# The Diagonal dimension of sub-$\mathrm{C}^{*}$-algebras

Kang Li  Department of Mathematics, FAU Erlangen–Nürnberg,
Cauerstrasse 11, 91058 Erlangen, Deutschland<EMAIL_ADDRESS>,  Hung-Chang
Liao  Department of Mathematics and Statistics, University of Ottawa,
585 King Edward Avenue, Ottawa, Ontario, KiN 6N5, Canada<EMAIL_ADDRESS>and
Wilhelm Winter  Mathematisches Institut der WWU Münster,
Einsteinstrasse 62, 48149 Münster, Deutschland<EMAIL_ADDRESS>
###### Abstract.

We introduce _diagonal dimension_ , a version of nuclear dimension for
diagonal sub-$\mathrm{C}^{*}$-algebras (sometimes also referred to as diagonal
$\mathrm{C}^{*}$-pairs). Our concept has good permanence properties and
detects more refined information than nuclear dimension. In many situations it
is precisely how dynamical information is encoded in an associated
$\mathrm{C}^{*}$-pair.

For free actions on compact Hausdorff spaces, diagonal dimension of the
crossed product with its canonical diagonal is bounded above by a product
involving Kerr’s tower dimension of the action and covering dimension of the
space. It is bounded below by the dimension of the space, by the asymptotic
dimension of the group, and by the fine tower dimension of the action. For a
locally compact, Hausdorff, étale groupoid, diagonal dimension of the groupoid
$\mathrm{C}^{*}$-algebra is bounded below by the dynamic asymptotic dimension
of the groupoid. For free Cantor dynamical systems, diagonal dimension
(defined at the level of the crossed product $\mathrm{C}^{*}$-algebra) and
tower dimension (an entirely dynamical notion) agree on the nose. Similarly,
for a finitely generated group diagonal dimension of its uniform Roe algebra
with the canonical diagonal agrees precisely with asymptotic dimension of the
group. This statement also holds for uniformly bounded metric spaces. We apply
the lower bounds above to a number of further examples which show how diagonal
dimension keeps track of information not seen by nuclear dimension.

today

## Introduction

With every topological dynamical system $G\curvearrowright X$, where $G$ is a
discrete group and $X$ is a compact Hausdorff space, one can naturally and
canonically associate $\mathrm{C}^{*}$-algebras via various crossed product
constructions.

Important and notoriously hard questions then are to what extent
$\mathrm{C}^{*}$-algebras remember the underlying dynamical systems and how to
extract dynamical information from the algebras. In this paper we will be
interested in a particular aspect of the second question: How can dimension
type information of dynamical systems be read of from the associated
$\mathrm{C}^{*}$-algebras?

We will see that one can expect only very limited answers from crossed
products $C(X)\rtimes G$ alone, and that one should be prepared to also keep
track of the canonical inclusion of $C(X)$ (which exists because we assume the
group to be discrete). Then the underlying space is recorded as the spectrum
of the abelian subalgebra, and our initial questions now ask what the position
of this subalgebra can tell us about the dynamics. The problem remains hard,
but a handle is provided by the set of _normalisers_ , i.e., by elements of
the crossed product which conjugate $C(X)$ into itself.

Rigidity questions as above from the $\mathrm{C}^{*}$-algebra point of view
are particularly relevant when the group or at least the action is amenable.
In this situation there is essentially only one crossed product, which is
_nuclear_ , and we have strong tools to analyse and compare such
$\mathrm{C}^{*}$-algebras. If the action is free and minimal, the crossed
product is simple and we even have far-reaching classification results
available on the $\mathrm{C}^{*}$-side. Such classification theorems allow us
to decide whether two crossed products, at least if they have _finite nuclear
dimension_ , are isomorphic as $\mathrm{C}^{*}$-algebras by just computing
their (ordered) K-theory and determining their tracial state spaces. Here,
tracial states correspond to invariant Borel probability measures on the
underlying topological space, and we have efficient tools to compute K-theory
(which is usually easier to describe in terms of the algebra than the
dynamical system). Nuclear dimension is a notion of noncommutative covering
dimension which is defined in terms of approximations of the
$\mathrm{C}^{*}$-algebra by a uniformly bounded number (of which we think as
colours) of finite dimensional $\mathrm{C}^{*}$-algebras. If the underlying
space is finite dimensional and the group is sufficiently nice, e.g. the
integers, then the crossed product does have finite nuclear dimension, so
classification as above applies.

Dimension type properties occur at the level of dynamical systems in many and
sometimes subtle and surprising ways. On the one hand there is covering
dimension of the space, and on the other hand the group (even if it is
discrete hence zero dimensional as a topological space) may have interesting
dimension like features, most notably asymptotic dimension in the sense of
Gromov. More surprisingly, these properties often interact and yield stunning
phenomena and applications, e.g. to embedding problems in topological dynamics
(see [31, 22]), or to the structure of flow spaces which in turn has been
relevant for the Farrell–Jones conjecture (see [4, 3]).

It is an intriguing consequence of classification that one can have free and
minimal, uniquely ergodic actions of the integers on non-homeomorphic compact
spaces with isomorphic crossed product $\mathrm{C}^{*}$-algebras. Moreover,
there are classes of dynamical systems with very different dimension type
properties, yet their crossed products all have the same nuclear dimension
(namely one). This shows that a crossed product $\mathrm{C}^{*}$-algebra111We
have _reduced_ crossed products in mind, and keep track of this point of view
in our notation, even though we will stick to situations where the groups or
at least the actions are amenable, hence the full and the reduced crossed
products coincide. $C(X)\rtimes_{\mathrm{r}}G$ in itself can only be expected
to carry limited information about the underlying dynamics, and it is one of
the reasons to also consider the canonical inclusion of $C(X)$ and the set of
normalisers.

When the action is free, the sub-$\mathrm{C}^{*}$-algebra $(C(X)\subset
C(X)\rtimes_{\mathrm{r}}G)$ is in fact a _diagonal_ , i.e., $C(X)$ is maximal
abelian, it is the image of a faithful conditional expectation, normalisers
generate all of $C(X)\rtimes_{\mathrm{r}}G$ as a $\mathrm{C}^{*}$-algebra, and
every pure state on $C(X)$ extends uniquely to $C(X)\rtimes_{\mathrm{r}}G$.
One can define diagonal pairs in this sense also when there is no underlying
dynamical system; the ambient $\mathrm{C}^{*}$-algebra of such a pair
$(D\subset A)$ is then a groupoid $\mathrm{C}^{*}$-algebra. We study
normalisers of sub-$\mathrm{C}^{*}$-algebras and maps between them in Section
1.

Our main definition (Definition 2.1 below) describes a noncommutative version
of covering dimension in terms of colouring numbers of completely positive
approximations, which at the same time keep track of the given abelian
subalgebra and its normalisers.

######  Definition A.

Let $(D\subset A)$ be a sub-$\mathrm{C}^{*}$-algebra with $D$ abelian. We say
$(D\subset A)$ has _diagonal dimension_ at most $d$,
$\mathrm{dim}_{\mathrm{diag}}(D\subset A)\leq d$, if for every finite subset
$\mathcal{F}\subset A$ and $\varepsilon>0$ there exist a finite-dimensional
$\mathrm{C}^{*}$-algebra $F=F^{(0)}\oplus\ldots\oplus F^{(d)}$ with a diagonal
subalgebra $D_{F}=D^{(0)}\oplus\ldots\oplus D^{(d)}$ and completely positive
maps

$A\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}F\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}A$

such that

  1. (1)

$\psi$ is contractive,

  2. (2)

$\|\varphi\psi(a)-a\|<\varepsilon$ for every $a\in\mathcal{F}$,

  3. (3)

for each $i=0,\ldots,d$, the map $\varphi|_{F^{(i)}}$ is completely positive
contractive with order zero, i.e., it preserves orthogonality,

  4. (4)

$\psi(D)\subset D_{F}$,

  5. (5)

for each $i$, $\varphi$ maps every normaliser of $D^{(i)}$ in $F^{(i)}$ to a
normaliser of $D$ in $A$.

When $D=\\{0\\}$, then conditions (4) and (5) are trivially satisfied and
diagonal dimension of $(D\subset A)$ precisely agrees with nuclear dimension
of $A$, so our notion indeed generalises nuclear dimension to
sub-$\mathrm{C}^{*}$-algebras. If $D$ contains an approximate unit for $A$,
then finite diagonal dimension implies that $(D\subset A)$ is a diagonal in
the sense explained above. We introduce this concept and derive its basic
properties in Section 2.

As one should expect, diagonal dimension has good permanence properties with
respect to direct sums, tensor products, hereditary subalgebras, unitisations,
quotients, inductive limits, and stabilisations. The zero-dimensional case can
be characterised in terms of AF algebras with canonical diagonals. We derive
these results in Sections 3 and 4.

We are excited about the notion of diagonal dimension because it allows to
recover purely dynamical information from purely $\mathrm{C}^{*}$-algebraic
data. Let us recall the notion of _tower dimension_ introduced by Kerr in [26,
Definition 4.3], in order to explain this in more detail (a full account is
given in Section 5). Suppose $G\curvearrowright X$ is an action so that for
every finite subset $E\subset G$ there are finite families $(V_{j})_{j\in J}$
of open subsets of $X$ and $(S_{j})_{j\in J}$ of finite subsets of $G$, and a
partition $J=J^{(0)}\sqcup\ldots\sqcup J^{(d)}$ such that

  1. (1)

$\bigcup_{j\in J}S_{j}V_{j}=X$,

  2. (2)

for every $x\in X$ there are $j\in J$ and $t\in S_{j}$ such that $x\in tV_{j}$
and $Et\subset S_{j}$,

  3. (3)

for each $i\in\\{0,\ldots,d\\}$ the sets $sV_{j}$ are pairwise disjoint for
$s\in S_{j}$, $j\in J^{(i)}$.

Then, the action is said to have tower dimension at most $d$, written
$\mathrm{dim}_{\mathrm{tow}}(X,G)\leq d$.

We can now state our first main result (Theorem 5.4 below), which relates the
dynamical and $\mathrm{C}^{*}$-algebraic notions of covering dimension
described above. (We use the suggestive $\dim^{+1}$ notation whenever our
statements involve products of dimensions, since then what matters is the
number of colours, i.e., the value of the dimension plus one.)

######  Theorem B.

Let $\alpha:G\curvearrowright X$ be an action of a countable, discrete,
amenable group on a compact Hausdorff space. Then

$\displaystyle\mathrm{dim}_{\mathrm{tow}}^{+1}(X,G)$
$\displaystyle\qquad\qquad\leq\mathrm{dim}_{\mathrm{diag}}^{+1}(C(X)\subset
C(X)\rtimes_{\mathrm{r}}G)$
$\displaystyle\qquad\qquad\qquad\qquad\leq\mathrm{dim}_{\mathrm{tow}}^{+1}(X,G)\cdot\dim^{+1}(X).$

In particular, if $X$ is zero-dimensional then

$\mathrm{dim}_{\mathrm{tow}}(X,G)=\mathrm{dim}_{\mathrm{diag}}(C(X)\subset
C(X)\rtimes_{\mathrm{r}}G).$

Let us now describe how diagonal dimension is relevant for coarse geometry,
which studies ‘large-scale’ properties of metric spaces. We restrict ourselves
to discrete metric spaces of bounded geometry, which cover the important
motivating examples of finitely generated discrete groups equipped with word-
length metrics. Following Gromov, such a space $X$ is said to have _asymptotic
dimension_ at most $d$, $\mathrm{asdim}(X)\leq d$, if for every $R\geq 1$
there is a cover $\mathcal{U}$ of $X$ such that the members of $\mathcal{U}$
have uniformly bounded diameter such that every $R$-ball in $X$ intersects at
most $d+1$ members of $\mathcal{U}$

The _uniform Roe algebra_ $\mathrm{C}^{*}_{\mathrm{u}}(X)$ associated with $X$
can then be defined as the $\mathrm{C}^{*}$-algebra generated by operators
with finite propagation on the Hilbert space $\ell^{2}(X)$; see [41]. In the
case of a finitely generated group $G$, it can be identified with the crossed
product $\ell^{\infty}(G)\rtimes_{\mathrm{r}}G$ (with the action being left
translation). It was shown only recently that the uniform Roe algebra
determines $X$ as a metric space up to _coarse equivalence_ (see [5]), a
notion under which asymptotic dimension is invariant. This in particular means
that the asymptotic dimension of $X$ is encoded in the uniform Roe algebra;
the question then is how to read of this information. In [52], abstract
properties of the sub-$\mathrm{C}^{*}$-algebra
$(\ell^{\infty}(X)\subset\mathrm{C}^{*}_{\mathrm{u}}(X))$ were identified
which ensure that every such _Roe Cartan subalgebra_ yields a coarse
equivalence of the underlying spaces. With diagonal dimension at hand we can
now read of the precise value of asymptotic dimension from the uniform Roe
algebra (combining Theorem 7.7 and Corollary 7.8) as follows:

######  Theorem C.

For $X$ a discrete metric space with bounded geometry, we have

$\mathrm{dim}_{\mathrm{diag}}(\ell^{\infty}(X)\subset\mathrm{C}^{*}_{\mathrm{u}}(X))=\mathrm{asdim}(X).$

Moreover, we have

$\mathrm{dim}_{\mathrm{diag}}(B\subset\mathrm{C}^{*}_{\mathrm{u}}(X))=\mathrm{asdim}(X)$

for every Roe Cartan subalgebra $(B\subset\mathrm{C}^{*}_{\mathrm{u}}(X))$.

Topological dynamical systems and metric spaces of bounded geometry can be
studied in the joint framework of (locally compact, Hausdorff, étale)
_groupoids_ ; see [38, 21] and our Section 6 below. With such a groupoid
$\mathcal{G}$ one can associate a sub-$\mathrm{C}^{*}$-algebra
$(C_{0}(\mathcal{G}^{(0)})\subset\mathrm{C}^{*}_{\mathrm{r}}(\mathcal{G}))$,
and it turns out that every diagonal sub-$\mathrm{C}^{*}$-algebra (hence in
particular every sub-$\mathrm{C}^{*}$-algebra with finite diagonal dimension)
indeed comes from a groupoid which is _principal_ (a notion analogous to
freeness for dynamical systems) and possibly _twisted_ (a notion which we will
not require in our applications); see Proposition 6.1:

######  Proposition D.

Let $(D_{A}\subset A)$ be a nondegenerate sub-$\mathrm{C}^{*}$-algebra with
finite diagonal dimension. Then there is an – up to isomorphism uniquely
determined – twisted, étale, locally compact, Hausdorff, principal groupoid
$(\mathcal{G},\Sigma)$ such that $(D_{A}\subset A)$ is isomorphic to
$(C_{0}(\mathcal{G}^{(0)})\subset\mathrm{C}_{\mathrm{r}}^{*}(\mathcal{G},\Sigma))$.

Guentner, Willett, and Yu have generalised asymptotic dimension to the notion
of _dynamic asymptotic dimension_ for groupoids (written
$\mathrm{dad}(\mathcal{G})$; spelling out the definition requires some
preparation and we postpone it to Definition 6.5 below). The concept has been
related to tower dimension in [26], and just like tower dimension, it provides
a lower bound for diagonal dimension of the associated
sub-$\mathrm{C}^{*}$-algebra; see Theorem 6.7 below:

######  Theorem E.

Let $\mathcal{G}$ be a locally compact, Hausdorff, étale groupoid. Then

$\mathrm{dad}(\mathcal{G})\leq\mathrm{dim}_{\mathrm{diag}}(C_{0}(\mathcal{G}^{(0)})\subset\mathrm{C}^{*}_{\mathrm{r}}(\mathcal{G})).$

The theorems above consist of upper and lower bounds for diagonal dimension.
The upper bounds yield a plethora of examples of sub-$\mathrm{C}^{*}$-algebras
with finite diagonal dimension. The proofs are essentially contained in
existing ones for nuclear dimension – we more or less just have to follow
those arguments and make sure they can be adjusted to also keep track of
diagonal subalgebras.

The lower bounds show that diagonal dimension is genuinely a more sensitive
invariant than nuclear dimension. The proofs, albeit sometimes technical to
write down, are natural in the sense that the covers required by tower
dimension or by (dynamic) asymptotic dimension can be constructed explicitly
from completely positive approximations in the sense of Definition A. This
process requires a certain amount of rigidity, which is ensured by the order
zero condition (3) in tandem with the normaliser condition (5) of the
definition.

Our examples for which diagonal dimension carries interesting information
range from Cantor actions of locally finite, or virtually nilpotent, or
Grigorchuk groups, over actions of amenable and residually finite groups on
profinite completions, to universal minimal flows with finite asymptotic
dimension. We work through these examples in Section 7.

### Acknowledgments

This work was partially funded by: the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy – EXC 2044 –
390685587, Mathematics Münster – Dynamics – Geometry – Structure; the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – SFB 878; the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
427320536 – SFB 1442; ERC Advanced Grant 834267 – AMAREC.

We are grateful to Selcuk Barlak, David Kerr, Xin Li, Stuart White, and Rufus
Willett for helpful comments and many inspiring discussions.

###### Contents

  1. 1 Sub-$\mathrm{C}^{*}$-algebras and normalisers
  2. 2 Diagonal dimension
  3. 3 Permanence properties
  4. 4 Dimension zero
  5. 5 Topological dynamical systems
  6. 6 Groupoids
  7. 7 Further Examples

##  1\. Sub-$\mathrm{C}^{*}$-algebras and normalisers

For a $\mathrm{C}^{*}$-algebra $A$, we write $A_{+}$ for the set of positive
elements in $A$ and $A^{1}$ and $A^{1}_{+}$ for the norm-closed unit balls of
$A$ and $A_{+}$, respectively.

######  Definition 1.1.

If $A$ is a $\mathrm{C}^{*}$-algebra containing another
$\mathrm{C}^{*}$-algebra $D$ we write $(D\subset A)$ for this setup and call
it a _sub- $\mathrm{C}^{*}$-algebra_, or a _pair_ of
$\mathrm{C}^{*}$-algebras, or just a _$\mathrm{C}^{*}$ -pair_. A
sub-$\mathrm{C}^{*}$-algebra $(D\subset A)$ is said to be _nondegenerate_ if
$D$ contains an approximate unit for $A$.

######  Remarks 1.2.

(i) Our notation is reminiscent of that for subfactors; it indicates that all
three pieces of data – $D$, $A$, and the position of $D$ in $A$ – carry
crucial information.

(ii) A sub-$\mathrm{C}^{*}$-algebra $(D\subset A)$ is nondegenerate if and
only if the positive open unit ball of $D$ forms an approximate unit for $A$.
If $(D\subset A)$ is a nondegenerate sub-$\mathrm{C}^{*}$-algebra and $A$ is
unital, then $D$ contains the unit of $A$.

######  Definition 1.3.

Let $(D\subset A)$ be a sub-$\mathrm{C}^{*}$-algebra. An element $a\in A$ is
called a _normaliser_ of $D$ in $A$ if $aDa^{*}+a^{*}Da\subset D$. We also say
$a$ _normalises_ $D$. The collection of normalisers of $D$ in $A$ is denoted
by $\mathcal{N}_{A}(D)$.

Note that $\mathcal{N}_{A}(D)$ is closed under multiplication, involution, and
norm-limits. It is in general not closed under addition.

######  Definition 1.4 (see [29, 39, 40]).

Let $(D\subset A)$ be a sub-$\mathrm{C}^{*}$-algebra. $D$ is a _Cartan
subalgebra_ of $A$ if

  1. (0)

$(D\subset A)$ is nondegenerate,

  2. (1)

$D$ is a maximal abelian ∗-subalgebra of $A$ (a _masa_ , for short),

  3. (2)

$D$ is _regular_ , in the sense that $\mathcal{N}_{A}(D)$ generates $A$ as a
$\mathrm{C}^{*}$-algebra, and

  4. (3)

there exists a _faithful conditional expectation_ from $A$ onto $D$, i.e., a
completely positive contractive map $\Phi:A\to D$ which is injective on
$A_{+}$ and satisfies $\Phi|_{D}=\mathrm{id}_{D}$.

