+\overline{\mathcal{A}_{I}}$ is an intermediate
$C^{*}$-algebra.

###### Proof.

To show that $\overline{\mathcal{A}_{I}}$ is a $C^{*}$-algebra, it is enough
to show that $\overline{\mathcal{A}_{I}}$ is closed under multiplication.
Towards that end, let us choose elements of the form
$\eta_{1}\tilde{a_{1}}\eta^{\prime}_{1},\eta_{2}\tilde{a_{2}}\eta^{\prime}_{2}\in\mathcal{A}_{I}$
with $\eta_{1},\eta^{\prime}_{1},\eta_{2},\eta^{\prime}_{2}\in I$ and
$\tilde{a_{1}},\tilde{a_{2}}\in\mathcal{A}\rtimes_{r}\Gamma$. Clearly,

$\eta_{1}\tilde{a_{1}}\eta_{1}^{\prime}\eta_{2}\tilde{a_{2}}\eta_{2}^{\prime}=\eta_{1}\left(\tilde{a_{1}}\eta_{1}^{\prime}\eta_{2}\tilde{a_{2}}\right)\eta_{2}^{\prime}\in\mathcal{A}_{I}.$

It now follows by a standard limiting argument that
$\overline{\mathcal{A}_{I}}$ is closed under multiplication. Let us first
check that $\mathcal{B}$ is norm closed. If
$\\{a_{n}+b_{n}\\}_{n\in\mathbb{N}}\in\mathcal{B}$ is such that $a_{n}\in
C_{r}^{*}(\Gamma)$, $b_{n}\in\overline{\mathcal{A}_{I}}$ and $a_{n}+b_{n}\to
c$. then $\mathbb{E}_{\varphi}(a_{n}+b_{n})\to\mathbb{E}_{\varphi}(c)$.
Therefore, $a_{n}+\mathbb{E}_{\varphi}(b_{n})\to\mathbb{E}_{\varphi}(c)$. As a
result, we see that $b_{n}-\mathbb{E}_{\varphi}(b_{n})\to
c-\mathbb{E}_{\varphi}(c)$. Since $b_{n}\in\overline{\mathcal{A}_{I}}$ and
$\mathbb{E}_{\varphi}\left(\overline{\mathcal{A}_{I}}\right)\subset\overline{\mathcal{A}_{I}}$,
we see that $b_{n}-\mathbb{E}_{\varphi}(b_{n})\in\overline{\mathcal{A}_{I}}$
and therefore, $c-\mathbb{E}_{\varphi}(c)\in\overline{\mathcal{A}_{I}}$.
Hence,
$c=\mathbb{E}_{\varphi}(c)+\left(c-\mathbb{E}_{\varphi}(c)\right)\in\mathcal{B}$.
Since $I$ is an ideal of $C_{r}^{*}(\Gamma)$, it follows that
$\lambda(s)\mathcal{A}_{I}\subset\mathcal{A}_{I}$ and
$\mathcal{A}_{I}\lambda(s)\subset\mathcal{A}_{I}$ for all $s\in\Gamma$. Hence,
$\mathcal{B}$ is closed under multiplication. ∎

######  Remark 5.3.

It is also true that

$\overline{\mathcal{A}_{I}}=\overline{\text{Span}\left\\{\eta
a\eta^{\prime}:\eta,\eta^{\prime}\in I,\leavevmode\nobreak\
a\in\mathcal{A}\right\\}}$

It is clear that

$\overline{\text{Span}\left\\{\eta a\eta^{\prime}:\eta,\eta^{\prime}\in
I,\leavevmode\nobreak\
a\in\mathcal{A}\right\\}}\subseteq\overline{\text{Span}\left\\{\eta\tilde{a}\eta^{\prime}:\eta,\eta^{\prime}\in
I,\leavevmode\nobreak\ \tilde{a}\in\mathcal{A}\rtimes_{r}\Gamma\right\\}}$

For the reverse inclusion, we observe that for $\eta,\eta^{\prime}\in I$,
$\eta(a\lambda(s))\eta^{\prime}=\eta a(\lambda(s)\eta^{\prime})$ and hence,
the claim follows. ∎