If, in addition, $D$ has the _unique extension property_ relative to $A$, that
is, every pure state on $D$ extends uniquely to a pure state on $A$, then $D$
is said to be a _diagonal_ in $A$. We note that condition (0) above was shown
in [36] to be redundant in connection with conditions (1) and (2).

######  Example 1.5.

Let $(D_{F}\subset F)$ be a sub-$\mathrm{C}^{*}$-algebra with $F$ (hence also
$D_{F}$) finite-dimensional. If $D_{F}$ is a masa, it is a diagonal. Moreover,
any two diagonals of $F$ are unitary conjugates of each other.

######  Definition 1.6.

If $F$ is a finite-dimensional $\mathrm{C}^{*}$-algebra and $D_{F}$ is a masa
of $F$, we say an element $v$ in $F$ is a _matrix unit with respect to
$D_{F}$_ if $v^{*}v$ and $vv^{*}$ are minimal projections in $D_{F}$. Note
that every matrix unit is a normaliser of $D_{F}$, and that, if $v$ is a
matrix unit, then so is $\lambda\cdot v$ for every $\lambda\in\mathbb{C}$ with
$|\lambda|=1$.

The set of normalisers is obviously closed under multiplication, but not at
all under addition. The next proposition, taken from [29, Example 2∘],
characterises normalisers of diagonals in finite-dimensional
$\mathrm{C}^{*}$-algebras. It nicely illustrates the role of orthogonality in
this context, and provides first evidence why order zero maps will play a
role. We will return to the matter in Proposition 1.9 and in Section 2.

######  Proposition 1.7.

If $F$ is a finite dimensional $\mathrm{C}^{*}$-algebra with a diagonal
$D_{F}$, then any normaliser for $D_{F}$ is a linear combination of pairwise
orthogonal matrix units with respect to $D_{F}$.

It is not hard to conclude from the proposition above that any _positive_
normaliser of $D_{F}$ in fact belongs to $D_{F}$. This fails if $D_{F}$ is not
maximal abelian (take for example $D_{F}=\\{0\\}$). However, any positive
normaliser will at least commute with $D_{F}$. This was shown as a general
fact also outside the finite dimensional setting in [36, Proposition 2.1]. We
include below a different proof which may be interesting in its own right. We
also use this to observe that continuous functions of positive normalisers are
again normalisers.

######  Lemma 1.8.

Let $(D\subset A)$ be a sub-$\mathrm{C}^{*}$-algebra with $D$ abelian. If $e$
is a positive contraction in $\mathcal{N}_{A}(D)$ then $e$ belongs to $A\cap
D^{\prime}$ (so if $D$ is a masa, then $e\in D$). Moreover, $f(e)$ normalises
$D$ for any continuous function $f$ on the spectrum of $e$ with $f(0)=0$.

###### Proof.

Assume for a contradiction that $\mathrm{C}^{*}(D,e)$ is not abelian. Then
there exists an irreducible representation

$\pi:\mathrm{C}^{*}(D,e)\longrightarrow\mathcal{B}(\mathcal{H})$

such that $\mathcal{H}\ncong\mathbb{C}$.

Now if $\pi(D)$ was contained in $\mathbb{C}\cdot\mathbf{1}_{\mathcal{H}}$,
then $\pi|_{\mathrm{C}^{*}(e)}$ was an irreducible representation of an
abelian $\mathrm{C}^{*}$-algebra on a Hilbert space of dimension strictly
greater than one; this is impossible, so
$\pi(D)\not\subset\mathbb{C}\cdot\mathbf{1}_{\mathcal{H}}$.

It follows that the spectrum of $\pi^{\sim}(D^{\sim})$ (where $D^{\sim}$ is
the smallest unitisation of $D$ and $\pi^{\sim}$ is the unitisation of $\pi$)
contains at least two points, at least one of which is also in the spectrum of
$\pi(D)$. But then there are $d\in D^{1}_{+}$ and $b\in(D^{\sim})^{1}_{+}$
such that $0\neq\pi(d)$, $0\neq\pi^{\sim}(b)$, and $db=0$.

For $k\in\mathbb{N}$ we now compute

$\displaystyle\|be^{k}d\|^{2}$ $\displaystyle=\|be^{k}d^{2}e^{k}b\|$
$\displaystyle\leq\|be^{k}de^{k}b\|$
$\displaystyle=\|be^{k}de^{k}b^{2}e^{k}de^{k}b\|^{\frac{1}{2}}$
$\displaystyle=\|b^{2}e^{k}de^{k}e^{k}de^{k}b^{2}\|^{\frac{1}{2}}$
$\displaystyle\leq\|b^{2}e^{k}de^{k}de^{k}b^{2}\|^{\frac{1}{2}}$
$\displaystyle=\|be^{k}de^{k}bde^{k}b^{2}\|^{\frac{1}{2}}$ $\displaystyle=0,$

where we have used that $e$ is a positive contraction normalising $D$, so that
in particular $e^{k}de^{k}\in D$ commutes with $b\in D^{\sim}$. As a
consequence we have

$b\,\mathrm{C}^{*}(D,e)\,d=\\{0\\}.$ (1.1)

Because both $\pi(d)$ and $\pi^{\sim}(b)$ are nonzero, so are the closed
linear subspaces $\overline{\pi(\mathrm{C}^{*}(D,e)d)\mathcal{H}}$ and
$\overline{\pi^{\sim}(b)\mathcal{H}}$ of $\mathcal{H}$. At the same time, they
are orthogonal by (1.1), and the subspace
$\overline{\pi(\mathrm{C}^{*}(D,e)d)\mathcal{H}}$ is invariant under
$\pi(\mathrm{C}^{*}(D,e))$. This shows that $\pi$ is not irreducible and we
have reached the desired contradiction, proving that $e$ and $D$ commute.

The second assertion is a trivial consequence of maximality together with the
first statement.

For the third statement, note that for any $d\in D$ and $0\neq
k,m\in\mathbb{N}$,

$e^{2k}de^{2m}=e^{k+m}de^{k+m}\in D$

since $e$ commutes with $D$ and $e^{k+m}$ is a normaliser of $D$. It follows
that for any even polynomial $p$ with vanishing constant term we have

$p(e)^{*}dp(e)=\bar{p}(e)dp(e)\in D.$

But any $f\in C_{0}(\sigma(e)\setminus\\{0\\})$ can be approximated in norm by
even polynomials, so $f(e)$ normalises $D$. ∎

In the remainder of this section we discuss circumstances under which
sub-$\mathrm{C}^{*}$-algebras and their normalisers are preserved under maps.

The first observation is that one cannot expect to say much without suitable
nondegeneracy conditions: If $(D\subset A)$ is a sub-$\mathrm{C}^{*}$-algebra
such that the normaliser $\mathcal{N}_{A}(D)$ is not all of $A$, then the
identity map $\mathrm{id}:(\\{0\\}\subset A)\to(D\subset A)$ preserves the
sub-$\mathrm{C}^{*}$-algebra structure, but
$A=\mathcal{N}_{A}(\\{0\\})\not\subset\mathcal{N}_{A}(D)$. Conversely,
$\mathrm{id}:(D\subset A)\to(\\{0\\}\subset A)$ sends $\mathcal{N}_{A}(D)$ to
$\mathcal{N}_{A}(\\{0\\})$, but not $D$ to $\\{0\\}$.

The second observation concerns the types of maps which may preserve
normalisers. Since the definition of normalisers involves the multiplicative
structure, one cannot expect general statements for maps which are just
linear. On the other hand, we will need to consider maps more general than
∗-homomorphisms. Lemma 1.8 suggests that positivity will play a role, and in
view of Proposition 1.7, it seems natural to consider maps preserving
orthogonality (but not necessarily the full multiplicative structure).

Recall that a completely positive (c.p. for short) map $\varphi:A\to B$
between $\mathrm{C}^{*}$-algebras is said to be _order zero_ if it preserves
orthogonality, i.e., $\varphi(a)\varphi(b)=0$ whenever $a,b\in A_{+}$ satisfy
$ab=0$. By the structure theorem for order zero maps (see [57, Theorem 3.3]),
every c.p. order zero map $\varphi:A\to B$ has the form
$\varphi(\,.\,)=h\pi_{\varphi}(\,.\,)$, where $\pi_{\varphi}$ is a
∗-homomorphism from $A$ into a larger $\mathrm{C}^{*}$-algebra (one can take
the bidual $B^{**}$, for example) and $h$ is a positive contraction in that
larger $\mathrm{C}^{*}$-algebra which commutes with the image of
$\pi_{\varphi}$. (If $A$ is unital then in fact we may take
$h=\varphi(\mathbf{1}_{A})$.) We say $\pi_{\varphi}$ is a _supporting
∗-homomorphism_ for $\varphi$.

######  Proposition 1.9.

Let $(D_{A}\subset A)$ and $(D_{B}\subset B)$ be two
sub-$\mathrm{C}^{*}$-algebras with $D_{A}$ and $D_{B}$ abelian. Let
$\varphi:A\to B$ be a positive linear map.

(i) If $(D_{B}\subset B)$ is nondegenerate and
$\varphi(\mathcal{N}_{A}(D_{A}))\subset\mathcal{N}_{B}(D_{B})$, then
$\varphi(D_{A})\subset D_{B}$.

(ii) If $\varphi$ is c.p. order zero with $\varphi(D_{A})\subset D_{B}$, and
if $D_{A}$ admits an approximate unit $(u_{\alpha})_{\alpha}$ for $A$ such
that $\varphi(u_{\alpha})D_{B}\varphi(u_{\alpha})\subset\varphi(D_{A})$ for
all $\alpha$, then
$\varphi(\mathcal{N}_{A}(D_{A}))\subset\mathcal{N}_{B}(D_{B})$.

(iii) If $(D_{A}\subset A)$ is nondegenerate and $\varphi$ is a ∗-homomorphism
such that $\varphi(D_{A})$ is a hereditary subalgebra of $D_{B}$, then
$\varphi(\mathcal{N}_{A}(D_{A}))\subset\mathcal{N}_{B}(D_{B})$.

###### Proof.

(i) (Cf. [30, Lemma 1.6].) By hypothesis, $\varphi$ maps positive normalisers
to positive normalisers. Now if $(u_{\beta})_{\beta}\subset D_{B}$ is an
approximate unit for $B$, we have for any $0\leq a\in D_{A}$

$\textstyle\varphi(a)=(\varphi(a)^{2})^{\frac{1}{2}}=(\lim_{\beta}\varphi(a)u_{\beta}\varphi(a))^{\frac{1}{2}}\subset
D_{B}.$

By linearity this implies $\varphi(D_{A})\subset D_{B}$.

(ii) Given $a\in\mathcal{N}_{A}(D_{A})$ and $d\in D_{B}$, we will show that
$\varphi(a)d\varphi(a)^{*}$ belongs to $D_{B}$. By assumption for each
$\alpha$ there exists an element $d_{\alpha}$ in $D_{A}$ such that
$\varphi(d_{\alpha})=\varphi(u_{\alpha})d\varphi(u_{\alpha})$. Let
$\pi_{\varphi}:A\to B^{**}$ be a supporting ∗-homomorphism for $\varphi$. Then

$\displaystyle\varphi(ad_{\alpha}a^{*})$
$\displaystyle=\pi_{\varphi}(a)\varphi(d_{\alpha})\pi_{\varphi}(a^{*})$
$\displaystyle=\pi_{\varphi}(a)\varphi(u_{\alpha})d\varphi(u_{\alpha})\pi_{\varphi}(a^{*})$
$\displaystyle=\varphi(au_{\alpha})d\varphi(u_{\alpha}a^{*}),$

which implies

$\textstyle\varphi(a)d\varphi(a)^{*}=\lim_{\alpha}\varphi(ad_{\alpha}a^{*})\in\varphi(D_{A})\subset
D_{B}.$

(iii) Since $\varphi(D_{A})\subset D_{B}$ is a hereditary subalgebra, we have

$\varphi(u_{\alpha})D_{B}\varphi(u_{\alpha})\subset\overline{\varphi(D_{A})D_{B}\varphi(D_{A})}=\varphi(D_{A})$

for any approximate unit $(u_{\alpha})_{\alpha}$ for $D_{A}$ and the assertion
follows from (ii). ∎

######  Remark 1.10.

One can prove a slightly more involved version of Proposition 1.9(iii) for
order zero maps. We restrict ourselves to the case of ∗-homomorphisms since it
is simpler and covers the application we particularly care about: If
$(D_{B}\subset B)$ is a sub-$\mathrm{C}^{*}$-algebra with $D_{B}$ abelian,
then $(D_{B}\subset\overline{D_{B}BD_{B}})$ is a nondegenerate
sub-$\mathrm{C}^{*}$-algebra which satisfies

$\mathcal{N}_{B}(D_{B})=\mathcal{N}_{\overline{D_{B}BD_{B}}}(D_{B}).$

This makes it possible to pass to nondegenerate sub-$\mathrm{C}^{*}$-algebras
without changing the involved sets of normalisers.

##  2\. Diagonal dimension

Below we define the diagonal dimension of a sub-$\mathrm{C}^{*}$-algebra as an
approximation property and establish some of its basic features. We derive an
equivalent characterisation which also applies to nuclear dimension and which
only involves the incoming maps. A large part of this section will be spent on
showing that finite diagonal dimension in fact implies that the subalgebra is
diagonal in the sense of Definition 1.4; see Theorem 2.10.

######  Definition 2.1.

Let $(D_{A}\subset A)$ be a sub-$\mathrm{C}^{*}$-algebra with $D_{A}$ abelian.
We say $(D_{A}\subset A)$ has _diagonal dimension_ at most $d$, written as
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)\leq d$, if for every finite
subset $\mathcal{F}\subset A$ and $\varepsilon>0$ there exist a finite-
dimensional $\mathrm{C}^{*}$-algebra $F$ with a masa $D_{F}$ and c.p. maps

$A\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}F\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}A$

such that

  1. (1)

$\psi$ is contractive,

  2. (2)

$\|\varphi\psi(a)-a\|<\varepsilon$ for every $a\in\mathcal{F}$,

  3. (3)

$F$ decomposes into $F=F^{(0)}\oplus\ldots\oplus F^{(d)}$ such that
$\varphi^{(i)}:=\varphi|_{F^{(i)}}$ is completely positive contractive (c.p.c.
for short) order zero for each $i=0,\ldots,d$,

  4. (4)

$\psi(D_{A})\subset D_{F}$,

  5. (5)

$\varphi$ maps every matrix unit with respect to $D_{F}$ into
$\mathcal{N}_{A}(D_{A})$.

As common for notions of dimension we write
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=d$ if $d$ is the least integer
such that $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)\leq d$. If no such $d$
exists we write $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=\infty$.

By a _system of c.p. approximations witnessing
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)\leq d$_ we mean a net
$(F_{\lambda},D_{F_{\lambda}},\allowbreak\psi_{\lambda},\varphi_{\lambda})_{\lambda\in\Lambda}$
of approximations as above with
$\varphi_{\lambda}\psi_{\lambda}\to\mathrm{id}_{A}$ in the point-norm
topology.

Similarly, a single approximation $(F,D_{F},\psi,\varphi)$ as in Definition
2.1 with respect to a given finite subset $\mathcal{F}\subset A$ and
$\varepsilon>0$ will be called a _c.p. approximation witnessing
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)\leq d$ for
$(\mathcal{F},\varepsilon)$_ (or for $\mathcal{F}$ within $\varepsilon$).

######  Remarks 2.2.

(i) Upon dropping conditions (4) and (5) of Definition 2.1 one recovers the
_nuclear dimension_ of $A$ (as in [58, Definition 2.1]), whence

$\mathrm{dim}_{\mathrm{nuc}}A\leq\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)$

for any sub-$\mathrm{C}^{*}$-algebra $(D_{A}\subset A)$.

If $D_{A}=\\{0\\}$ then conditions (4) and (5) are automatically satisfied, so
that nuclear dimension and diagonal dimension agree in this situation,

$\mathrm{dim}_{\mathrm{nuc}}A=\mathrm{dim}_{\mathrm{diag}}(\\{0\\}\subset A).$

If $(D_{A}\subset A)$ is nondegenerate and
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)$ is finite, then for a system of
approximations
$(F_{\lambda},D_{F_{\lambda}},\allowbreak\psi_{\lambda},\varphi_{\lambda})_{\lambda\in\Lambda}$
as in Definition 2.1 we have $\psi_{\lambda}(D_{A})\subset D_{F_{\lambda}}$
and (by Proposition 1.9(i)) $\varphi_{\lambda}(D_{F_{\lambda}})\subset D_{A}$.
Therefore, $(D_{F_{\lambda}},\psi_{\lambda}|{D_{A}},\varphi|_{F_{\lambda}})$
is a system of c.p. approximations witnessing

$\mathrm{dim}_{\mathrm{nuc}}D_{A}\leq\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A).$

(It will follow from Theorem 3.1(iii) that
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset\overline{D_{A}AD_{A}})\leq\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A)$, whence non-degeneracy is a red herring in the statement above.)

It also follows directly from the definitions that

$\mathrm{dim}_{\mathrm{nuc}}(C_{0}(X))=\mathrm{dim}_{\mathrm{diag}}(C_{0}(X)\subset
C_{0}(X))$

for any abelian $\mathrm{C}^{*}$-algebra $C_{0}(X)$.

(ii) If $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=d$ then the maps
$\varphi$ in Definition 2.1 satisfy $\|\varphi\|\leq d+1$ (but are not
necessarily contractive). Moreover, when $(D_{A}\subset A)$ is nondegenerate,
the same argument as in [58, Remark 2.2 (iv)] shows that in Definition 2.1 one
may assume that the composition $\varphi\psi$ is contractive. On the other
hand, given a positive contraction $h\in A$ we can define

$\displaystyle\hat{F}:=\mathrm{her}(\psi(h))\subset F,$
$\displaystyle\hat{\psi}:=\psi(h)^{-\frac{1}{2}}\psi(\,.\,)\psi(h)^{-\frac{1}{2}}:A\longrightarrow\hat{F},$
$\displaystyle\hat{\varphi}:=\varphi(\psi(h)^{\frac{1}{2}}\,.\,\psi(h)^{\frac{1}{2}}):\hat{F}\longrightarrow
A.$ (2.1)

Then $\hat{\varphi}$ becomes contractive and
$\hat{\varphi}\hat{\psi}(a)=\varphi\psi(a)$ for all $a\in A$ satisfying
$ha=a=ah$ (see the proof of [58, Proposition 4.3]). Note that, however, the
map $\hat{\varphi}$ in general is no longer a sum of order zero maps.

(iii) The proof of [58, Proposition 3.2] shows that (possibly after throwing
away some summands of $F$) we may assume the maps $\psi_{\lambda}$ in a system
of c.p. approximations witnessing $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A)\leq d$ to be almost order zero, i.e.,

$\|\psi_{\lambda}(a)\psi_{\lambda}(b)\|\longrightarrow 0$

whenever $a,b\in A_{+}$ satisfy $ab=0$.

Moreover, by [56, Proposition 4.2] and its proof, if $A$ is unital then for
any system of c.p. approximations witnessing
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)\leq d$ we have

$\|\varphi_{\lambda}^{(i)}\psi_{\lambda}^{(i)}(\mathbf{1}_{A})\,a-\varphi_{\lambda}^{(i)}\psi_{\lambda}^{(i)}(a)\|\longrightarrow
0$

for all $a\in A$ and $i\in\\{0,\ldots,d\\}$ (the separability assumption in
[56, Proposition 4.2] is not essential at this point).

(iv) Although we required $D_{A}$ to be abelian in Definition 2.1, this is in
fact automatic when $(D_{A}\subset A)$ has finite diagonal dimension. Indeed,
let
$(F_{\lambda},D_{F_{\lambda}},\psi_{\lambda},\varphi_{\lambda})_{\lambda\in\Lambda}$
be a system of c.p. approximations witnessing
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)\leq d$. By Remark 2.2(iii) we
may assume the maps $\psi_{\lambda}$ to be almost order zero. In particular
the map

$\bar{\psi}:D_{A}\longrightarrow\prod\nolimits_{\Lambda}D_{F_{\lambda}}\big{/}\bigoplus\nolimits_{\Lambda}D_{F_{\lambda}}$

induced by the $\psi_{\lambda}|_{D_{A}}$ has order zero. Since
$\varphi_{\lambda}\psi_{\lambda}\to\mathrm{id}_{A}$ pointwise, the map
$\bar{\psi}$ is faithful. Now a supporting ∗-homomorphism $\pi_{\bar{\psi}}$
maps $D_{A}$ into an abelian $\mathrm{C}^{*}$-algebra. As $\bar{\psi}$ is
faithful, the map $\pi_{\bar{\psi}}$ is necessarily an embedding. This shows
that $D_{A}$ is abelian.

(v) In Definition 2.1 one may replace condition (5) by the – formally
stronger, but in fact equivalent – condition

  1. (5’)

$\varphi^{(i)}(\mathcal{N}_{F^{(i)}}(D_{F^{(i)}}))\subset\mathcal{N}_{A}(D_{A}),\,i=0,\ldots,d$.

In this way one can avoid using matrix units in the definition and phrase it
more symmetrically. We nonetheless preferred to use (5) in 2.1, since matrix
units will be used heavily in proofs.

Note that the condition in (5’) is required for each colour $i$ separately.
This cannot be improved: For example, the sub-$\mathrm{C}^{*}$-algebra
$(D_{A}\subset A)=(C([0,1],D_{2})\subset C([0,1],M_{2}))$ (with $(D_{2}\subset
M_{2})$ the standard diagonal) has diagonal dimension 1, but using Proposition
1.7 one can show that diagonal dimension 1 cannot be witnessed by
approximations with
$\varphi(\mathcal{N}_{F}(D_{F}))\subset\mathcal{N}_{A}(D_{A})$.

(vi) If $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)$ is finite, then $D_{A}$
is regular in $A$: Indeed, if
$(F_{\lambda},D_{F_{\lambda}},\psi_{\lambda},\varphi_{\lambda})$ is a system
of c.p. approximations witnessing $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A)=d$, then the span of the union
$\bigcup_{\lambda}\varphi_{\lambda}(F_{\lambda})$ is dense in $A$. Since
$\varphi_{\lambda}(v)$ belongs to $\mathcal{N}_{A}(D_{A})$ for each matrix
unit $v$ in $F_{\lambda}$ and these elements span the subspace
$\varphi_{\lambda}(F_{\lambda})$, we see that $A$ is generated by
$\mathcal{N}_{A}(D_{A})$.

Next we give characterisations of nuclear dimension and of diagonal dimension
which do not involve the maps $\psi:A\to F$. In [44, Theorem 6.2], Sato gave
an explicit one-sided characterisation of decomposition rank (cf. [28]) for
unital separable $\mathrm{C}^{*}$-algebras in terms of sequence algebras. This
result could be adapted to characterise nuclear dimension as well, and also to
cover the nonunital case. Such a result would then also yield the statement
about nuclear dimension in the proposition below, but it is not at all obvious
how to handle diagonal dimension along these lines. It is also worth
mentioning that our method is different in the sense that it bypasses the
heavy machinery involving Connes’ theorem used in [44]. In the same vein, the
characterisation of nuclear dimension from Proposition 2.3 easily passes to
quotients, thus giving a proof that finite nuclear dimension passes to
quotients which does not involve the – notoriously heavy – respective
statement for nuclearity.

######  Proposition 2.3.

For a $\mathrm{C}^{*}$-algebra $A$, $\mathrm{dim}_{\mathrm{nuc}}A\leq d$ if
and only if the following holds:

If $\mathcal{F}\subset A^{1}$ is a finite subset admitting a positive
contraction $h\in A^{1}_{+}$ such that $ha=ah=a$ for all $a\in\mathcal{F}$,
then for every $\varepsilon>0$ there is a linear map $\varphi:F\to A$ such
that

  1. (1)

$F=F^{(0)}\oplus\ldots\oplus F^{(d)}$ is a finite dimensional
$\mathrm{C}^{*}$-algebra,

  2. (2)

$\varphi^{(i)}:=\varphi|_{F^{(i)}}$ is c.p.c. order zero for each $i$,

  3. (3)

for every $a\in\mathcal{F}\cup\\{h\\}$ there is
$b_{a}=b_{a}^{(0)}\oplus\ldots\oplus b_{a}^{(d)}\in F^{1}$ with

$\|\varphi(b_{a})-a\|<\varepsilon$

and, for each $i$ and each $a\in\mathcal{F}$,

$\|\varphi^{(i)}(b^{(i)}_{h})a-\varphi^{(i)}(b_{a}^{(i)})\|<\varepsilon.$

When $(D_{A}\subset A)$ is a nondegenerate sub-$\mathrm{C}^{*}$-algebra with
$D_{A}$ abelian, then we have $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A)\leq d$ if and only if the following holds:

If $\mathcal{F}\subset A^{1}$ is a finite subset admitting a positive
contraction $h\in(D_{A})^{1}_{+}$ such that $ha=ah=a$ for all
$a\in\mathcal{F}$, then for every $\varepsilon>0$ there is a linear map
$\varphi:F\to A$ satisfying conditions (1), (2) and (3) as above and, in
addition, there is a masa $D_{F}=D_{F}^{(0)}\oplus\ldots\oplus
D_{F}^{(d)}\subset F=F^{(0)}\oplus\ldots\oplus F^{(d)}$ such that for each
$i\in\\{0,\ldots,d\\}$

$\varphi^{(i)}(\mathcal{N}_{F^{(i)}}(D_{F^{(i)}}))\subset\mathcal{N}_{A}(D_{A})$
(2.2)

and such that $b^{(i)}_{h}$ can be chosen to lie in $D_{F^{(i)}}$.

Note that, if $A$ is unital, the element $h$ above can simply be taken to be
the unit of $A$.

###### Proof.

For the forward implication of the statement about nuclear dimension let us
first consider the case where $A$ is unital. We may then assume
$\mathbf{1}_{A}\in\mathcal{F}$, which implies
$h=\mathbf{1}_{A}\in\mathcal{F}$. Then take a c.p.c. approximation
$(F,\psi,\varphi)$ witnessing $\mathrm{dim}_{\mathrm{nuc}}A\leq d$ for
$\mathcal{F}$ within $\varepsilon$; by [56, Proposition 4.2] (also cf. Remark
2.2(iii)) we may assume that the approximation in addition satisfies
$\|\varphi^{(i)}\psi^{(i)}(\mathbf{1}_{A})\,a-\varphi^{(i)}\psi^{(i)}(a)\|<\varepsilon$.
With $b_{a}:=\psi(a)$ for $a\in\mathcal{F}$ this yields the characterisation
of nuclear dimension in the proposition.

If $A$ is not unital, essentially the same argument works upon modifying the
proof of [56, Proposition 4.2] to yield a nonunital version as follows: First,
replace $\mathbf{1}_{A}$ by $h$ in all places and delete the (unique
occurrence of the) word “unital”. Second, choose the system
$(F_{p},\psi_{p},\varphi_{p})$ so that (22) of [56, Proposition 4.2] holds for
$b\in\mathcal{F}\cup\\{h\\}\cup(\mathcal{F}\cup\\{h\\})^{2}$ (as opposed to
all $b\in A$). Third, change the definition of the maps $\hat{\psi}_{p}$ to

$\hat{\psi}_{p}(\,.\,):=\psi_{p}(h)^{-\frac{1}{2}}\psi_{p}(h^{\frac{1}{2}}\,.\,h^{\frac{1}{2}})\psi_{p}(h)^{-\frac{1}{2}}.$

In this way, (23) and (24) of [56, Proposition 4.2] hold for $b\in\mathcal{F}$
and $\mathbf{1}_{A}$ replaced by $h$, so that we can indeed reach
$\|\varphi^{(i)}\psi^{(i)}(h)\,a-\varphi^{(i)}\psi^{(i)}(a)\|<\varepsilon$ for
$a\in\mathcal{F}$. With $b_{a}:=\psi(a)$ for $a\in\mathcal{F}$ this yields the
characterisation of nuclear dimension in the proposition also in the nonunital
case.

We now prove the reverse.

Let $\mathcal{F}\subset A^{1}_{+}$ finite and $\eta>0$ be given. We have to
produce a c.p.c. approximation for $\mathcal{F}$ within $\eta$ witnessing
$\mathrm{dim}_{\mathrm{nuc}}(A)\leq d$. (The assumption $\mathcal{F}\subset
A^{1}_{+}$ causes no loss of generality.) By a routine argument (essentially,
conjugating the elements of $\mathcal{F}$ with an idempotent approximate
unit), we may moreover assume that there is $h\in A^{1}_{+}$ such that
$ha=ah=a$ for all $a\in\mathcal{F}$. Choose

$0<\varepsilon<\eta/3$ (2.3)

so small that the conditions in (3) will imply

$\|(\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}}\,a\,(\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}}-\varphi^{(i)}(b_{a}^{(i)})\|<\frac{\eta^{2}}{9(d+1)^{2}}$
(2.4)

(this is possible since $(\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}}$ can be
approximated by polynomials in $\varphi^{(i)}(b^{(i)}_{h})$, which in turn
will almost commute with $a$; see [55, Proposition 1.8], for instance).

Now take $\varphi:F\to A$ as in the proposition for this $\mathcal{F}$, $h$,
and $\varepsilon$. It follows from the structure theorem that an order zero
map (just like a ∗-homomorphism) out of a simple C∗-algebra is either zero or
injective. Therefore, by dropping some matrix summands of $F$ if necessary, we
may assume each of the order zero maps $\varphi^{(i)}$ to be injective.

Now if $E$ is a matrix block of $F^{(i)}$, then $\mathbf{1}_{E}$ is a minimal
central projection in $F^{(i)}$ and the map

$\bar{\varphi}_{E}:=\|\varphi^{(i)}(\mathbf{1}_{E})\|^{-1}\cdot\varphi^{(i)}|_{E}(\,.\,):E\longrightarrow
A$

is isometric. Note that we can also write this map as

$\bar{\varphi}_{E}(x)=\varphi^{(i)}(d_{E}^{-1}\,x)$ (2.5)

for $x\in E$, where
$d_{E}:=\|\varphi^{(i)}(\mathbf{1}_{E})\|\cdot\mathbf{1}_{E}\in Z(F^{(i)})$ is
an element in the centre of $F^{(i)}$ which is invertible in $E$. Again by the
structure theorem for order zero maps there is an isomorphism
$\theta:\mathrm{C}^{*}(\bar{\varphi}_{E}(E))\to C_{0}(W,E)$ for some closed
subset $W\subset(0,1]$ such that
$\theta\circ\bar{\varphi}_{E}(x)=\mathrm{id}_{W}\cdot x$ for $x\in E$. Since
we forced $\bar{\varphi}_{E}$ to be isometric, this implies that $1\in W$, and
we have a ∗-homomorphism

$\bar{\psi}_{E}:=\mathrm{ev}_{1}\circ\theta:\mathrm{C}^{*}(\bar{\varphi}_{E}(E))\longrightarrow
E$

satisfying

$\bar{\psi}_{E}\circ\bar{\varphi}_{E}=\mathrm{id}_{E}.$ (2.6)

Now we may use Arveson’s theorem to find a c.p.c. extension

$\tilde{\psi}_{E}:A\longrightarrow E$

of $\bar{\psi}_{E}$ to all of $A$.

We may do the same for each matrix block of each $F^{(i)}$ to obtain
invertible elements $d^{(i)}\in Z(F^{(i)})_{+}^{1}$ such that the c.p. order
zero maps

$\bar{\varphi}^{(i)}(\,.\,):=\varphi^{(i)}((d^{(i)})^{-1}\,.\,):F^{(i)}\longrightarrow
A$ (2.7)

are isometric, and such that there are ∗-homomorphisms

$\bar{\psi}^{(i)}:\mathrm{C}^{*}(\bar{\varphi}^{(i)}(F^{(i)}))\longrightarrow
F^{(i)}$

which extend to c.p.c. maps $\tilde{\psi}^{(i)}:A\to F^{(i)}$ with
$\tilde{\psi}^{(i)}\circ\bar{\varphi}^{(i)}=\mathrm{id}_{F^{(i)}}$. Define
projections

$q^{(i)}:=\chi_{(\frac{\eta}{3(d+1)},1]}(d^{(i)})\in Z(F^{(i)})$ (2.8)

(with $\chi_{(\frac{\eta}{3(d+1)},1]}$ the characteristic function on the
interval $(\frac{\eta}{3(d+1)},1]$) and note that

$\|\varphi^{(i)}(q^{(i)}x)-\varphi^{(i)}(x)\|\leq\frac{\eta}{3(d+1)}\|x\|$
(2.9)

for $x\in F^{(i)}$. We now define c.p. maps $\psi^{(i)}:A\to F^{(i)}$ by

$\psi^{(i)}(\,.\,):=q^{(i)}(d^{(i)})^{-1}\bar{\psi}^{(i)}((\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}}\;.\;(\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}})$
(2.10)

and $\psi:=\oplus_{i=0}^{d}\psi^{(i)}:A\to F$. The $\psi^{(i)}$ and therefore
also $\psi$ are contractive since, for any $b\in A^{1}_{+}$,

$\displaystyle\psi^{(i)}(b)$
$\displaystyle\stackrel{{\scriptstyle\eqref{2-3-1}}}{{\leq}}q^{(i)}(d^{(i)})^{-1}\bar{\psi}^{(i)}(\varphi^{(i)}(b^{(i)}_{h}))$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{2-3-1}}}}{{=}}q^{(i)}(d^{(i)})^{-1}\bar{\psi}^{(i)}\varphi^{(i)}((d^{(i)})^{-1}d^{(i)}b^{(i)}_{h})$
$\displaystyle\stackrel{{\scriptstyle\;\eqref{2-3-7-1}\;}}{{=}}q^{(i)}(d^{(i)})^{-1}\bar{\psi}^{(i)}\bar{\varphi}^{(i)}(d^{(i)}b^{(i)}_{h})$
$\displaystyle\stackrel{{\scriptstyle\;\eqref{2-3-5}\;}}{{=}}q^{(i)}(d^{(i)})^{-1}d^{(i)}b^{(i)}_{h}$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{2-3-1}}}}{{\leq}}\mathbf{1}_{F^{(i)}}.$

We now compute for $a\in\mathcal{F}$

$\displaystyle\|a-\varphi\psi(a)\|$
$\displaystyle\stackrel{{\scriptstyle\ref{prop:dim-without-
psi}(3)}}{{\leq}}\Big{\|}\sum\nolimits_{i}\varphi^{(i)}(b_{a}^{(i)})-\varphi^{(i)}\psi^{(i)}(a)\Big{\|}+\varepsilon$
$\displaystyle\stackrel{{\scriptstyle\eqref{2-3-1}}}{{=}}\Big{\|}\sum\nolimits_{i}\varphi^{(i)}(b_{a}^{(i)})$
$\displaystyle\mathrel{\phantom{\stackrel{{\scriptstyle\eqref{2-3-1}}}{{=}}}}{}-\varphi^{(i)}(q^{(i)}(d^{(i)})^{-1}\bar{\psi}^{(i)}((\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}}\;a\;(\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}}))\Big{\|}$
$\displaystyle\mathrel{\phantom{\stackrel{{\scriptstyle\eqref{2-3-1}}}{{=}}}}{}+\varepsilon$
$\displaystyle\stackrel{{\scriptstyle\;\eqref{2-3-2}\;}}{{\leq}}\Big{\|}\sum\nolimits_{i}\varphi^{(i)}(b_{a}^{(i)})-\varphi^{(i)}(q^{(i)}(d^{(i)})^{-1}\bar{\psi}^{(i)}(\varphi^{(i)}(b^{(i)}_{a})))\Big{\|}$
$\displaystyle\mathrel{\phantom{\stackrel{{\scriptstyle\eqref{2-3-1}}}{{=}}}}{}+(d+1)\Big{(}\frac{\eta}{3(d+1)}\Big{)}^{-1}\frac{\eta^{2}}{9(d+1)^{2}}+\varepsilon$
$\displaystyle\stackrel{{\scriptstyle\;\eqref{2-3-4}\;}}{{=}}\Big{\|}\sum\nolimits_{i}\varphi^{(i)}(b_{a}^{(i)})-\varphi^{(i)}(q^{(i)}(d^{(i)})^{-1}\bar{\psi}^{(i)}(\bar{\varphi}^{(i)}(d^{(i)}b^{(i)}_{a})))\Big{\|}$
$\displaystyle\mathrel{\phantom{\stackrel{{\scriptstyle\eqref{2-3-1}}}{{=}}}}{}+\frac{\eta}{3}+\varepsilon$
$\displaystyle\stackrel{{\scriptstyle\;\eqref{2-3-5}\;}}{{=}}\Big{\|}\sum\nolimits_{i}\varphi^{(i)}(b_{a}^{(i)})-\varphi^{(i)}(q^{(i)}b^{(i)}_{a})\Big{\|}+\frac{\eta}{3}+\varepsilon$
$\displaystyle\stackrel{{\scriptstyle\;\eqref{2-3-6}\;}}{{\leq}}(d+1)\frac{\eta}{3(d+1)}+\frac{\eta}{3}+\varepsilon$
$\displaystyle\stackrel{{\scriptstyle\;\eqref{2-3-7}\;}}{{<}}\eta.$

This shows that $(F,\psi,\varphi)$ is a c.p. approximation witnessing
$\mathrm{dim}_{\mathrm{nuc}}A\leq d$ for $\mathcal{F}$ within $\eta$.

We now turn to the statement about diagonal dimension. The forward implication
works exactly as above, upon noting that the existence of $D_{F}$ and (2.2) is
already built into the initial approximations for $(D_{A}\subset A)$ and
survives the modification along the lines of [56, Proposition 4.2] carried out
at the beginning of this proof; cf. Remark 2.2(iii).

The reverse implication essentially follows from the construction above; it
only remains to show that $\psi(D_{A})\subset D_{F}$ provided we have
$\varphi^{(i)}(\mathcal{N}_{F^{(i)}}(D_{F^{(i)}}))\subset\mathcal{N}_{A}(D_{A})$
and $b^{(i)}_{h}\in D_{F^{(i)}}$ for some masa $D_{F}\subset F$.

To see this, suppose for a contradiction that $\psi(D_{A})\not\subset D_{F}$.
Since $D_{F}\subset F$ is maximal abelian, this implies there are $a\in D_{A}$
and a rank one projection $e\in D_{F^{(i)}}$ for some $i$ such that

$e\psi^{(i)}(a)\neq\psi^{(i)}(a)e.$

Since $q^{(i)}$ and $d^{(i)}$ are central in $F^{(i)}$, by (2.10) this
furthermore implies that

$e\,\bar{\psi}^{(i)}((\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}}\;a\;(\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}})\neq\bar{\psi}^{(i)}((\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}}\;a\;(\varphi^{(i)}(b^{(i)}_{h}))^{\frac{1}{2}})\,e,$

which also reads as

$e\,\bar{\psi}^{(i)}(\varphi^{(i)}(b^{(i)}_{h})\;a)\neq\bar{\psi}^{(i)}(a\;\varphi^{(i)}(b^{(i)}_{h}))\,e,$

because $a$ and $\varphi^{(i)}(b^{(i)}_{h})$ lie in $D_{A}$, where we have
used that by Proposition 1.9(i) $\varphi^{(i)}(D_{F^{(i)}})\subset D_{A}$ for
each $i$. Next note that by (2.6)

$e=\bar{\psi}^{(i)}\bar{\varphi}^{(i)}(e)$

and that $\bar{\varphi}^{(i)}(e)$ lies in the multiplicative domain of
$\bar{\psi}^{(i)}$. But then we have

$\bar{\psi}^{(i)}(\bar{\varphi}^{(i)}(e)\,\varphi^{(i)}(b^{(i)}_{h})\;a)\neq\bar{\psi}^{(i)}(a\;\varphi^{(i)}(b^{(i)}_{h})\,\bar{\varphi}^{(i)}(e)).$

Since $\bar{\varphi}^{(i)}(D_{F^{(i)}})=\varphi^{(i)}(D_{F^{(i)}})\subset
D_{A}$ we have now reached a contradiction to
$a,\varphi^{(i)}(b^{(i)}_{h}),\bar{\varphi}^{(i)}(e)\in D_{A}$ and $D_{A}$
being abelian. ∎

######  Remark 2.4.

In Definition 2.1, instead of asking for condition (4) (i.e.,
$\psi(D_{A})\subset D_{F}$), at least in the unital case one can equivalently
ask for the _a priori_ weaker condition

  1. (4’)

$\psi(\mathbf{1}_{A})\subset D_{F}$.