######  Remark 5.4.

We can also show that $\mathcal{B}$ is closed in $\|.\|$ without taking the
help of $\mathbb{E}_{\varphi}$. Let us denote
$\mathcal{D}=\overline{\mathcal{B}}$. Since $\mathcal{B}$ is closed under
multiplication, so is $\mathcal{D}$. Consequently, $\mathcal{D}$ is a
$C^{*}$-algebra. Moreover, $\overline{\mathcal{A}_{I}}$ is a closed two-sided
ideal of $\mathcal{D}$. Using [Mur90, Theorem 3.1.7], we see that
$\overline{\mathcal{A}_{I}}+C_{r}^{*}(\Gamma)$ is a $C^{*}$-subalgebra.

######  Proposition 5.5.

interm notcrossedprod-Ga Let $\Gamma$ be a discrete group and $\mathcal{A}$, a
$\Gamma$-$C^{*}$-algebra. Let $I$ be a non-trivial closed $\Gamma$-invariant
two-sided ideal in $C_{r}^{*}(\Gamma)$. Let $\varphi$ be a faithful
$\Gamma$-invariant state on $\mathcal{A}$. Let
$\mathcal{B}=C_{r}^{*}(\Gamma)+\overline{\mathcal{A}_{I}}$. Then,
$\mathcal{B}\cap\mathcal{A}=\mathbb{C}$.

###### Proof.

It follows from algebra that $\mathcal{B}$ is an intermediate $C^{*}$-algebra.
Let us now suppose that $\varphi$ is a $\Gamma$-invariant state on
$\mathcal{A}$ and $\mathbb{E}_{\varphi}$, the associated conditional
expectation onto $C_{r}^{*}(\Gamma)$. We first claim that
$\mathbb{E}_{\varphi}\left(\overline{\mathcal{A}_{I}}\right)\subset\overline{\mathcal{A}_{I}}$.
For this to hold, it is enough to show that
$\mathbb{E}_{\varphi}\left(\mathcal{A}_{I}\right)\subset\mathcal{A}_{I}$ after
which the claim would follow by the density of $\mathcal{A}_{I}$ inside
$\overline{\mathcal{A}_{I}}$ and the continuity of $\mathbb{E}_{\varphi}$.
Towards that end, for an element of the form
$\sum_{i=1}^{n}\eta_{i}\tilde{a_{i}}\eta^{\prime}_{i}\in\mathcal{A}_{I}$, we
see that

$\mathbb{E}_{\varphi}\left(\sum_{i=1}^{n}\eta_{i}\tilde{a_{i}}\eta^{\prime}_{i}\right)=\sum_{i=1}^{n}\eta_{i}\mathbb{E}_{\varphi}(\tilde{a_{i}})\eta^{\prime}_{i}$

Since $\eta^{\prime}_{i}\in I$ and $\mathbb{E}_{\varphi}(\tilde{a_{i}})\in
C_{r}^{*}(\Gamma)$, we see that
$\mathbb{E}_{\varphi}(\tilde{a_{i}})\eta^{\prime}_{i}\in I$ and hence,

$\sum_{i=1}^{n}\eta_{i}\mathbb{E}_{\varphi}(\tilde{a_{i}})\eta^{\prime}_{i}=\sum_{i=1}^{n}\eta_{i}\mathbf{1}_{\mathcal{A}}\mathbb{E}_{\varphi}(\tilde{a_{i}})\eta^{\prime}_{i}\in\mathcal{A}_{I}.$

We now show that $\overline{\mathcal{A}_{I}}\cap\mathcal{A}={0}$. Assume that
$0\neq a\in\overline{\mathcal{A}_{I}}\cap\mathcal{A}$. By replacing $a$ with
$a^{*}a$, we can assume that $a\geq 0$. In particular, $\varphi(a)>0$. Let
$0<\epsilon<1$ be given. We can find an element of the form
$\sum_{i=1}^{n}\eta_{i}\tilde{a_{i}}\eta^{\prime}_{i}\in\mathcal{A}_{I}$ such
that