This follows from Proposition 2.3 in connection with Remark 2.2(iii). It will
be useful when we derive permanence properties for diagonal dimension in
Section 3.

Diagonal dimension is particularly relevant in the nondegenerate case. The
proposition below gives a useful characterisation of this situation.

######  Proposition 2.5.

Let $(D_{A}\subset A)$ be a sub-$\mathrm{C}^{*}$-algebra with $D_{A}$ abelian.
If $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)<\infty$, then the following
are equivalent:

  1. (i)

$(D_{A}\subset A)$ is nondegenerate;

  2. (ii)

every system of approximations
$(F_{\lambda},D_{F_{\lambda}},\psi_{\lambda},\varphi_{\lambda})_{\lambda\in\Lambda}$
witnessing $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)<\infty$ satisfies
$\varphi_{\lambda}(D_{F_{\lambda}})\subset D_{A}$ for all $\lambda$;

  3. (iii)

there exists a system of approximations
$(F_{\lambda},D_{F_{\lambda}},\psi_{\lambda},\varphi_{\lambda})_{\lambda\in\Lambda}$
witnessing $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)<\infty$ such that
$\varphi_{\lambda}(D_{F_{\lambda}})\subset D_{A}$ for all $\lambda$.

###### Proof.

(i) $\Longrightarrow$ (ii) follows from Proposition 1.9(i) in connection with
condition (5’) of 2.2(v).

(ii) $\Longrightarrow$ (iii) is trivial.

(iii) $\Longrightarrow$ (i): Let $(D_{A})_{+}^{<1}$ be the upward-directed set
of all positive elements in $D_{A}$ with norm strictly less than one. We claim
that $(D_{A})^{<1}_{+}$ is an approximate unit for $A$. To see this, let
$\mathcal{F}\subset A^{1}_{+}$ be a finite subset and $\varepsilon>0$. There
is $h\in A^{1}_{+}$ such that for each $a\in\mathcal{F}$ there is a $b_{a}\in
A^{1}_{+}$ with $hb_{a}=b_{a}h=b_{a}$ and $\|a-b_{a}\|<\varepsilon$. For each
$\lambda$ define c.p.c. maps $\hat{\psi}_{\lambda}:A\to\hat{F}_{\lambda}$ and
$\hat{\varphi}_{\lambda}:\hat{F}_{\lambda}\to A$ as in Remark 2.2(ii). We then
have $\|\hat{\varphi}_{\lambda}\hat{\psi}_{\lambda}(b_{a})-b_{a}\|\to 0$,
$\|\hat{\varphi}_{\lambda}\hat{\psi}_{\lambda}(b_{a}^{2})-b_{a}^{2}\|\to 0$
and
$\|\hat{\varphi}_{\lambda}(\mathbf{1}_{\hat{F}_{\lambda}})-h\|=\|\varphi_{\lambda}\psi_{\lambda}(h)-h\|\to
0$ for each $a\in\mathcal{F}$. By [28, Lemma 3.6] we have for each
$a\in\mathcal{F}$

$\|\hat{\varphi}_{\lambda}\hat{\psi}_{\lambda}(b_{a})-\hat{\varphi}_{\lambda}(\mathbf{1}_{\hat{F}_{\lambda}})\hat{\varphi}_{\lambda}\hat{\psi}_{\lambda}(b_{a})\|\longrightarrow
0.$

As a consequence, there is an index $\lambda\in\Lambda$ such that

$\|\hat{\varphi}_{\lambda}(\mathbf{1}_{\hat{F}_{\lambda}})a-a\|\leq
5\varepsilon$

for all $a\in\mathcal{F}$.

Furthermore we have

$0\leq\hat{\varphi}_{\lambda}(\mathbf{1}_{\hat{F}_{\lambda}})=\varphi_{\lambda}\psi_{\lambda}(h)\leq\varphi_{\lambda}(\mathbf{1}_{F_{\lambda}}),$

and so $\hat{\varphi}_{\lambda}(\mathbf{1}_{\hat{F}_{\lambda}})$ lies in the
hereditary subalgebra generated by
$\varphi_{\lambda}(\mathbf{1}_{F_{\lambda}})$. But then there is
$d\in\mathrm{C}^{*}(\varphi_{\lambda}(\mathbf{1}_{F_{\lambda}}))_{+}$ with
$\|d\|<1$ such that

$\|d\hat{\varphi}_{\lambda}(\mathbf{1}_{\hat{F}_{\lambda}})-\hat{\varphi}_{\lambda}(\mathbf{1}_{\hat{F}_{\lambda}})\|<\varepsilon,$

whence

$\|da-a\|\leq\|d\hat{\varphi}_{\lambda}(\mathbf{1}_{\hat{F}_{\lambda}})a-\hat{\varphi}_{\lambda}(\mathbf{1}_{\hat{F}_{\lambda}})a\|+10\varepsilon<11\varepsilon.$

By the hypothesis of (iii) we have
$\varphi_{\lambda}(\mathbf{1}_{F_{\lambda}})\in D_{A}$, whence $d\in D_{A}$
and we have shown that $D_{A}^{<1}$ is an approximate unit for $A$. ∎

In the remainder of this section we show that a nondegenerate
sub-$\mathrm{C}^{*}$-algebra $(D_{A}\subset A)$ with finite diagonal dimension
is indeed a diagonal (Theorem 2.10). The proof amounts to verifying the
conditions of Definition 1.4 one by one. We are grateful to Selçuk Barlak and
Xin Li for drawing our attention to the unique extension property and for
showing us the argument for the proposition below.

######  Proposition 2.6.

Let $(D_{A}\subset A)$ be a nondegenerate sub-$\mathrm{C}^{*}$-algebra with
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=d<\infty$. Then $D_{A}\subset A$
is a masa with the unique extension property.

###### Proof.

We only verify the unique extension property, since by [2, Corollary 2.7] and
[2, Remark 2.6(iii)] this will imply that $D_{A}$ is a masa in $A$. (There is
also a direct argument which we do not spell out at this point, because it
proceeds along similar lines as Proposition 2.7.)

Suppose $\varrho_{1},\varrho_{2}\in\mathrm{PS}(A)$ are pure states on $A$
extending a pure state $\sigma\in\mathrm{PS}(D_{A})$ on $D_{A}$. Then $D_{A}$
is in the multiplicative domain of the $\varrho_{i}$ since $\sigma$ is a
character on $D$, i.e., $\varrho_{i}(bd)=\varrho_{i}(b)\varrho_{i}(d)$ for
$b\in A$, $d\in D_{A}$. (The statement is an easy consequence of Stinespring’s
theorem; cf. [28, Lemma 3.5].) But then for any $b\in A$, $d\in D_{A}$ and
$i=1,2$ one has

$\varrho_{i}(bd)=\varrho_{i}(b)\varrho_{i}(d)=\varrho_{i}(d)\varrho_{i}(b)=\varrho_{i}(db),$

which means that the $\varrho_{i}$ vanish on $\text{span}[D_{A},A]$.
Therefore, in order to prove $\varrho_{1}=\varrho_{2}$ it suffices to show
that the linear subspace $D_{A}+\text{span}[D_{A},A]$ is dense in $A$.

So let $a\in A$ and $\varepsilon>0$ be given. Find a c.p. approximation
$(F,D_{F},\psi,\varphi)$ witnessing $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A)=d$ for $(\\{a\\},\varepsilon)$. If we identify $(D_{F}\subset F)$ with the
direct sum
$\big{(}\bigoplus_{j=1}^{N}D_{r^{(j)}}\subset\bigoplus_{j=1}^{N}M_{r^{(j)}}\big{)}$
and write $\\{e^{(j)}_{k,\ell}\\}$ for the standard matrix units, then

$\psi(a)=\sum\nolimits_{j=1}^{N}\sum\nolimits_{k,\ell=1}^{r^{(j)}}\psi(a)^{(j)}_{k,\ell}\cdot
e^{(j)}_{k,\ell},$

where each complex number $\psi(a)^{(j)}_{k,\ell}$ is the $(k,\ell)$-matrix
entry of the $j$th summand of $\psi(a)$. For each $j\in\\{1,\ldots,N\\}$ and
$k,\ell\in\\{1,\ldots,r^{(j)}\\}$ with $k\neq\ell$,

$\displaystyle\varphi(e^{(j)}_{k,\ell})$
$\displaystyle=\varphi(e^{(j)}_{k,k}e^{(j)}_{k,\ell})-\varphi(e^{(j)}_{k,\ell}e^{(j)}_{k,k})$
$\displaystyle=\varphi^{\frac{1}{2}}(e^{(j)}_{k,k})\varphi^{\frac{1}{2}}(e^{(j)}_{k,\ell})-\varphi^{\frac{1}{2}}(e^{(j)}_{k,\ell})\varphi^{\frac{1}{2}}(e^{(j)}_{k,k})\in[D_{A},A]$

(here $\varphi^{\frac{1}{2}}$ is defined using the functional calculus for
order zero maps; cf. [57]). Since

$\displaystyle\varphi\psi(a)$
$\displaystyle=\sum\nolimits_{j=1}^{N}\sum\nolimits_{k,\ell=1}^{r^{(j)}}\psi(a)^{(j)}_{k,\ell}\cdot\varphi(e^{(j)}_{k,\ell})$
$\displaystyle=\sum\nolimits_{j=1}^{N}\sum\nolimits_{k=1}^{r^{(j)}}\psi(a)^{(j)}_{k,k}\cdot\varphi(e^{(j)}_{k,k})+\sum\nolimits_{j=1}^{N}\sum\nolimits_{k\neq\ell}\psi(a)^{(j)}_{k,\ell}\cdot\varphi(e^{(j)}_{k,\ell}),$

and since $\varphi(e^{(j)}_{k,k})\in D_{A}$ by Proposition 2.5, the element
$\varphi\psi(a)$ belongs to the linear subspace $D_{A}+\text{span}[D_{A},A]$.
∎

We now turn to the conditional expectation from $A$ onto $D_{A}$. Just like
maximality of the abelian subalgebra one can deduce its existence from the
unique extension property (see [2, Corollary 2.7]). Below we give a direct
argument which yields a concrete formula for the expectation; this will allow
us to also prove faithfulness in Proposition 2.9.

######  Proposition 2.7.

Let $(D_{A}\subset A)$ be a nondegenerate sub-$\mathrm{C}^{*}$-subalgebra with
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=d<\infty$. Let
$(F_{\lambda},D_{F_{\lambda}},\psi_{\lambda},\varphi_{\lambda})_{\lambda\in\Lambda}$
be a system of c.p. approximations witnessing
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=d$, and for each $\lambda$ let
$E_{\lambda}$ be the (unique) conditional expectation from $F_{\lambda}$ onto
$D_{F_{\lambda}}$.

Then the map $\Phi:A\to D_{A}$ given by the formula

$\Phi(a)=\lim_{\lambda}\varphi_{\lambda}E_{\lambda}\psi_{\lambda}(a)$ (2.11)

is a well-defined conditional expectation from $A$ onto $D_{A}$. The
conditional expectation is uniquely determined and does not depend on the
particular choice of the system
$(F_{\lambda},D_{F_{\lambda}},\psi_{\lambda},\varphi_{\lambda})_{\lambda\in\Lambda}$.

###### Proof.

Consider the quotient $\mathrm{C}^{*}$-algebra

$Q:=\prod\nolimits_{\Lambda}D_{A}\big{/}\bigoplus\nolimits_{\Lambda}D_{A}$

and note that the $\mathrm{C}^{*}$-norm on $Q$ is given by
$\|[(a_{\lambda})_{\lambda\in\Lambda}]\|_{Q}=\limsup_{\lambda}\|a_{\lambda}\|$.

Note also that we have a c.p. map

$\bar{\Phi}:A\longrightarrow Q$

given by

$\bar{\Phi}(a):=[(\varphi_{\lambda}E_{\lambda}\psi_{\lambda}(a))_{\lambda\in\Lambda}]$

(each composition $\varphi_{\lambda}E_{\lambda}\psi_{\lambda}$ is c.p. and has
norm at most $d+1$, and we have $\varphi_{\lambda}(D_{F_{\lambda}})\subset
D_{A}$ by Proposition 1.9(i)).

Now since
$E_{\lambda}|_{D_{F_{\lambda}}}\circ\psi_{\lambda}|_{D_{A}}=\psi_{\lambda}|_{D_{A}}$,
and since $\varphi_{\lambda}\psi_{\lambda}\to\mathrm{id}_{A}$, we see that for
$d\in D_{A}$

$\bar{\Phi}(d)=\iota_{D_{A}}(d),$ (2.12)

where $\iota_{D_{A}}:D_{A}\hookrightarrow Q$ is the canonical embedding. Since
$D_{A}$ contains an approximate unit for $A$ this also implies that
$\bar{\Phi}$ is in fact contractive. Moreover, since $\bar{\Phi}$ is
multiplicative on $D_{A}$, Stinespring’s theorem implies that $D_{A}$ is in
the multiplicative domain of $\bar{\Phi}$, i.e.,

$\bar{\Phi}(ad)=\bar{\Phi}(a)\bar{\Phi}(d)$

for $a\in A$, $d\in D_{A}$; cf. [28, Lemma 3.5].

Now fix some index $\lambda_{0}\in\Lambda$ and let $v\in
F^{(i)}_{\lambda_{0}}$ be some off-diagonal matrix unit (so that $vv^{*}$ and
$v^{*}v$ are orthogonal rank one projections in $D_{F_{\lambda_{0}}^{(i)}}$).
Define the c.p.c. order zero map $(\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}$
using order zero functional calculus (cf. [57]), then

$\varphi^{(i)}_{\lambda_{0}}(v)=(\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}(vv^{*})(\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}(v)(\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}(v^{*}v)$

and

$(\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}(D_{F_{\lambda_{0}}^{(i)}})\subset
D_{A}.$

It follows that

$\displaystyle\bar{\Phi}(\varphi^{(i)}_{\lambda_{0}}(v))$
$\displaystyle=\bar{\Phi}((\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}(vv^{*}))\,\bar{\Phi}((\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}(v))\,\bar{\Phi}((\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}(v^{*}v))$
$\displaystyle=\iota_{D_{A}}((\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}(vv^{*})\,(\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}(v^{*}v))\,\bar{\Phi}((\varphi^{(i)}_{\lambda_{0}})^{\frac{1}{3}}(v))$
$\displaystyle=0.$

From this and (2.12) we conclude

$\bar{\Phi}\circ\varphi_{\lambda_{0}}=\iota_{D_{A}}\circ\varphi_{\lambda_{0}}\circ
E_{\lambda_{0}}.$

Since $\lambda_{0}$ was arbitrary and
$A=\overline{\bigcup_{\lambda}\varphi_{\lambda}\psi_{\lambda}(A)}$ we now have

$\bar{\Phi}(A)\subset\iota_{D_{A}}(D_{A})$

and, for $a\in A$,

$\displaystyle\bar{\Phi}(a)$
$\displaystyle=\bar{\Phi}(\lim_{\lambda}\varphi_{\lambda}\psi_{\lambda}(a))$
$\displaystyle=\lim_{\lambda}\bar{\Phi}\varphi_{\lambda}\psi_{\lambda}(a)$
$\displaystyle=\lim_{\lambda}\iota_{D_{A}}\varphi_{\lambda}E_{\lambda}\psi_{\lambda}(a)$
$\displaystyle=\iota_{D_{A}}\lim_{\lambda}\varphi_{\lambda}E_{\lambda}\psi_{\lambda}(a),$

where for the second equality we have used continuity of $\bar{\Phi}$ and for
the last equality we have used that $\iota_{D_{A}}$ is an isometry. But this
means that

$(a\mapsto\Phi(a):=\lim_{\lambda}\varphi_{\lambda}E_{\lambda}\psi_{\lambda}(a))$

is indeed a well-defined c.p.c. map $A\to\iota_{D_{A}}(D_{A})\cong D_{A}$.
$\Phi$ is a conditional expectation since $\Phi|_{D_{A}}=\text{id}_{D_{A}}$.
As a consequence of the unique extension property (cf. Proposition 2.6) ,
there can be only one conditional expectation from $A$ onto $D_{A}$. ∎

Let us next turn to faithfulness of $\Phi$. The proof is nontrivial and
ultimately relies on the rigidity provided by the normaliser condition 2.1(5).
We try to highlight this phenomenon in the following Lemma, which we think is
worth pointing out since it turns approximate commutation into exact
commutation, independently of the matrix sizes. This is special since
commutativity relations are generally not at all robust under small
permutations, as was demonstrated for example by Voiculescu with his famous
almost commuting unitaries.

######  Lemma 2.8.

Let $r\in\mathbb{N}$, $0\leq\alpha<{1}/{144}$, and let $q\in M_{r}$ be a rank
one projection such that, for every $c\in D_{r}^{1}$, $\|qc-cq\|\leq\alpha$.

Then, there is a uniquely determined rank one projection $d\in D_{r}$ with
$\|d-q\|\leq 6\alpha^{\frac{1}{2}}$.

###### Proof.

Let $e_{k}\in D_{r}$, $k=1,\ldots,r$ be the standard rank one projections and
set

$p_{i}:=\sum\nolimits_{k=1}^{i}e_{k}\in D_{r},\;p_{0}:=0.$

For every $i=1,\ldots,r$ by our hypothesis we have

$\textstyle\|p_{i}qp_{i}-p_{i}qp_{i}p_{i}qp_{i}\|\leq\alpha\leq{1}/{4}.$

But then the interval
$(\frac{1}{2}-(\frac{1}{4}-\alpha)^{\frac{1}{2}},\frac{1}{2}+(\frac{1}{4}-\alpha)^{\frac{1}{2}})$
has empty intersection with the spectrum of $p_{i}qp_{i}$. As a consequence,
$\|qp_{i}q\|=\|p_{i}qp_{i}\|$ is either smaller than
$\frac{1}{2}-(\frac{1}{4}-\alpha)^{\frac{1}{2}}$ or larger than
$\frac{1}{2}+(\frac{1}{4}-\alpha)^{\frac{1}{2}}$.

Next observe that $0=\|qp_{0}q\|\leq\|qp_{1}q\|\leq\ldots\leq\|qp_{r}q\|=1$,
and so there is some $\bar{\imath}\in\\{1,\ldots,r\\}$ such that
$\|qp_{\bar{\imath}-1}q\|\leq\frac{1}{2}-(\frac{1}{4}-\alpha)^{\frac{1}{2}}$
and
$\|qp_{\bar{\imath}}q\|\geq\frac{1}{2}+(\frac{1}{4}-\alpha)^{\frac{1}{2}}$.
Now

$\textstyle\|qe_{\bar{\imath}}q\|=\|qp_{\bar{\imath}}q-qp_{\bar{\imath}-1}q\|\geq
2({1}/{4}-\alpha)^{\frac{1}{2}},$

whence ($q$ has rank one)

$\textstyle\|q-qe_{\bar{\imath}}q\|=1-\|qe_{\bar{\imath}}q\|\leq
1-2({1}/{4}-\alpha)^{\frac{1}{2}}.$

Since $\|qe_{\bar{\imath}}q\|=\|e_{\bar{\imath}}qe_{\bar{\imath}}\|$, in the
same manner one gets

$\textstyle\|e_{\bar{\imath}}-e_{\bar{\imath}}qe_{\bar{\imath}}\|\leq
1-2({1}/{4}-\alpha)^{\frac{1}{2}}.$

We can now estimate

$\displaystyle\|q-e_{\bar{\imath}}\|$
$\displaystyle\leq\|q-qe_{\bar{\imath}}q\|+\|qe_{\bar{\imath}}q-e_{\bar{\imath}}qe_{\bar{\imath}}\|+\|e_{\bar{\imath}}qe_{\bar{\imath}}-e_{\bar{\imath}}\|$
$\displaystyle\textstyle\leq
1-2({1}/{4}-\alpha)^{\frac{1}{2}}+2\alpha+1-2({1}/{4}-\alpha)^{\frac{1}{2}}$
$\displaystyle\leq 2\alpha^{\frac{1}{2}}+2\alpha+2\alpha^{\frac{1}{2}}$
$\displaystyle\textstyle\leq 6\alpha^{\frac{1}{2}}$ $\displaystyle<{1}/{2}$

and take $d:=e_{\bar{\imath}}$.

For $i\neq\bar{\imath}$ we have
$\|q-e_{i}\|\geq\|e_{i}-e_{\bar{\imath}}\|-\|q-e_{\bar{\imath}}\|\geq
1-6\alpha^{\frac{1}{2}}>{1}/{2}$, so $e_{\bar{\imath}}$ is the only rank one
projection in $D_{r}$ with $\|q-e_{\bar{\imath}}\|\leq 6\alpha^{\frac{1}{2}}$.
∎

######  Proposition 2.9.

Let $(D_{A}\subset A)$ be a nondegenerate sub-$\mathrm{C}^{*}$-algebra with
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=d<\infty$. Then the conditional
expectation $\Phi:A\to D_{A}$, as defined in Proposition 2.7, is faithful.

###### Proof.

Let
$(F_{\lambda},D_{F_{\lambda}},\psi_{\lambda},\varphi_{\lambda})_{\lambda\in\Lambda}$
be a system of c.p approximations witnessing
$\mathrm{dim}_{\mathrm{diag}}(A,D_{A})=d$. We may assume that the
$F_{\lambda}$ are nonzero (if $A=\\{0\\}$ there is nothing to show); by
Remarks 2.2(ii) and (iii) we may also assume that the compositions
$\varphi_{\lambda}\psi_{\lambda}$ are contractive and that the
$\psi_{\lambda}$ are approximately order zero, so that the induced map
$\bar{\psi}:A\to\prod F_{\lambda}/\bigoplus F_{\lambda}$ is c.p.c. order zero
with $\bar{\psi}(D_{A})\subset\prod D_{F_{\lambda}}/\bigoplus
D_{F_{\lambda}}$. We need to show that $\Phi(a)\neq 0$ for any positive
nonzero $a\in A$; we may assume $\|a\|=1$.

Choose $\lambda_{0}\in\Lambda$ such that

$\|\varphi_{\lambda_{0}}\psi_{\lambda_{0}}(a)-a\|<\frac{1}{16(d+1)^{3}}.$
(2.13)

Since $\varphi_{\lambda_{0}}$ is a sum of $d+1$ c.p.c. order zero maps, there
is a matrix summand $(M_{R},D_{R})$ of $(F_{\lambda_{0}},D_{F_{\lambda_{0}}})$
such that

$\|\varphi_{\lambda_{0}}(\mathbf{1}_{R}\psi_{\lambda_{0}}(a))\|\geq\frac{1}{2(d+1)},$

and so

$\|\mathbf{1}_{R}\psi_{\lambda_{0}}(a)\|\geq\frac{1}{2(d+1)},\quad\mbox{ and
}\quad\|\varphi_{\lambda_{0}}(\mathbf{1}_{R})\|\geq\frac{1}{2(d+1)}.$ (2.14)

Moreover, $\varphi_{\lambda_{0}}|_{M_{R}}$ is c.p.c. order zero with
$\varphi_{\lambda_{0}}(D_{R})\subset D_{A}$, and we have just seen that
$\|\varphi_{\lambda_{0}}|_{M_{R}}\|\geq{1}/{(2(d+1))}$.

Note that $\bar{\psi}\varphi_{\lambda_{0}}|_{M_{R}}:M_{R}\to\prod
F_{\lambda}/\bigoplus F_{\lambda}$ is c.p.c. order zero and that, for each
$\lambda$, $\psi_{\lambda}\varphi_{\lambda_{0}}(D_{R})\subset
D_{F_{\lambda}}$, whence $\bar{\psi}\varphi_{\lambda_{0}}(D_{R})\subset\prod
D_{F_{\lambda}}/\bigoplus D_{F_{\lambda}}\subset\prod F_{\lambda}/\bigoplus
F_{\lambda}$.

Also, since
$\|\varphi_{\lambda}\psi_{\lambda}(\varphi_{\lambda_{0}}(e_{k,l}))-\varphi_{\lambda_{0}}(e_{k,l})\|\to
0$ (where $e_{k,l}$ denote the standard matrix units of $M_{R}$), and since
each
$\varphi_{\lambda}\psi_{\lambda}=\sum_{i=0}^{d}\varphi_{\lambda}^{(i)}\psi_{\lambda}^{(i)}$,
there is $\lambda_{1}\in\Lambda$ such that for each $\lambda\geq\lambda_{1}$
there is $i\in\\{0,\ldots,d\\}$ with

$\textstyle\|\psi_{\lambda}^{(i)}\varphi_{\lambda_{0}}(e_{k,l})\|\geq\|\varphi_{\lambda}^{(i)}\psi_{\lambda}^{(i)}\varphi_{\lambda_{0}}(e_{k,l})\|\geq{1}/{(4(d+1)^{2})}.$
(2.15)

Now for each $\lambda\geq\lambda_{1}$ there is a rank one projection
$d_{1,\lambda}\in D_{F_{\lambda}}$ such that

$\textstyle
d_{1,\lambda}\psi_{\lambda}\varphi_{\lambda_{0}}(e_{1,1})=\psi_{\lambda}\varphi_{\lambda_{0}}(e_{1,1})d_{1,\lambda}=\|\psi_{\lambda}\varphi_{\lambda_{0}}(e_{1,1})\|\cdot
d_{1,\lambda}$

and such that

$\|\varphi_{\lambda}(d_{1,\lambda})\|\geq\frac{1}{4(d+1)^{2}}.$ (2.16)

Set

$d_{1}:=[(d_{1,\lambda})_{\lambda}]\in\prod D_{F_{\lambda}}\big{/}\bigoplus
D_{F_{\lambda}}$

(the values of $d_{1,\lambda}$ for $\lambda\leq\lambda_{1}$ do not matter in
the quotient), then $d_{1}$ is a projection and we have

$d_{1}\geq
d_{1}\bar{\psi}\varphi_{\lambda_{0}}(e_{1,1})=\bar{\psi}\varphi_{\lambda_{0}}(e_{1,1})d_{1}\geq\frac{1}{4(d+1)^{2}}\cdot
d_{1}.$ (2.17)

This also implies that there is

$d_{1}\leq u\leq 4(d+1)^{2}\cdot d_{1}\in\prod D_{F_{\lambda}}\big{/}\bigoplus
D_{F_{\lambda}}$

with

$u\bar{\psi}\varphi_{\lambda_{0}}(e_{1,1})=\bar{\psi}\varphi_{\lambda_{0}}(e_{1,1})u=d_{1}.$

Moreover, $u$ has a lift $(u_{\lambda})_{\lambda}\in\prod D_{F_{\lambda}}$
with $d_{1,\lambda}\leq u_{\lambda}\leq 4(d+1)^{2}\cdot d_{1,\lambda}$ for
$\lambda\geq\lambda_{1}$.

For $k=2,\ldots,R$ we now define

$d_{k}:=\bar{\psi}\varphi_{\lambda_{0}}(e_{k,1})ud_{1}u\bar{\psi}\varphi_{\lambda_{0}}(e_{1,k})\in\prod
F_{\lambda}\big{/}\bigoplus F_{\lambda},$ (2.18)

then

$\displaystyle d_{k}^{2}$
$\displaystyle=\bar{\psi}\varphi_{\lambda_{0}}(e_{k,1})ud_{1}u\bar{\psi}\varphi_{\lambda_{0}}(e_{1,k})\bar{\psi}\varphi_{\lambda_{0}}(e_{k,1})ud_{1}u\bar{\psi}\varphi_{\lambda_{0}}(e_{1,k})$
$\displaystyle=\bar{\psi}\varphi_{\lambda_{0}}(e_{k,1})ud_{1}u\bar{\psi}\varphi_{\lambda_{0}}(e_{1,1})\bar{\psi}\varphi_{\lambda_{0}}(e_{1,1})ud_{1}u\bar{\psi}\varphi_{\lambda_{0}}(e_{1,k})$
$\displaystyle=\bar{\psi}\varphi_{\lambda_{0}}(e_{k,1})ud_{1}u\bar{\psi}\varphi_{\lambda_{0}}(e_{1,k})$
$\displaystyle=d_{k},$ (2.19)

where we have used that $\bar{\psi}\varphi_{\lambda_{0}}$ is order zero and
that $d_{1}$ is a projection. Therefore, each $d_{k}$ is a projection which is
Murray–von Neumann equivalent to $d_{1}$ via the partial isometry
$v_{k}:=u\bar{\psi}\varphi_{\lambda_{0}}(e_{1,k})$.

If $(\tilde{d}_{k,\lambda})_{\lambda}\in\prod F_{\lambda}$ is a positive
contractive lift for $d_{k}$, then we may take
$\bar{d}_{k,\lambda}:=\chi_{[1/2,1]}(\tilde{d}_{k,\lambda})\in F_{\lambda}$
for each $\lambda$; it is clear that $(\bar{d}_{k,\lambda})_{\lambda}$ is a
lift of $d_{k}$ as well. Moreover, the $\bar{d}_{k,\lambda}$ are projections
of rank one for $\lambda\geq\lambda_{2}$ for some sufficiently large
$\lambda_{2}$, since by (2.18) and (2.19) the $\bar{d}_{k,\lambda}$ and
$d_{1,\lambda}$ are Murray–von Neumann equivalent for $\lambda$ large enough.

For each $k=2,\ldots,R$ we now have lifts of the $d_{k}$ in $\prod
F_{\lambda}$ consisting of rank one projections (at least for $\lambda$ large
enough). However, we will need lifts in $\prod D_{\lambda}$, as we have for
$d_{1}$. Arranging this is our next task.

Claim: For each $k=2,\ldots,R$ and $\lambda\in\Lambda$ there are projections
$d_{k,\lambda}\in D_{F_{\lambda}}$ of rank at most one such that
$(d_{k,\lambda})_{\lambda}$ lifts $d_{k}$.

To prove the claim, note first that

$\|\bar{d}_{k,\lambda}-\psi_{\lambda}\varphi_{\lambda_{0}}(e_{k,1})u_{\lambda}^{2}\psi_{\lambda}\varphi_{\lambda_{0}}(e_{1,k})\|\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}0,$

whence

$\|\varphi_{\lambda}(\bar{d}_{k,\lambda})-\varphi_{\lambda}(\psi_{\lambda}\varphi_{\lambda_{0}}(e_{k,1})u_{\lambda}^{2}\psi_{\lambda}\varphi_{\lambda_{0}}(e_{1,k}))\|\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}0.$

But

$\|\varphi_{\lambda}(\psi_{\lambda}\varphi_{\lambda_{0}}(e_{k,1})u_{\lambda}^{2}\psi_{\lambda}\varphi_{\lambda_{0}}(e_{1,k}))-\varphi_{\lambda_{0}}(e_{k,1})\varphi_{\lambda}(u_{\lambda}^{2})\varphi_{\lambda_{0}}(e_{1,k})\|\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}0$

by [28, Lemma 3.6] (the $\psi_{\lambda}\varphi_{\lambda_{0}}(e_{k,l})$ are
approximately in the multiplicative domain of the $\varphi_{\lambda}$), and
$\varphi_{\lambda_{0}}(e_{k,1})\varphi_{\lambda}(u_{\lambda}^{2})\varphi_{\lambda_{0}}(e_{1,k})\in
D_{A}$ for each $\lambda$, since the $\varphi_{\lambda_{0}}(e_{1,k})$
normalise $D_{A}$. This in particular implies that for every given

$0<\beta<\frac{1}{144\cdot 16(d+1)^{4}}$

there is $\lambda_{3}\geq\lambda_{2}\in\Lambda$ such that, for every
$\lambda\geq\lambda_{3}$ and every $c\in D_{F_{\lambda}}^{1}$ in the same
matrix summand (say $B_{\lambda}$) as the rank one projection
$\bar{d}_{k,\lambda}$, we have

$\|(\varphi_{\lambda}|_{B_{\lambda}})^{2}([\bar{d}_{k,\lambda},c])\|=\|[\varphi_{\lambda}(\bar{d}_{k,\lambda}),\varphi_{\lambda}(c)]\|<\beta.$
(2.20)

Here we have used that $\varphi_{\lambda}$ is order zero when restricted to
any matrix summand of $F_{\lambda}$, so that
$(\varphi_{\lambda}|_{B_{\lambda}})^{2}(\bar{d}_{k,\lambda}\,c)=\varphi_{\lambda}(\bar{d}_{k,\lambda})\,\varphi_{\lambda}(c)$.
For each $k\in\\{2,\ldots,R\\}$ and $\lambda\geq\lambda_{3}$,
$\bar{d}_{k,\lambda}$ is in the same matrix summand $B_{\lambda}$ as
$d_{1,\lambda}$, and so for any $x\in B_{\lambda}$ by (2.16) we have

$\textstyle\|(\varphi_{\lambda}|_{B_{\lambda}})^{2}(x)\|=\|\varphi_{\lambda}(x)\|\|\varphi_{\lambda}(\mathbf{1}_{B_{\lambda}})\|\geq\|x\|\cdot{1}/{(16(d+1)^{4})}.$
(2.21)

From (2.20) and (2.21) we see that for $\lambda\geq\lambda_{3}$ and $c\in
D_{F_{\lambda}}^{1}$ as above

$\textstyle\|[\bar{d}_{k,\lambda},c]\|\leq\beta\cdot 16(d+1)^{4}<{1}/{144}.$

Now Lemma 2.8 implies that for every $\lambda\geq\lambda_{3}$ there is a
uniquely determined rank one projection $d_{k,\lambda}\in D_{F_{\lambda}}$
such that

$\|d_{k,\lambda}-\bar{d}_{k,\lambda}\|\leq 24\beta^{\frac{1}{2}}(d+1)^{2}.$

By decreasing $\beta$ (and since the $d_{k,\lambda}$ are uniquely determined)
it also follows that
$\|[\bar{d}_{k,\lambda},d_{k,\lambda}]\|\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}0$.
But almost commuting projections which are less than $1$ apart almost agree,
whence
$\|\bar{d}_{k,\lambda}-d_{k,\lambda}\|\stackrel{{\scriptstyle\lambda}}{{\longrightarrow}}0$.
We have now found projections $d_{k,\lambda}\in D_{F_{\lambda}}$ of rank at
most one (at least for $\lambda\geq\lambda_{3}$ – for all other $\lambda$ we
may take $d_{k,\lambda}=0$) such that $[(d_{k,\lambda})_{\lambda}]=d_{k}$, and
so the claim above is established.

We are now ready to define a map

$\Delta:\prod F_{\lambda}\big{/}\bigoplus F_{\lambda}\longrightarrow\prod
F_{\lambda}\big{/}\bigoplus F_{\lambda}$

by

$\Delta(\,.\,):=\Big{(}\sum\nolimits_{k=1}^{R}d_{k}\Big{)}(\,.\,)\Big{(}\sum\nolimits_{k=1}^{R}d_{k}\Big{)}.$

By our construction, and since $\bar{\psi}\varphi_{\lambda_{0}}|_{M_{R}}$ is
order zero, we have for $m,n\in\\{1,\ldots,R\\}$

$\displaystyle\Big{(}\sum\nolimits_{k=1}^{R}d_{k}\Big{)}\bar{\psi}\varphi_{\lambda_{0}}(e_{m,n})$
$\displaystyle=d_{m}\bar{\psi}\varphi_{\lambda_{0}}(e_{m,n})$
$\displaystyle=\bar{\psi}\varphi_{\lambda_{0}}(e_{m,n})d_{n}$
$\displaystyle=\bar{\psi}\varphi_{\lambda_{0}}(e_{m,n})\Big{(}\sum\nolimits_{k=1}^{R}d_{k}\Big{)}.$

It follows that $\Delta\bar{\psi}\varphi_{\lambda_{0}}|_{M_{R}}$ is also
c.p.c. order zero, and from (2.17) we see that for $x\in M_{R}$

$\|\Delta\bar{\psi}\varphi_{\lambda_{0}}(x)\|\geq\mu\|x\|\geq\frac{1}{4(d+1)^{2}}\|x\|.$
(2.22)

.

By the formula (2.11) for the conditional expectation $\Phi$ in connection
with (2.16) and the fact that $\varphi|_{B_{\lambda}}$ is order zero we have

$\displaystyle\|\Phi(a)\|$
$\displaystyle\stackrel{{\scriptstyle\eqref{cond_exp_formula}}}{{=}}\lim_{\lambda}\|\varphi_{\lambda}E_{\lambda}\psi_{\lambda}(a)\|$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{cond_exp_formula}}}}{{\geq}}\max_{k}\limsup_{\lambda}\|\varphi_{\lambda}(d_{k,\lambda}\psi_{\lambda}(a)d_{k,\lambda})\|$
$\displaystyle\stackrel{{\scriptstyle\eqref{faithful-4}}}{{\geq}}\frac{1}{4(d+1)^{2}}\max_{k}\|d_{k}\bar{\psi}(a)d_{k}\|$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{cond_exp_formula}}}}{{\geq}}\frac{1}{4(d+1)^{2}}\max_{k}\|d_{k}\bar{\psi}(a)\|^{2}.$
(2.23)

Suppose for a contradiction that

$\max_{k}\|d_{k}\bar{\psi}(a)\|\leq\frac{1}{R\cdot 16\cdot(d+1)^{3}},$ (2.24)

then

$\Big{\|}\sum\nolimits_{k=1}^{R}d_{k}\bar{\psi}(a)\Big{\|}\leq\frac{1}{16\cdot(d+1)^{3}},$
(2.25)

whence

$\|\Delta\bar{\psi}\varphi_{\lambda_{0}}(\mathbf{1}_{R}\psi_{\lambda_{0}}(a))\|\leq\|\Delta\bar{\psi}\varphi_{\lambda_{0}}(\psi_{\lambda_{0}}(a))\|\stackrel{{\scriptstyle\eqref{2-9-6},\eqref{2-9-5}}}{{<}}\frac{2}{16\cdot(d+1)^{3}}$

and, with (2.22),

$\frac{1}{4(d+1)^{2}}\cdot\|\mathbf{1}_{R}\psi_{\lambda_{0}}(a)\|<\frac{2}{16\cdot(d+1)^{3}}.$
(2.26)

From this we get

$\frac{1}{8(d+1)^{3}}\stackrel{{\scriptstyle\eqref{2-9-7}}}{{\leq}}\frac{1}{4(d+1)^{2}}\|\mathbf{1}_{R}\psi_{\lambda_{0}}(a)\|\stackrel{{\scriptstyle\eqref{2-9-8}}}{{<}}\frac{2}{16\cdot(d+1)^{3}},$

a contradiction. This means (2.24) cannot hold, i.e.,

$\max_{k}\|d_{k}\bar{\psi}(a)\|>\frac{1}{R\cdot 16\cdot(d+1)^{3}}.$ (2.27)

We therefore have

$\|\Phi(a)\|\stackrel{{\scriptstyle\eqref{2-9-9},\eqref{2-9-11}}}{{\geq}}\frac{1}{4(d+1)^{2}}\cdot\frac{1}{(R\cdot
16\cdot(d+1)^{3})^{2}}>0.$

∎

By Propositions 2.6, 2.7, and 2.9, finite diagonal dimension of a
nondegenerate sub-$\mathrm{C}^{*}$-algebra implies the unique extension
property and the existence of a unique and faithful conditional expectation.
Regularity was observed in Remark 2.2(vi). Therefore the conditions of
Definition 1.4 are satisfied and we get:

######  Theorem 2.10.

Let $(D_{A}\subset A)$ be a nondegenerate sub-$\mathrm{C}^{*}$-algebra with
finite diagonal dimension. Then $D_{A}$ is a diagonal in $A$.