$\left\|a-\sum_{i=1}^{n}\eta_{i}\tilde{a_{i}}\eta^{\prime}_{i}\right\|<\varphi(a)\epsilon$

Applying the conditional expectation $\mathbb{E}_{\varphi}$, we see that

$\left\|\varphi(a)-\sum_{i=1}^{n}\eta_{i}\mathbb{E}_{\varphi}(\tilde{a_{i}})\eta^{\prime}_{i}\right\|<\varphi(a)\epsilon$

Let us observe that

$\sum_{i=1}^{n}\eta_{i}\mathbb{E}_{\varphi}(\tilde{a_{i}})\eta^{\prime}_{i}\in
I,$

where $I$ is a non-trivial ideal in $C_{r}^{*}(\Gamma)$. Hence,

$\left\|1-\frac{\sum_{i=1}^{n}\eta_{i}\mathbb{E}_{\varphi}(\tilde{a_{i}})\eta^{\prime}_{i}}{\varphi(a)}\right\|<\epsilon<1$

Since
$\sum_{i=1}^{n}\eta_{i}\mathbb{E}_{\varphi}(\tilde{a_{i}})\eta^{\prime}_{i}\in
I$, it follows that
$\frac{\sum_{i=1}^{n}\eta_{i}\mathbb{E}_{\varphi}(\tilde{a_{i}})\eta^{\prime}_{i}}{\varphi(a)}\in
I$. Now,
$\frac{\sum_{i=1}^{n}\eta_{i}\mathbb{E}_{\varphi}(\tilde{a_{i}})\eta^{\prime}_{i}}{\varphi(a)}$
being close to $1$ is an invertible operator in $I$. Hence,
$I=C_{r}^{*}(\Gamma)$. This is a contradiction. Therefore, $a=0$.

It is clear that $C_{r}^{*}(\Gamma)\cap\mathcal{A}=\mathbb{C}$. Let us now
show that $\mathcal{A}\cap\mathcal{B}=\mathbb{C}$. Let
$\tilde{a}\in\mathcal{B}\cap\mathcal{A}$. Then, we can find $a\in
C_{r}^{*}(\Gamma)$ and $b\in\overline{\mathcal{A}_{I}}$ such that
$\tilde{a}=a+b$. Hence, $b=\tilde{a}-a$. Applying the canonical conditional
expectation $\mathbb{E}$, we get that $\mathbb{E}(b)=\tilde{a}-\tau_{0}(a)$.
In particular, $\tilde{a}=\mathbb{E}(b)+\tau_{0}(a)$. Therefore,
$b=\mathbb{E}(b)+\tau_{0}(a)-a$. Since $b\in\overline{\mathcal{A}_{I}}$ and
$\mathbb{E}_{\varphi}(\overline{\mathcal{A}_{I}})\subset\overline{\mathcal{A}_{I}}$,
we see that $b-\mathbb{E}_{\varphi}(b)\in\overline{\mathcal{A}_{I}}$. Now,

$\displaystyle b-\mathbb{E}_{\varphi}(b)$
$\displaystyle=\mathbb{E}(b)+\tau_{0}(a)-a-\mathbb{E}_{\varphi}\left(\mathbb{E}(b)+\tau_{0}(a)-a\right)$
$\displaystyle=\mathbb{E}(b)-\varphi\left(\mathbb{E}(b)\right)\in\mathcal{A}$

Since $\overline{\mathcal{A}_{I}}\cap\mathcal{A}=\\{0\\}$ and
$(b-\mathbb{E}_{\varphi}(b))\in\overline{\mathcal{A}_{I}}$, it follows that
$b=\mathbb{E}_{\varphi}(b)$. As a consequence, we see that
$\tilde{a}=a+b=a+\mathbb{E}_{\varphi}(b)$. Applying the canonical conditional
expectation on both sides, we see that

$\tilde{a}=\mathbb{E}(\tilde{a})=\mathbb{E}(a+\mathbb{E}_{\varphi}(b))=\tau_{0}\left(a+\mathbb{E}_{\varphi}(b)\right)\in\mathbb{C}.$

∎

For the group $\mathbb{Z}/2\mathbb{Z}$ with two elements, it is possible to
construct an intermediate algebra that is not a crossed product in a canonical
way without using the ideals from $C_{r}^{*}(\mathbb{Z}/2\mathbb{Z})$.