##  3\. Permanence properties

In this section we study permanence properties of diagonal dimension. Our
proofs largely follow those of the respective statements for nuclear
dimension. However, since the approximating maps have to preserve subalgebras
exactly and not just approximately, we have to make some additional effort
when it comes to quotients and inductive limits. On the other hand, for
hereditary subalgebras the argument becomes slightly easier due to the built-
in rigidity of diagonal dimension. For the reader’s convenience we collect all
the permanence properties in the theorem below. The proofs will be given in
the corresponding subsections below.

######  Theorem 3.1.

Let $(D_{A}\subset A)$ and $(D_{B}\subset B)$ be two
sub-$\mathrm{C}^{*}$-algebras with $D_{A}$ and $D_{B}$ abelian. Then we have
the following permanence properties.

  1. (i)

Direct sums:

$\displaystyle\mathrm{dim}_{\mathrm{diag}}(D_{A}\oplus D_{B}\subset A\oplus
B)$ $\displaystyle=\max\\{\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A),\mathrm{dim}_{\mathrm{diag}}(D_{B}\subset B)\\}.$

  2. (ii)

Tensor products:222In case the dimensions take finite values the involved
algebras are nuclear, so there is no need to specify which tensor product we
are working with. Also, recall that we follow Gábor Szabó’s notation and write
$\dim^{+1}(\,.\,)$ for $\dim(\,.\,)+1$ for various dimension theories whenever
there are products involved.

$\displaystyle\mathrm{dim}_{\mathrm{diag}}^{+1}(D_{A}\otimes D_{B}\subset
A\otimes B)$ $\displaystyle\leq\mathrm{dim}_{\mathrm{diag}}^{+1}(D_{A}\subset
A)\cdot\mathrm{dim}_{\mathrm{diag}}^{+1}{(D_{B}\subset B)}.$

  3. (iii)

Hereditary subalgebras: Suppose $B\subset A$ and $D_{B}\subset D_{A}$ are
hereditary subalgebras and that $(D_{B}\subset B)$ is nondegenerate. Then

$\mathrm{dim}_{\mathrm{diag}}(D_{B}\subset
B)\leq\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A).$

  4. (iv)

Unitisations: Suppose $(D_{A}\subset A)$ is nondegenerate and let $A^{\sim}$
be the smallest unitisation of $A$. Under the canonical identification of the
unitisation $D_{A}^{\sim}$ with $D_{A}+{\mathbb{C}}\mathbf{1}_{A^{\sim}}$ we
then have

$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A)=\mathrm{dim}_{\mathrm{diag}}(D_{A}^{\sim}\subset A^{\sim}).$

  5. (v)

Quotients: Suppose $(D_{A}\subset A)$ is nondegenerate, and suppose there is a
surjective ∗-homomorphism $\pi:A\to B$ such that $\pi(D_{A})=D_{B}$. Then

$\mathrm{dim}_{\mathrm{diag}}(D_{B}\subset
B)\leq\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A).$

  6. (vi)

Inductive limits: Let $((D_{i}\subset A_{i}),\varrho_{i,j})_{i,j\in I}$ be an
inductive system of sub-$\mathrm{C}^{*}$-algebras with $D_{i}$ abelian and
nondegenerate, and such that $\varrho_{i,j}$ maps $\mathcal{N}_{A_{i}}(D_{i})$
into $\mathcal{N}_{A_{j}}(D_{j})$ for all $i$ and $j$. Then $(\varinjlim
D_{i}\subset\varinjlim A_{i})$ is canonically a sub-$\mathrm{C}^{*}$-algebra
and we have

$\mathrm{dim}_{\mathrm{diag}}(\varinjlim D_{i}\subset\varinjlim
A_{i})\leq\liminf_{i\in I}(\mathrm{dim}_{\mathrm{diag}}(D_{i}\subset A_{i})).$

  7. (vii)

Stabilisations: Diagonal dimension is invariant under stabilisation with
matrices or algebras of compact operators with their canonical diagonals. More
precisely,

$\displaystyle\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)$
$\displaystyle=\mathrm{dim}_{\mathrm{diag}}(D_{A}\otimes D_{n}\subset A\otimes
M_{n})$ $\displaystyle=\mathrm{dim}_{\mathrm{diag}}(D_{A}\otimes
c_{0}({\mathbb{N}})\subset A\otimes\mathcal{K}(\ell^{2}({\mathbb{N}}))).$

Proof of Theorem 3.1(i): Direct sums.

The argument is exactly the same as for nuclear dimension; see [58,
Proposition 2.3] and [53, Proposition 2.10]. Conditions 2.1(4) and (5) will
automatically be satisfied since we may take $D_{F_{A}\oplus
F_{B}}=D_{F_{A}}\oplus D_{F_{B}}$ and we have $\mathcal{N}_{A\oplus
B}(D_{A}\oplus D_{B})=\mathcal{N}_{A}(D_{A})\oplus\mathcal{N}_{B}(D_{B})$.
$\square$

Proof of Theorem 3.1(ii): Tensor products.

Again the proof is the same as for nuclear dimension; see [58, Proposition
2.3] and [54, Proposition 3.1.4]. Condition 2.1(4) will be satisfied since
$\psi(D_{A}\otimes D_{B})\subset D_{F_{A}}\otimes D_{F_{B}}=D_{F_{A}\otimes
F_{B}}$. Condition 2.1(5) follows automatically since matrix units of
$(D_{F_{A}}\otimes D_{F_{B}}\subset F_{A}\otimes F_{B})$ are tensor products
of matrix units of $(D_{F_{A}}\subset F_{A})$ and $(D_{F_{B}}\subset F_{B})$,
respectively, hence map to tensor products of normalisers under the order zero
maps $\varphi_{A}^{(i)}\otimes\varphi_{B}^{(j)}$, which in turn are again
normalisers in $\mathcal{N}_{A\otimes B}(D_{A}\otimes D_{B})$. $\square$

Proof of Theorem 3.1(iii): Hereditary subalgebras.

Let $\mathcal{F}\subset B^{1}_{+}$ finite and $0<\varepsilon<1$ be given. We
may assume without loss of generality that there is $h\in(D_{B})^{1}_{+}$ such
that $hb=b$ for all $b\in\mathcal{F}$.

Fix $\eta:=\left(\frac{\varepsilon}{3d+4}\right)^{4}$ and take an
approximation $(F,D_{F},\psi,\varphi)$ witnessing
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)\leq d$ for
$\mathcal{F}\cup\\{h\\}$ within $\eta$.

Set $\beta:=2\eta^{\frac{1}{2}}$ and define projections
$q:=\chi_{(\beta,1]}(\psi(h))\in D_{F}$ and
$q^{(i)}:=\chi_{(\beta,1]}(\psi^{(i)}(h))\in D_{F^{(i)}}$ (where
$\chi_{(\beta,1]}$ denotes the characteristic function of the interval
$(\beta,1]$).

Define $\bar{F}:=qFq$ with summands $\bar{F}^{(i)}:=q^{(i)}F^{(i)}q^{(i)}$,
$D_{\bar{F}}:=qD_{F}q$, and a map
$\bar{\psi}(\,.\,):=q\psi(\,.\,)q:B\to\bar{F}$ with summands
$\bar{\psi}^{(i)}:B\to\bar{F}^{(i)}$. Note that $D_{\bar{F}}$ is diagonal in
$\bar{F}$, that $\bar{\psi}$ is c.p.c., and that $\bar{\psi}(D_{B})\subset
D_{\bar{F}}$.

With $f_{\beta}\in C_{0}((0,1])$ given by $f_{\beta}(t):=(t-\beta)_{+}$ define
c.p.c. order zero maps
$\bar{\varphi}^{(i)}:=f_{\beta}(\varphi^{(i)})|_{\bar{F}^{(i)}}:\bar{F}^{(i)}\to
A$ and a c.p. map $\bar{\varphi}:=\sum_{i=0}^{d}\bar{\varphi}^{(i)}:\bar{F}\to
A$.

Note that for $b\in\mathcal{F}$ we have

$\displaystyle\|\bar{\psi}(b)-\psi(b)\|$
$\displaystyle=\|q\psi(hbh)q-\psi(hbh)\|$
$\displaystyle\leq\|q\psi(hbh)(\mathbf{1}-q)\|+\|(\mathbf{1}-q)\psi(hbh)\|$
$\displaystyle\leq 2\|(\mathbf{1}-q)\psi(hbh)\|$
$\displaystyle=2\|(\mathbf{1}-q)\psi(hbh)^{2}(\mathbf{1}-q)\|^{\frac{1}{2}}$
$\displaystyle\leq 2\|(\mathbf{1}-q)\psi(hbh)(\mathbf{1}-q)\|^{\frac{1}{2}}$
$\displaystyle\leq 2\|(\mathbf{1}-q)\psi(h)(\mathbf{1}-q)\|^{\frac{1}{2}}$
$\displaystyle\leq 2\beta^{\frac{1}{2}}.$

As a consequence,

$\displaystyle\|b-\bar{\varphi}\bar{\psi}(b)\|$
$\displaystyle<\|\varphi\psi(b)-\bar{\varphi}\bar{\psi}(b)\|+\eta$
$\displaystyle\leq\|\varphi\psi(b)-\varphi\bar{\psi}(b)\|+\eta+(d+1)\beta$
$\displaystyle\leq(d+1)2\beta^{\frac{1}{2}}+\eta+(d+1)\beta$
$\displaystyle<(3d+4)\eta^{\frac{1}{4}}$ $\displaystyle=\varepsilon,$

and so $(\bar{F},\bar{\psi},\bar{\varphi})$ is indeed a c.p. approximation for
$\mathcal{F}$ within $\varepsilon$.

Since matrix units of $(D_{\bar{F}^{(i)}}\subset\bar{F}^{(i)})$ are also
matrix units of $(D_{F^{(i)}}\subset F^{(i)})$, the conditions on
$\varphi^{(i)}$ (see condition (5) of Definition 2.1) imply that for every
matrix unit $e$ of $(D_{\bar{F}^{(i)}}\subset\bar{F}^{(i)})$ the image
$\varphi^{(i)}(e)$ normalises $D_{A}$. Since $D_{B}\subset D_{A}$ is
hereditary, it follows that $\varphi^{(i)}(e)$ normalises $D_{B}$. Similarly,
$\varphi^{(i)}(q^{(i)})$ normalises $D_{B}$ for each $i$, and so does every
element of $\mathrm{C}^{*}(\varphi^{(i)}(q^{(i)}))_{+}$ by Lemma 1.8. But by
the structure theorem for order zero maps and the definition of order zero
functional calculus we have

$\bar{\varphi}^{(i)}(e)=f_{\beta}(\varphi^{(i)})(e)\in\overline{\mathrm{C}^{*}(\varphi^{(i)}(q^{(i)}))_{+}\varphi^{(i)}(e)},$

and so $\bar{\varphi}^{(i)}(e)$ normalises $D_{B}$ for every $i$ and every
matrix unit $e$ of $(D_{\bar{F}^{(i)}}\subset\bar{F}^{(i)})$.

It remains to check that $\bar{\varphi}(\bar{F})\subset B$; for this it will
suffice to prove that $\bar{\varphi}^{(i)}(q^{(i)})\in B$ for each $i$.

Define $\bar{D}^{(i)}:=\mathrm{C}^{*}(\varphi^{(i)}(q^{(i)}),D_{B})$ and note
that by Lemma 1.8 $\bar{D}^{(i)}$ is commutative. Moreover, we have

$0\leq\bar{\varphi}^{(i)}(q^{(i)})=f_{\beta}(\varphi^{(i)}(q^{(i)}))\in\bar{D}^{(i)}_{+}.$

Let $\varrho$ be a character on $\bar{D}^{(i)}$, and suppose
$\varrho(\bar{\varphi}^{(i)}(q^{(i)}))>0$. Then

$\textstyle\beta\leq\varrho(\varphi^{(i)}(q^{(i)}))\leq\frac{1}{\beta}\varrho(\varphi^{(i)}(\psi^{(i)}(h))),$

whence

$\beta^{2}\leq\varrho(\varphi\psi(h))\leq\varrho(h)+\eta<\varrho(h)+\beta^{2}.$

It follows that $\varrho(h)>0$, and since $\varrho$ was arbitrary and
$B\subset A$ is hereditary this now implies that
$\bar{\varphi}^{(i)}(q^{(i)})\in\overline{h\bar{D}^{(i)}h}\subset\overline{hAh}\subset
B$, as desired. $\square$

Proof of Theorem 3.1(iv): Unitisations.

Since $A$ and $D_{A}$ are ideals of $A^{\sim}$ and $D_{A}^{\sim}$,
respectively, Theorem 3.1(iii) shows that
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A)\leq\mathrm{dim}_{\mathrm{diag}}(D_{A}^{\sim}\subset A^{\sim})$.

To prove the reverse inequality, we essentially follow the argument for
decomposition rank in [28, Proposition 3.11]. We spell out the details, since
here we cannot assume the downwards map $\psi$ to be almost multiplicative,
and at the same time we have to keep track of the diagonal.

It is enough to show that given $0<\varepsilon\leq 1$ and
$b_{1},\ldots,b_{m}\in A_{+}^{1}$ we can find a c.p. approximation
$(\bar{F},D_{\bar{F}},\bar{\psi},\bar{\varphi})$ witnessing
$\mathrm{dim}_{\mathrm{diag}}(D_{A}^{\sim}\subset
A^{\sim})\leq\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=:d<\infty$ for
$\\{b_{1},\ldots,b_{m},\mathbf{1}_{A^{\sim}}\\}$ within $\varepsilon$. By
nondegeneracy of $D_{A}$, upon slightly perturbing the $b_{j}$ we may moreover
assume that there exist positive contractions $h_{0},h_{1}\in D_{A}$ such that
$h_{0}h_{1}=h_{1}$ and $h_{1}b_{j}=b_{j}$ for each $j\in\\{0,\ldots,m\\}$.

Choose $0<\gamma<(\epsilon/6)^{4}$, and set $\delta:=\gamma^{4}/4$. Take a
c.p. approximation

$\Big{(}F=\bigoplus\nolimits_{i=0}^{d}F^{(i)},D_{F}=\bigoplus\nolimits_{i=0}^{d}D_{F^{(i)}},\psi,\varphi\Big{)}$

witnessing $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=d$ for
$\\{h_{0},h_{1},b_{1},\ldots,b_{m}\\}$ within $\delta$. Set

$p:=\chi_{[\delta^{\frac{1}{2}},1]}(\psi(h_{1}))\in F,$

where $\chi_{[\delta^{\frac{1}{2}},1]}$ denotes the characteristic function of
the interval $[\delta^{\frac{1}{2}},1]$.

Notice that we have $p\leq\delta^{-\frac{1}{2}}\cdot\psi(h_{1})$. Let
$p^{(0)}:=\mathbf{1}_{F^{(0)}}p$ and
$q:=p^{(0)}+\sum_{i=1}^{d}\mathbf{1}_{F^{(i)}}$. Observing that
$\varphi(p^{(0)})$, $\varphi(1_{F})$, $h_{0}$, and $\mathbf{1}_{A^{\sim}}$ all
pairwise commute, we then have

$\displaystyle\|\varphi(p^{(0)})(\mathbf{1}_{A^{\sim}}-h_{0})\|$
$\displaystyle=\|(\mathbf{1}_{A^{\sim}}-h_{0})^{\frac{1}{2}}\varphi(p^{(0)})(\mathbf{1}_{A^{\sim}}-h_{0})^{\frac{1}{2}}\|$
$\displaystyle\leq\|(\mathbf{1}_{A^{\sim}}-h_{0})^{\frac{1}{2}}\varphi(p)(\mathbf{1}_{A^{\sim}}-h_{0})^{\frac{1}{2}}\|$
$\displaystyle\leq\|(\mathbf{1}_{A^{\sim}}-h_{0})^{\frac{1}{2}}\varphi\psi(h_{1})(\mathbf{1}_{A^{\sim}}-h_{0})^{\frac{1}{2}}\|\cdot\frac{1}{\delta^{\frac{1}{2}}}$
$\displaystyle<\|(\mathbf{1}_{A^{\sim}}-h_{0})^{\frac{1}{2}}h_{1}(\mathbf{1}_{A^{\sim}}-h_{0})^{\frac{1}{2}}\|\cdot\frac{1}{\delta^{\frac{1}{2}}}+\delta^{\frac{1}{2}}$
$\displaystyle=\delta^{\frac{1}{2}}$ (3.1)

and

$\displaystyle\|\varphi(p^{(0)})(\mathbf{1}_{A^{\sim}}-\varphi(q))\|$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{3-1-unitisations}}}}{{=}}\|\varphi(p^{(0)})(\mathbf{1}_{A^{\sim}}-\varphi(\mathbf{1}_{F})+\varphi(\mathbf{1}_{F^{(0)}}-p^{(0)}))\|$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{3-1-unitisations}}}}{{=}}\|\varphi(p^{(0)})(\mathbf{1}_{A^{\sim}}-\varphi(\mathbf{1}_{F}))\|$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{3-1-unitisations}}}}{{=}}\|\varphi(p^{(0)})^{\frac{1}{2}}(\mathbf{1}_{A^{\sim}}-\varphi(\mathbf{1}_{F}))\varphi(p^{(0)})^{\frac{1}{2}}\|$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{3-1-unitisations}}}}{{\leq}}\|\varphi(p^{(0)})^{\frac{1}{2}}(\mathbf{1}_{A^{\sim}}-\varphi\psi(h_{0}))\varphi(p^{(0)})^{\frac{1}{2}}\|$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{3-1-unitisations}}}}{{<}}\|\varphi(p^{(0)})^{\frac{1}{2}}(\mathbf{1}_{A^{\sim}}-h_{0})\varphi(p^{(0)})^{\frac{1}{2}}\|+\delta$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{3-1-unitisations}}}}{{=}}\|\varphi(p^{(0)})(\mathbf{1}_{A^{\sim}}-h_{0})\|+\delta$
$\displaystyle\stackrel{{\scriptstyle\eqref{3-1-unitisations}}}{{<}}2\delta^{\frac{1}{2}}.$
(3.2)

Define a continuous function $g\in C([0,1])$ by

$g(t)=\begin{cases}0&0\leq t\leq 2^{\frac{1}{2}}\delta^{\frac{1}{4}}\\\
t&2\delta^{\frac{1}{4}}\leq t\leq 1\\\
\text{linear}&2^{\frac{1}{2}}\delta^{\frac{1}{4}}<t<2\delta^{\frac{1}{4}}\end{cases}$

and note that (by (3.2), with functional calculus and since $\varphi(p^{(0)})$
and $\mathbf{1}_{A^{\sim}}-\varphi(q)$ commute)

$g(\varphi(p^{(0)}))\perp g(\mathbf{1}_{A^{\sim}}-\varphi(q))$

and that

$\|g(\varphi(p^{(0)}))-\varphi(p^{(0)})\|,\|g(\mathbf{1}_{A^{\sim}}-\varphi(q))-(\mathbf{1}_{A^{\sim}}-\varphi(q))\|<2^{\frac{1}{2}}\delta^{\frac{1}{4}}=\gamma.$

Applying order zero functional calculus to the c.p.c. order zero map
$\varphi|_{p^{(0)}Fp^{(0)}}:p^{(0)}Fp^{(0)}\to A$ we obtain a c.p.c. order
zero map

$\hat{\varphi}^{(0)}:=g(\varphi|_{p^{(0)}Fp^{(0)}}):p^{(0)}Fp^{(0)}\to\overline{g(\varphi(p^{(0)}))Ag(\varphi(p^{(0)}))}$

satisfying

  1. (1)

$\hat{\varphi}^{(0)}(p^{(0)}D_{F}p^{(0)})\subset\overline{g(\varphi(p^{(0)}))D_{A}g(\varphi(p^{(0)}))}$,

  2. (2)

$\hat{\varphi}^{(0)}(v)D_{A}\hat{\varphi}^{(0)}(v)^{*}\subset\overline{g(\varphi(p^{(0)}))D_{A}g(\varphi(p^{(0)}))}$
for every matrix unit $v$ in $p^{(0)}Fp^{(0)}$ with respect to
$p^{(0)}D_{F}p^{(0)}$, and

  3. (3)

$\|\hat{\varphi}^{(0)}(x)-\varphi(x)\|\leq
2^{\frac{1}{2}}\delta^{\frac{1}{4}}=\gamma$ for any positive contraction $x$
in $p^{(0)}Fp^{(0)}$.