######  Example 5.6 (Intermediate algebra for $\mathbb{Z}/2\mathbb{Z}$).

Let $\Gamma=\mathbb{Z}/2\mathbb{Z}=\\{e,s\\}$ and $X$, a compact Hausdorff
$\Gamma$-space which admits a $\Gamma$-invariant measure with full support.
Assume that $X$ has more than two points. Let $\mathcal{A}=C(X)$. Let

$J=\left\\{a_{e}+a_{s}\lambda(s):a_{e},a_{s}\in\mathbb{C},\leavevmode\nobreak\
a_{e}+a_{s}=0\right\\}$

be the non-trivial augmentation ideal. Let
$\mathcal{B}=C_{r}^{*}(\Gamma)+\overline{\mathcal{A}_{J}}$. In the light of
algebra and interm, it is enough to show that
$\mathbb{E}(\mathcal{B})\neq\mathbb{C}$. Since $|X|>2$, there exist $x_{1}$
and $x_{2}\in X$ such that $x_{1}\not\in\Gamma x_{2}$. Let $U$ and $V$ be two
open neighborhoods containing $x_{1}$ and $\Gamma x_{2}$ respectively such
that $U\cap V=\emptyset$. Let $f\in C(X)$ be such that $0\leq f\leq 1$,
$f(x_{1})=1$ and $\text{Supp}(f)\subset U$. Then,
$a=(\lambda(e)-\lambda(s))f(\lambda(e)-\lambda(s))=f-f\lambda(s)-s.f\lambda(s)+s.f$.
So, applying the canonical conditional expectation $\mathbb{E}$ on it, we see
that $\mathbb{E}(a)=f+s.f$. Let us now observe that
$\mathbb{E}(a)(x_{1})=f(x_{1})+f(s^{-1}x_{1})\geq 1$. On the other hand,
$\mathbb{E}(a)(x_{2})=f(x_{2})+f(sx_{2})=0$ since $x_{2},sx_{2}\in V$ and
$V\cap U=\emptyset$.

The following example shows that the assumption on $X$ in the above example is
necessary as long as the action is non-trivial.

######  Example 5.7.

ex:necessaryassumption Let $X=\\{x_{1},x_{2}\\}$ be a two point space and
$\Gamma=\\{e,s\\}$ denote $\mathbb{Z}/2\mathbb{Z}$. In this case,
$C(X)\rtimes_{r}\Gamma$ can be identified with $\mathbb{M}_{2}(\mathbb{C)}$
via the map

$u\lambda(e)+v\lambda(s)\mapsto\begin{bmatrix}u(x_{1})&v(x_{1})\\\
v(x_{2})&u(x_{2})\end{bmatrix}.$

Moreover, under this identification, we see that:

$C^{*}_{r}(\Gamma)\mapsto\left\\{\left.\begin{bmatrix}z&w\\\
w&z\end{bmatrix}\right|z,w\in{\mathbb{C}}\right\\},\qquad
C(X)\mapsto\left\\{\left.\begin{bmatrix}z&0\\\
0&w\end{bmatrix}\right|z,w\in{\mathbb{C}}\right\\}.$

It can now be easily verified that there is no intermediate algebra between
$C_{r}^{*}(\Gamma)$ and $C(X)\rtimes_{r}\Gamma$.

We now proceed to deal with all the other non-$C^{*}$-simple groups. We start
with the infinite i.c.c groups.

###  5.3. Intermediate algebras for i.c.c. group actions.

Let us recall that an infinite group $\Gamma$ is i.c.c.. if the conjugacy
class of every non-trivial group element is infinite.

######  Lemma 5.8.

lem:anelementseparating Let $\Gamma$ be an i.c.c. group. Let
$F\subset\Gamma\setminus\\{e\\}$ be a finite subset. Then, there exists an
element $t\in\Gamma$ such that $tFt^{-1}\cap F=\emptyset$.

###### Proof.

Write $F=\\{s_{i}:1\leq i\leq n\\}$. Let
$\Gamma(s_{i},s_{j})=\\{t\in\Gamma:ts_{i}t^{-1}=s_{j}\\}$. If the claim
doesn’t hold, $\Gamma=\cup_{i,j}\Gamma(s_{i},s_{j})$. Whenever
$\Gamma(s_{i},s_{j})$ is non-empty, it is a left coset of
$\Gamma(s_{i},s_{i})$. It is known that, as a consequence, at least one of the
subgroups $\Gamma(s_{i},s_{i})$ is of finite index (see the first few
paragraphs of [Neu54]). Consequently, $s_{i}$ has a finite conjugacy class,
contradicting our assumption that $\Gamma$ is an i.c.c. group. ∎

######  Theorem 5.9.

i.c.c-intermediate Let $\Gamma$ be an i.c.c. group and $I\leq
C_{r}^{*}(\Gamma)$, a non-trivial closed two-sided ideal. Let
$\mathcal{A}\neq\mathbb{C}$ be a unital $\Gamma$-$C^{*}$-algebra with a
faithful $\Gamma$-invariant state $\varphi$. Then,
$\mathcal{B}=C_{r}^{*}(\Gamma)+\overline{A_{I}}$ is not a crossed product in a
canonical way.

###### Proof.

The fact that $\mathcal{B}$ is an intermediate algebra is a consequence of
algebra. It also follows from interm that
$\mathcal{B}\cap\mathcal{A}=\mathbb{C}$ so that
$\mathcal{B}\lneqq\mathcal{A}\rtimes_{r}\Gamma$. We show that
$\overline{\mathbb{E}\left(\mathcal{A}_{I}\right)}=\mathcal{A}$. This will
show that $C^{*}_{r}(\Gamma)\lneqq\mathcal{B}$ and complete the proof.

Let $a\in\mathcal{A}$ and $\eta\in I$ be fixed. Without any loss of
generality, assume that $\|a\|=\|\eta\|=1$. Moreover, replacing $\eta$ by
$\eta^{*}\eta$ if required, we can assume that $\tau_{0}(\eta)\neq 0$. Let
$1>\epsilon>0$. We can find a finite subset $F\subset\Gamma\setminus\\{e\\}$
such that

$\left\|\eta-\sum_{s\in
F}c_{s}\lambda(s)-\tau_{0}(\eta)\right\|<\frac{\epsilon}{4}.$

This, in particular, implies that

$\left\|\sum_{s\in
F}c_{s}\lambda(s)+\tau_{0}(\eta)\right\|\leq\left\|\sum_{s\in
F}c_{s}\lambda(s)+\tau_{0}(\eta)-\eta\right\|+\|\eta\|<2.$

Let $t=t(F,\epsilon)\in\Gamma$ (guaranteed by lem:anelementseparating) be such
that $tFt^{-1}\cap F=\emptyset$. Then,

$\left\|\lambda(t)\eta^{*}\lambda(t^{-1})-\sum_{s\in
F}\overline{c_{s}}\lambda(ts^{-1}t^{-1})-\overline{\tau_{0}(\eta)}\right\|<\frac{\epsilon}{4}.$

Let us also observe that

$\displaystyle\mathbb{E}\left(\left(\sum_{s\in
F}c_{s}\lambda(s)+\tau_{0}(\eta)\right)a\left(\sum_{s\in
F}\overline{c_{s}}\lambda(ts^{-1}t^{-1})+\overline{\tau_{0}(\eta)}\right)\right)$
$\displaystyle=\mathbb{E}\left(\sum_{s,u\in
F}c_{s}\overline{c_{u}}(s.a)\lambda(stu^{-1}t^{-1})+\sum_{s\in
F}c_{s}\overline{\tau_{0}(\eta)}(s.a)\lambda(s)+\sum_{u\in