Here, for (2) we have used Proposition 1.9(ii); (3) follows from the
properties of order zero functional calculus.

We are now ready to construct the desired c.p. approximation for
$(D_{A}^{\sim}\subset A^{\sim})$. Let

$\bar{F}:=qFq\oplus{\mathbb{C}}=p^{(0)}Fp^{(0)}\oplus\Big{(}\bigoplus\nolimits_{i=1}^{d}F^{(i)}\Big{)}\oplus{\mathbb{C}}$

and

$D_{\bar{F}}:=qD_{F}q\oplus{\mathbb{C}}.$

Define a map $\bar{\psi}:A^{\sim}\to\bar{F}$ by

$\bar{\psi}(a+\lambda\mathbf{1}_{A^{\sim}}):=q\psi(a)q+\lambda(q\oplus\mathbf{1}_{{\mathbb{C}}}).$

Then $\bar{\psi}$ is unital and completely positive (see, for example, [10,
Proposition 2.2.1]) and satisfies $\bar{\psi}(D_{A}^{\sim})\subset
D_{\bar{F}}$.

Define $\bar{\varphi}:\bar{F}\to A^{\sim}$ by

$\bar{\varphi}|_{p^{(0)}Fp^{(0)}}:=\hat{\varphi}^{(0)},\;\bar{\varphi}|_{\bigoplus_{i=1}^{d}F^{(i)}}:=\varphi|_{\bigoplus_{i=1}^{d}F^{(i)}},\;\bar{\varphi}(\mathbf{1}_{{\mathbb{C}}}):=g(\mathbf{1}_{A^{\sim}}-\varphi(q)).$

Then by construction and

Then $\bar{\varphi}^{(i)}:=\bar{\varphi}|_{F^{(i)}}$, $i=1,\ldots,d$, are all
c.p.c. order zero;
$\bar{\varphi}^{(0)}:=\bar{\varphi}|_{p^{(0)}Fp^{(0)}\oplus{\mathbb{C}}}$ is
also c.p.c. order zero since $\hat{\varphi}^{(0)}$ is, and since

$\bar{\varphi}^{(0)}(\mathbf{1}_{\bar{F}^{(0)}})=\hat{\varphi}^{(0)}(p^{(0)})=g(\varphi(p^{(0)}))\perp
g(\mathbf{1}_{A^{\sim}}-\varphi(q))=\bar{\varphi}^{(0)}(\mathbf{1}_{{\mathbb{C}}}).$

It follows from Proposition 1.9(ii) that
$\mathcal{N}_{A}(D_{A})\subset\mathcal{N}_{A^{\sim}}(D_{A}^{\sim})$, and so we
see that $\bar{\varphi}(v)\in\mathcal{N}_{A^{\sim}}(D_{A}^{\sim})$ for each
matrix unit in $\bar{F}$ with respect to $D_{\bar{F}}$.

It remains to show that $\bar{\varphi}\bar{\psi}$ approximates the identity
map on the elements $b_{1},\ldots,b_{m}$ and on $\mathbf{1}_{A^{\sim}}$. For
the unit, we have

$\displaystyle\|\bar{\varphi}\bar{\psi}(\mathbf{1}_{A^{\sim}})-\mathbf{1}_{A^{\sim}}\|$
$\displaystyle=\|\bar{\varphi}(q\oplus\mathbf{1}_{{\mathbb{C}}})-\mathbf{1}_{A^{\sim}}\|$
$\displaystyle=\Big{\|}\hat{\varphi}^{(0)}(p^{(0)})+\varphi\Big{(}\sum\nolimits_{i=1}^{d}\mathbf{1}_{F^{(i)}}\Big{)}+g(\mathbf{1}_{A^{\sim}}-\varphi(q))-\mathbf{1}_{A^{\sim}}\Big{\|}$
$\displaystyle<\|\varphi(q)+g(\mathbf{1}_{A^{\sim}}-\varphi(q))-\mathbf{1}_{A^{\sim}}\|+(2\gamma)^{\frac{1}{2}}$
$\displaystyle<\gamma+(2\gamma)^{\frac{1}{2}}<\varepsilon.$

For the elements $b_{j}$, we first compute

$\displaystyle\|\varphi((\mathbf{1}_{F^{(0)}}-p^{(0)})\psi(b_{j}))\|$
$\displaystyle=\|\varphi((\mathbf{1}_{F^{(0)}}-p^{(0)})\psi(b_{j}))\cdot\varphi(\psi(b_{j})(\mathbf{1}_{F^{(0)}}-p^{(0)}))\|^{\frac{1}{2}}$
$\displaystyle\leq\|\varphi((\mathbf{1}_{F^{(0)}}-p^{(0)})\psi(b_{j})^{2}(\mathbf{1}_{F^{(0)}}-p^{(0)}))\|^{\frac{1}{2}}$
$\displaystyle\leq\|\varphi((\mathbf{1}_{F^{(0)}}-p^{(0)})\psi(h_{1})(\mathbf{1}_{F^{(0)}}-p^{(0)}))\|^{\frac{1}{2}}$
$\displaystyle\leq\delta^{\frac{1}{4}}.$

It follows that

$\|\varphi(p^{(0)}\psi(b_{j}))-\varphi(p^{(0)}\psi(b_{j})p^{(0)})\|\leq
2\delta^{\frac{1}{4}},$

whence

$\|\varphi(p^{(0)}\psi(b_{j})p^{(0)})-\varphi(\mathbf{1}_{F^{(0)}}\psi(b_{j})\mathbf{1}_{F^{(0)}})\|\leq
3\delta^{\frac{1}{4}}.$

We now show the approximation for $b_{j}$:

$\displaystyle\|\bar{\varphi}\bar{\psi}(b_{j})-b_{j}\|$
$\displaystyle=\|\bar{\varphi}(q\psi(b_{j})q)-b_{j}\|$
$\displaystyle=\Big{\|}\bar{\varphi}^{(0)}(p^{(0)}\psi(b_{j})p^{(0)})+\varphi\Big{(}\sum\nolimits_{i=1}^{d}\mathbf{1}_{F^{(0)}}\psi(b_{j})\Big{)}-b_{j}\Big{\|}$
$\displaystyle\leq\Big{\|}\varphi(p^{(0)}\psi(b_{j})p^{(0)})+\varphi\Big{(}\sum\nolimits_{i=1}^{d}\mathbf{1}_{F^{(0)}}\psi(b_{j})\Big{)}-b_{j}\Big{\|}+(2\gamma)^{\frac{1}{2}}$
$\displaystyle\leq\|\varphi\psi(b_{j})-b_{j}\|+(2\gamma)^{\frac{1}{2}}+3\delta^{\frac{1}{4}}$
$\displaystyle<\delta+(2\gamma)^{\frac{1}{2}}+3\delta^{\frac{1}{4}}$
$\displaystyle\leq 6\gamma^{\frac{1}{4}}$ $\displaystyle<\varepsilon.$

This completes the proof. $\square$

Proof of Theorem 3.1(v): Quotients.

Upon replacing $D_{A}$, $A$, $D_{B}$, $B$ and $\pi$ by their smallest
unitisations, in view of Theorem 3.1(iv) we may assume that the algebras and
the quotient map are unital. Now let a finite subset
$\mathbf{1}_{B}\in\mathcal{F}_{B}\subset B^{1}_{+}$ and $\varepsilon>0$ be
given. Choose a finite subset $\mathbf{1}_{A}\in\mathcal{F}_{A}\subset
A^{1}_{+}$ lifting $\mathcal{F}_{B}$. Now if
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=d<\infty$ choose
$D_{F}=D_{F}^{(0)}\oplus\ldots\oplus D_{F}^{(d)}\subset
F=F^{(0)}\oplus\ldots\oplus F^{(d)}$ and $\varphi:F\to A$ as in Proposition
2.3 (with $\mathcal{F}_{A}$ in place of $\mathcal{F}$). It is then clear that
with $B$ in place of $A$ and $\pi\circ\varphi$ in place of $\varphi$
conditions (i), (ii) and (iii) of Proposition 2.3 are satisfied. Morover,
since quotient maps send normalisers to normalisers, condition (2.2) also
holds _mutatis mutandis_ (i.e., with $A$ in place of $B$ and
$\pi\circ\varphi^{(i)}$ in place of $\varphi^{(i)}$), and so the reverse
direction of Proposition 2.3 entails that
$\mathrm{dim}_{\mathrm{diag}}(D_{B}\subset B)\leq d$. $\square$

Proof of Theorem 3.1(vi): Inductive limits.

Let us first reduce to the unital case. We have canonical inclusions of ideals
$D_{i}\subset D_{i}^{\sim}$ and $A_{i}\subset A_{i}^{\sim}$, and likewise for
the inductive limits
$\varinjlim(D_{i},\varrho_{i,j})\subset(\varinjlim(D_{i},\varrho_{i,j}))^{\sim}$
and
$\varinjlim(A_{i},\varrho_{i,j})\subset(\varinjlim(A_{i},\varrho_{i,j}))^{\sim}$.
Denoting the canonical unitisation of $\varrho_{i,j}$ by
$\varrho_{i,j}^{\sim}$, there are canonical isomorphisms
$(\varinjlim(D_{i},\varrho_{i,j}))^{\sim}\cong\varinjlim(D_{i}^{\sim},\varrho_{i,j}^{\sim})$
and
$(\varinjlim(A_{i},\varrho_{i,j}))^{\sim}\cong\varinjlim(A_{i}^{\sim},\varrho_{i,j}^{\sim})$.
Now from 3.1(iv) it follows that

$\mathrm{dim}_{\mathrm{diag}}(\varinjlim D_{i}\subset\varinjlim
A_{i})=\mathrm{dim}_{\mathrm{diag}}(\varinjlim D_{i}^{\sim}\subset\varinjlim
A_{i}^{\sim}).$

Of course we also have

$\liminf_{i\in I}(\mathrm{dim}_{\mathrm{diag}}(D_{i}\subset
A_{i}))=\liminf_{i\in I}(\mathrm{dim}_{\mathrm{diag}}(D_{i}^{\sim}\subset
A_{i}^{\sim})).$

Next observe that by Proposition 1.9(i) we have $\varrho_{i,j}(D_{i})\subset
D_{j}$, hence also $\varrho_{i,j}^{\sim}(D_{i}^{\sim})\subset D_{j}^{\sim}$.
But then from Proposition 1.9(ii) it follows that the $\varrho_{i,j}^{\sim}$
map normalisers to normalisers.

From this discussion we see that it suffices to prove the assertion for the
unitised system, i.e.

$\mathrm{dim}_{\mathrm{diag}}(\varinjlim D_{i}^{\sim}\subset\varinjlim
A_{i}^{\sim})\leq\liminf_{i\in
I}(\mathrm{dim}_{\mathrm{diag}}(D_{i}^{\sim}\subset A_{i}^{\sim})),$

and that the unitised system still satisfies the crucial hypothesis, namely
that the connecting maps preserve normalisers. In conclusion, we may impose
the additional assumption that the algebras $D_{i}$ and $A_{i}$, as well as
the connecting maps $\varrho_{i,j}$ are all unital to begin with.

Let $\mathbf{1}_{\varinjlim A_{i}}\in\mathcal{F}\subset(\varinjlim
A_{i})^{1}_{+}$ be a finite subset and let $\varepsilon>0$ be given. For each
$j\in I$, let $\varrho_{j}:A_{j}\to\varinjlim A_{i}$ denote the limit map.
Since $\bigcup_{j}\varrho_{j}(A_{j})$ is dense in $\varinjlim A_{i}$, we may
as well assume that $\mathcal{F}\subset\varrho_{j}(A_{j})$ for some $j\in I$.
But then $\mathcal{F}$ has a finite preimage
$\bar{\mathcal{F}}\subset(A_{j})^{1}_{+}$. Now apply Proposition 2.3 to
$(D_{j}\subset A_{j})$ to find $D_{F}\subset F=F^{(0)}\oplus\ldots\oplus
F^{(d)}$ (where $\mathrm{dim}_{\mathrm{diag}}(D_{j}\subset A_{j})\leq
d<\infty$), a set $\\{b_{a}\mid a\in\bar{\mathcal{F}}\\}\subset F^{1}_{+}$ and
a map $\varphi=\sum_{i=0}^{d}\varphi^{(i)}:F\to A_{j}$ satisfying 2.3(i),
(ii), (iii) (with $(D_{j}\subset A_{j})$ in place of $(D_{A}\subset A)$ and
with $\bar{\mathcal{F}}$ in place of $\mathcal{F}$) and

$\varphi^{(i)}(\mathcal{N}_{F^{(i)}}(D_{F^{(i)}}))\subset\mathcal{N}_{A_{j}}(D_{j}).$

But then the same set $\\{b_{a}\mid a\in\bar{\mathcal{F}}\\}$ and the map
$\varrho_{j}\circ\varphi:F\to\varinjlim A_{i}$ satisfy 2.3(i), (ii), (iii)
(this time with $(\varinjlim D_{i}\subset\varinjlim A_{i})$ in place of
$(D_{A}\subset A)$ and with $\mathcal{F}$). Moreover, since the connecting
maps preserve normalisers it is straightforward to see that so do the
$\varrho_{j}$, i.e.,
$\varrho_{j}(\mathcal{N}_{A_{j}}(D_{j}))\subset\mathcal{N}_{\varinjlim
A_{i}}(\varinjlim D_{i})$. Therefore also the $\varrho_{j}\circ\varphi^{(i)}$
preserve normalisers, thus confirming (2.2). Now since
$\mathcal{F}\subset(\varinjlim A_{i})^{1}_{+}$ and $\varepsilon>0$ were
arbitrary, Proposition 2.3 shows that

$\mathrm{dim}_{\mathrm{diag}}(\varinjlim D_{i}\subset\varinjlim
A_{i})\leq\liminf_{i\in I}(\mathrm{dim}_{\mathrm{diag}}(D_{i}\subset A_{i})),$

as desired. $\square$

Proof of Theorem 3.1(vii): Stabilisations.

Taking the identity map as approximation we clearly have
$\mathrm{dim}_{\mathrm{diag}}(D_{n}\subset M_{n})=0$. From 3.1(ii) it follows
that $\mathrm{dim}_{\mathrm{diag}}(D_{A}\otimes D_{n}\subset A\otimes
M_{n})\leq\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)$. The estimate
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A)\leq\mathrm{dim}_{\mathrm{diag}}(D_{A}\otimes D_{n}\subset A\otimes M_{n})$
follows from the permanence property for hereditary subalgebras 3.1(iii).

The equality $\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset
A)=\mathrm{dim}_{\mathrm{diag}}(D_{A}\otimes c_{0}({\mathbb{N}})\subset
A\otimes\mathcal{K}(\ell^{2}({\mathbb{N}})))$ follows in the same manner after
noticing that
$(c_{0}({\mathbb{N}})\subset\mathcal{K}(\ell^{2}({\mathbb{N}})))$ can be
written as inductive limit of sub-C∗-algebras $(D_{n}\subset M_{n})$ with
upper left corner embeddings, so that
$\mathrm{dim}_{\mathrm{diag}}(c_{0}({\mathbb{N}})\subset\mathcal{K}(\ell^{2}({\mathbb{N}})))=0$
by 3.1(vi). $\square$

######  Remark 3.2.

In [32] Matsumoto defined the concept of relative Morita equivalence for
relative $\sigma$-unital sub-$\mathrm{C}^{*}$-algebras (see [32, Definition
2.1 and Definition 3.5]). A relative $\sigma$-unital
sub-$\mathrm{C}^{*}$-algebras is always nondegenerate. Examples include all
unital sub-$\mathrm{C}^{*}$-algebras and
$(c_{0}({\mathbb{N}})\subset\mathcal{K}(\ell^{2}({\mathbb{N}})))$ (see [32,
Example 2.5] for more details). It follows from 3.1(vii) and [32, Theorem 1.1]
that the diagonal dimension of relative $\sigma$-unital
sub-$\mathrm{C}^{*}$-algebras is invariant under relative Morita equivalences.

##  4\. Dimension zero

Recall that a $\mathrm{C}^{*}$-algebra has nuclear dimension zero precisely
when it is locally finite dimensional (see [58, Remark 2.2(iii)] and [53,
Theorem 3.4]). In the separable case this is equivalent to being an AF
algebra, i.e., an inductive limit of finite dimensional
$\mathrm{C}^{*}$-algebras. In this section we give the respective
characterisations for (nondegenerate) sub-$\mathrm{C}^{*}$-algebras with
diagonal dimension zero.

######  Theorem 4.1.

Let $(D_{A}\subset A)$ be a nondegenerate sub-$\mathrm{C}^{*}$-algebra with
$D_{A}$ abelian. Then the following are equivalent:

  1. (i)

$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=0$.

  2. (ii)

For every finite subset $\mathcal{F}\subset A^{1}_{+}$ and $\varepsilon>0$
there exists a finite-dimensional $\mathrm{C}^{*}$-algebra $F\subset A$
together with a diagonal subalgebra $D_{F}\subset F$ such that

    1. (1)

$\mathrm{dist}(\mathcal{F},F^{1}_{+})<\varepsilon$ and

    2. (2)

$\mathcal{N}_{F}(D_{F})\subset\mathcal{N}_{A}(D_{A})$.

If $A$ is separable, then the conditions above are equivalent to:

  1. (iii)

There exists an increasing sequence of finite-dimensional
$\mathrm{C}^{*}$-subalgebras $F_{n}$ of $A$ together with diagonal subalgebras
$D_{F_{n}}$ of $F_{n}$ such that

$\overline{\bigcup\nolimits_{n=1}^{\infty}F_{n}}=A,\quad\overline{\bigcup\nolimits_{n=1}^{\infty}D_{F_{n}}}=D_{A},$
(4.1)

and such that each inclusion $F_{n}\hookrightarrow F_{n+1}$ maps
$\mathcal{N}_{F_{n}}(D_{F_{n}})$ into $\mathcal{N}_{F_{n+1}}(D_{F_{n+1}})$.

###### Proof.

(i) $\Longrightarrow$ (ii): Assume first that $A$ is unital. Take
$\mathbf{1}_{A}\in\mathcal{F}\subset A^{1}_{+}$ and $0<\varepsilon<1$ as in
(ii) and apply Proposition 2.3 with $\varepsilon/2$ in place of $\varepsilon$
to obtain a $D_{F}\subset F$, a c.p.c. order zero map $\varphi:F\to A$ (since
we assume the diagonal dimension to be zero there is only one colour $i$) and
a set $\\{b_{a}\mid a\in\mathcal{F}\\}\subset F^{1}_{+}$ as in 2.3 (but in
2.3(3) replacing $\varepsilon$ with $\varepsilon/2$). We then have

$(1-\varepsilon/2)\cdot\mathbf{1}_{A}\leq\varphi(b_{\mathbf{1}_{A}})\leq\varphi(\mathbf{1}_{F})\leq\mathbf{1}_{A}.$

Therefore, the spectrum of $\varphi(\mathbf{1}_{F})$ does not contain zero and
we may apply order zero functional calculus to $\varphi$ with the constant
function $1$. This will yield a unital c.p. order zero map $\bar{\varphi}:F\to
A$ with $\|\bar{\varphi}(x)-\varphi(x)\|\leq\varepsilon/2$ for all $x\in
F^{1}_{+}$. But unital order zero maps are just ∗-homomorphisms, and we may
take $\bar{\varphi}(F)\subset A$ as our finite dimensional subalgebra in
4.1(ii) above. Condition 4.1(ii)(2) will be satisfied because of (2.2) in
connection with Lemma 1.8 (the latter ensures that not only $\varphi$ but in
fact $\bar{\varphi}$ maps matrix units to normalisers).

Now suppose $A$ is not unital. Our assumption
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=0$ implies that both $A$ and
$D_{A}$ are locally finite dimensional, and since $(D_{A}\subset A)$ is
nondegenerate this in particular entails that $D_{A}$ contains an approximate
unit for $A$ which consists of projections.

Now if $\mathcal{F}\subset A^{1}_{+}$ and $\varepsilon$ are as in 4.1(ii),
since we want to approximate elements of $\mathcal{F}$ we may as well assume
that there is a projection $p\in D_{A}\cap\mathcal{F}$ with $pa=a$ for all
$a\in\mathcal{F}$. Set $(D_{B}\subset B):=(pD_{A}p\subset pAp)$, then by
Theorem 3.1(iii) on hereditary subalgebras we have
$\mathrm{dim}_{\mathrm{diag}}(D_{B}\subset B)=0$. Since we already know (i)
$\Longrightarrow$ (ii) in the unital case, this means that the assertion of
4.1(ii) holds with $B$ in place of $A$. By Proposition 1.9(iii) we have
$\mathcal{N}_{B}(D_{B})\subset\mathcal{N}_{A}(D_{A})$, so in fact 4.1(ii)
holds as it stands.

(ii) $\Longrightarrow$ (i): If $A$ is unital this is immediate from
Proposition 2.3 (note that condition (1) of 4.1(ii) implies condition (3) of
2.3, albeit with $2\varepsilon$ instead of $\varepsilon$, and that
automatically $\mathbf{1}_{F}=\mathbf{1}_{A}\in D_{A}$ provided
$\varepsilon<1/2$.).

Now again suppose $A$ is not unital. Given a finite subset $\mathcal{F}\subset
A^{1}_{+}$ and $\varepsilon>0$, take $D_{F}\subset F\subset A$ as in 4.1(ii).
Define $\bar{F}:=F\oplus\mathbb{C}\cdot(\mathbf{1}_{A^{\sim}}-\mathbf{1}_{F})$
and
$D_{\bar{F}}:=D_{F}\oplus\mathbb{C}\cdot(\mathbf{1}_{A^{\sim}}-\mathbf{1}_{F})$,
then $\bar{F}\subset A^{\sim}$ and $D_{\bar{F}}\subset\bar{F}$ is a diagonal.
Note that (1) $\mathrm{dist}(a,\bar{F})<\varepsilon$ for all
$a\in\mathcal{F}\cup\\{\mathbf{1}_{A^{\sim}}\\}$ and that (2) every matrix
unit in $\bar{F}$ with respect to $D_{\bar{F}}$ belongs to the normaliser
$\mathcal{N}_{A^{\sim}}(D_{A}^{\sim})$.

But this means that 4.1(ii) holds for $A^{\sim}$ in place of $A$ (the special
form $\mathcal{F}\cup\\{\mathbf{1}_{A^{\sim}}\\}$ of the finite subset is no
loss of generality), so by our initial observation ((ii) $\Longrightarrow$ (i)
in the unital case) we have $\mathrm{dim}_{\mathrm{diag}}(D^{\sim}_{A}\subset
A^{\sim})=0$. Now 4.1(i) follows from Theorem 3.1(iv) on the diagonal
dimension of unitisations.

(iii) $\Longrightarrow$ (ii): Given $\mathcal{F}$ and $\varepsilon$, since

$\mathrm{dist}\Big{(}\mathcal{F},\overline{\bigcup\nolimits_{n=1}^{\infty}F_{n}}\Big{)}=\mathrm{dist}(\mathcal{F},A)=0,$

there is some $\bar{n}$ such that
$\mathrm{dist}(\mathcal{F},(F_{\bar{n}})^{1}_{+})<\varepsilon$. Moreover, the
inclusions
$\mathcal{N}_{F_{n}}(D_{F_{n}})\subset\mathcal{N}_{F_{n+1}}(D_{F_{n+1}})$
together with (4.1) imply
$\mathcal{N}_{F_{\bar{n}}}(D_{F_{\bar{n}}})\subset\mathcal{N}_{A}(D_{A})$.

(ii) $\Longrightarrow$ (iii): This is slightly more involved, since we have to
turn approximate inclusions of finite dimensional $\mathrm{C}^{*}$-algebras
into exact ones, while at the same time respecting the diagonals.

Step 1. Consider the set of finite dimensional sub-$\mathrm{C}^{*}$-algebras

$\displaystyle\Lambda$ $\displaystyle:=\\{(D_{F}\subset F)\mid F\subset
A,\,D_{F}\subset F\mbox{ is a diagonal, and}\,$
$\displaystyle\phantom{{}:={}\\{}\mathcal{N}_{F}(D_{F})\subset\mathcal{N}_{A}(D_{A})\\}.$
(4.2)

Note that every $(D_{F}\subset F)\in\Lambda$ has a set $S_{F}\subset
F\setminus D_{F}$ of matrix units which is _tight_ in the following sense:
Whenever $e\in F\setminus D_{F}$ is a matrix unit, then precisely one of $e$
or $e^{*}$ can be written in a unique manner as a product of a scalar and at
most $\dim D_{F}-1$ elements of $S_{F}$. (If $(D_{F}\subset
F)\cong(D_{r}\subset M_{r})$, then $S_{M_{r}}:=\\{e_{i,i+1}\mid i=1,\ldots
r-1\\}$ is tight in this sense; if $F$ is a direct sum of matrix blocks, take
$S_{F}$ to be a disjoint union of tight sets of matrix units for the
individual summands.)

We claim that if $(D_{E}\subset E)\in\Lambda$ is another finite dimensional
sub-$\mathrm{C}^{*}$-algebra which contains a set $\\{v_{e}\mid e\in S_{F}\\}$
of partial isometries normalising $D_{E}$ and such that, for each $e\in
S_{F}$, $\|v_{e}-e\|<\frac{1}{4}$, then the assignment $e\mapsto v_{e}$ for
$e\in S_{F}$ extends to a ∗-homomorphism $\alpha:F\to E$ which maps
$\mathcal{N}_{F}(D_{F})$ into $\mathcal{N}_{E}(D_{E})$.

To prove the claim, observe first that any two elements in the set
$\\{e^{*}e,ee^{*}\mid e\in S_{F}\\}\subset D_{F}$ of range and orthogonal
projections either agree or are orthogonal. And since
$\|v_{e}^{*}v_{e}-e^{*}e\|,\,\|v_{e}v_{e}^{*}-ee^{*}\|<1$ for each $e\in
S_{F}$, the pairwise commuting projections in
$\\{v_{e}^{*}v_{e},\,v_{e}v_{e}^{*}\mid e\in S_{F}\\}\subset D_{E}$ satisfy
the same relations as those in $\\{e^{*}e,ee^{*}\mid e\in S_{F}\\}$. From this
and the unique decomposition of matrix units in $F$ as products of elements
from $S_{F}$ it follows that products in $v_{e}$ and $v_{e}^{*}$ satisfy the
same relations as products in $e$ and $e^{*}$, whence the assignment $e\mapsto
v_{e}$ indeed yields a ∗-homomorphism $\alpha$ as claimed. Every matrix unit
in $F$ maps to a product of elements $v_{e}$ or $v_{e}^{*}$, hence maps to
$\mathcal{N}_{E}(D_{E})$. It now follows from Proposition 1.7 that indeed
$\alpha(\mathcal{N}_{F}(D_{F}))\subset\mathcal{N}_{E}(D_{E})$.

Step 2. After these preparations we now turn to constructing nested
sub-$\mathrm{C}^{*}$-algebras as in (iii). We start by inductively
constructing $(D_{E_{n}}\subset E_{n})\in\Lambda$ and suitable maps between
them; the nested $(D_{F_{n}}\subset F_{n})$ will then arise as small
perturbations.

Let $\\{a_{0},a_{1},a_{2},\ldots\\}\subset A_{+}^{1}$ be a countable dense
subset of the positive unit ball. We may assume $a_{0}=0$ and define

$(D_{E_{0}}\subset E_{0}):=(\\{0\\}\subset\\{0\\})\in\Lambda.$

Now suppose $(D_{E_{n}}\subset E_{n})\in\Lambda$ has been defined. Take some

$0<\gamma_{n}\leq\frac{1}{2^{n}\cdot\dim^{+1}(D_{E_{n}})\cdot\dim^{+1}(E_{n})}$

(here $\dim$ stands for vector space dimension) and some

$0<\zeta_{n}<\frac{\gamma_{n}^{2}}{144}\leq\frac{1}{144},$ (4.3)

and choose $(D_{E_{n+1}}\subset E_{n+1})\in\Lambda$ such that

$\operatorname{dist}(\\{a_{0},\ldots,a_{n}\\}\cup(E_{n})^{1}_{+},(E_{n+1})_{+}^{1}),\,\operatorname{dist}((E_{n})^{1},\,(E_{n+1})^{1})<\zeta_{n}.$
(4.4)

Induction yields a sequence $((D_{E_{n}}\subset
E_{n}))_{n\in\mathbb{N}}\subset\Lambda$.

Step 3. Next, for each $n$ we construct ∗-homomorphisms $\alpha_{n}:E_{n}\to
E_{n+1}$ such that
$\alpha_{n}(\mathcal{N}_{E_{n}}(D_{E_{n}}))\subset\mathcal{N}_{E_{n+1}}(D_{E_{n+1}})$
for each $n$.

Since $E_{n+1}$ is finite dimensional, the identity map extends to some
conditional expectation $\psi:A\to E_{n+1}$. Note that $D_{E_{n+1}}$ lies in
the multiplicative domain of $\psi$, and that we know from (4.2) and Lemma 1.8
that $D_{E_{n}}\cup D_{E_{n+1}}\subset D_{A}$. Therefore, for each $p\in
D_{E_{n}}$ and $q\in D_{E_{n+1}}$ we have
$q\psi(p)=\psi(q)\psi(p)=\psi(qp)=\psi(pq)=\psi(p)\psi(q)=\psi(p)q$. Now since
$D_{E_{n+1}}$ is maximal abelian in $E_{n+1}$ this implies that

$\psi(D_{E_{n}})\subset D_{E_{n+1}}.$

With $\Phi_{E_{n+1}}:E_{n+1}\to D_{E_{n+1}}$ denoting the canonical
conditional expectation we then in particular have

$\Phi_{E_{n+1}}\circ\psi|_{D_{E_{n}}}=\psi|_{D_{E_{n}}}.$ (4.5)

Now choose a tight set $S_{E_{n}}$ of matrix units of $(D_{E_{n}}\subset
E_{n})$. For each matrix unit $e$ of $S_{E_{n}}$, by (4.4) there is some
$w_{e}\in(E_{n+1})^{1}$ with

$\|w_{e}-e\|<\zeta_{n}$ (4.6)

Noting that

$\psi(w_{e}^{*}w_{e})=w_{e}^{*}w_{e}$ (4.7)

we compute

$\displaystyle\|\Phi_{E_{n+1}}(w_{e}^{*}w_{e})-e^{*}e\|$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{4-1-5}}}}{{\leq}}\|\Phi_{E_{n+1}}(w_{e}^{*}w_{e})-\Phi_{E_{n+1}}(\psi(w_{e}^{*}w_{e}))\|$
$\displaystyle\phantom{{}\stackrel{{\scriptstyle\eqref{4-1-5}}}{{\leq}}{}}+\|\Phi_{E_{n+1}}(\psi(w_{e}^{*}w_{e}))-\Phi_{E_{n+1}}(\psi(e^{*}e))\|$
$\displaystyle\phantom{{}\stackrel{{\scriptstyle\eqref{4-1-5}}}{{\leq}}{}}+\|\Phi_{E_{n+1}}(\psi(e^{*}e))-\psi(e^{*}e)\|$
$\displaystyle\phantom{{}\stackrel{{\scriptstyle\eqref{4-1-5}}}{{\leq}}{}}+\|\psi(e^{*}e)-\psi(w_{e}^{*}w_{e})\|$
$\displaystyle\phantom{{}\stackrel{{\scriptstyle\eqref{4-1-5}}}{{\leq}}{}}+\|\psi(w_{e}^{*}w_{e})-e^{*}e\|$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{4-1-5}}}}{{<}}0+2\zeta_{n}+0+2\zeta_{n}+2\zeta_{n}$
$\displaystyle\textstyle\stackrel{{\scriptstyle\eqref{4-1-5}}}{{<}}1/2,$ (4.8)

where for the estimates of the five summands we have used (in this order)
(4.7), (4.6), (4.5), (4.6), and (4.7) and (4.6). This shows that the
projection $\chi_{(\frac{1}{2},1]}(\Phi_{E_{n+1}}(w_{e}^{*}w_{e}))\in
D_{E_{n+1}}$ has norm distance less than $1$ from $e^{*}e$, and since both are
projections in the commutative $\mathrm{C}^{*}$-algebra $D_{A}$, they have to
agree. The respective statements hold for $ee^{*}$, and in particular this
entails that $D_{E_{n}}\subset D_{E_{n+1}}$.

At this point we could deform $w_{e}$ into a partial isometry using functional
calculus. However, without further ado this will not necessarily normalise
$D_{E_{n+1}}$, as is required to apply Step 1 above. Fortunately we can
arrange for a more refined deformation. To this end, write
$ee^{*}=q_{1}+\ldots+q_{m}$ as a sum of pairwise orthogonal rank one
projections $q_{i}\in D_{E_{n+1}}$ and for each $i$ set $p_{i}:=e^{*}q_{i}e$.
We have $p_{i}\in D_{A}$ since $e$ normalises $D_{A}$, and similarly to (4.8)
we estimate

$\|\Phi_{E_{n+1}}(w_{e}^{*}q_{i}w_{e})-p_{i}\|=\|\Phi_{E_{n+1}}(w_{e}^{*}q_{i}w_{e})-e^{*}q_{i}e\|<6\zeta_{n}<1/2,$

whence $\Phi_{E_{n+1}}(w_{e}^{*}q_{i}w_{e})$ is close to a projection in
$D_{E_{n+1}}$, which then has to agree with $p_{i}$. This shows $p_{i}\in
D_{E_{n+1}}$. From

$\|p_{i}-p_{i}w_{e}^{*}q_{i}w_{e}p_{i}\|=\|p_{i}e^{*}q_{i}ep_{i}-p_{i}w_{e}^{*}q_{i}w_{e}p_{i}\|\stackrel{{\scriptstyle\eqref{4-1-2}}}{{<}}2\zeta_{n}$
(4.9)

we see that $p_{i}$ is a rank one projection, and that there is

$1-2\zeta_{n}\leq\nu\leq 1$ (4.10)

such that

$v_{i}:=\nu^{-\frac{1}{2}}\cdot q_{i}w_{e}p_{i}\in E_{n+1}$ (4.11)

is a matrix unit with $v_{i}^{*}v_{i}=p_{i}$ and $v_{i}v_{i}^{*}=q_{i}$, and
with

$\displaystyle\|v_{i}-q_{i}ep_{i}\|$
$\displaystyle\leq(\nu^{-\frac{1}{2}}-1)+\zeta_{n}\leq((1-2\zeta_{n})^{-\frac{1}{2}}-1)+\zeta_{n}\leq
3\zeta_{n}^{\frac{1}{2}},$ (4.12)

where the first estimate uses (4.11) and (4.6), the second uses (4.10), and
the third estimate holds by a small calculation and the fact that
$\zeta_{n}<\frac{1}{2}$ (see (4.3)).

For $i\neq j$ we have $q_{i}ep_{j}=q_{i}ee^{*}q_{j}e=q_{i}q_{j}ee^{*}e=0$ and
$q_{i}v_{i}p_{j}=v_{i}p_{i}p_{j}=v_{i}e^{*}q_{i}ep_{j}=0$. Therefore, for

$v_{e}:=\sum\nolimits_{i}v_{i}$

we have

$\displaystyle\|v_{e}-e\|$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{4-1-8}}}}{{=}}\Big{\|}\Big{(}\sum\nolimits_{i}q_{i}\Big{)}(v_{e}-e)\Big{(}\sum\nolimits_{i}p_{i}\Big{)}\Big{\|}$
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{4-1-8}}}}{{=}}\max_{i}\|v_{i}-q_{i}ep_{i}\|$
$\displaystyle\stackrel{{\scriptstyle\eqref{4-1-8}}}{{\leq}}3\zeta_{n}^{\frac{1}{2}}$
$\displaystyle\stackrel{{\scriptstyle\,\eqref{4-1-5}\,}}{{<}}1/4.$

Finally, since $v_{i}\in\mathcal{N}_{E_{n+1}}(D_{E_{n+1}})$, it follows from
Proposition 1.7 that also $v_{e}\in\mathcal{N}_{E_{n+1}}(D_{E_{n+1}})$.

We are now in position to apply the claim in Step 1 above (with $E_{n}$ in
place of $F$ and $E_{n+1}$ in place of $E$), which makes sure that the
assignment $e\mapsto v_{e}$ extends to a ∗-homomorphism $\alpha_{n}:E_{n}\to
E_{n+1}$ such that
$\alpha_{n}(\mathcal{N}_{E_{n}}(D_{E_{n}}))\subset\mathcal{N}_{E_{n+1}}(D_{E_{n+1}})$.
Note that for each $x\in E_{n}^{1}$ we have

$\textstyle\|\alpha_{n}(x)-x\|\leq\dim(D_{E_{n}})\cdot\dim(E_{n})\cdot
3\zeta_{n}^{\frac{1}{2}}\leq 1/2^{n}$ (4.13)

and that $\alpha_{n}|_{D_{E_{n}}}=\mathrm{id}_{D_{E_{n}}}$.

Step 4. It only remains to turn the sequence $((D_{E_{n}}\subset
E_{n}))_{n\in\mathbb{N}}$ into a nested one.

For each $n\in\mathbb{N}$ and $x\in E_{n}$ observe that
$(\alpha_{m}\ldots\alpha_{n}(x))_{m\geq n}$ is a Cauchy sequence in $A$ by
(4.13), so we may define a map $\beta_{n}:E_{n}\to A$ by setting
$\beta_{n}(x):=\lim_{m}\alpha_{m}\ldots\alpha_{n}(x)$. It is then clear that
$\beta_{n}$ is a ∗-homomorphism, that
$\beta_{n}|_{D_{E_{n}}}=\mathrm{id}_{D_{E_{n}}}$, and that the
$\beta_{m}(E_{m})$ are increasing with
$\overline{\bigcup_{m}\beta_{m}(E_{m})}=A$. Moreover, for any
$x\in\mathcal{N}_{E_{n}}(D_{E_{n}})$ the image $\beta_{n}(x)$ normalises
$\beta_{n+1}(D_{E_{n+1}})$ in $\beta_{n+1}({E_{n+1}})$ , i.e.,
$\beta_{n}(\mathcal{N}_{E_{n}}(D_{E_{n}}))\subset\mathcal{N}_{\beta_{n+1}(E_{n+1})}(\beta_{n+1}(D_{E_{n+1}}))$.
We may therefore set $F_{n}:=\beta_{n}(E_{n})$ and
$D_{F_{n}}:=\beta_{n}(D_{E_{n}})=D_{E_{n}}$, and are done. ∎

######  Remark 4.2.

It follows from Theorem 4.1 in connection with [37, Theorem 5.7] that if $A$
is a separable AF algebra, then $A$ admits a diagonal $D_{A}$ (called a
_regular canonical masa_ in [37]) which satisfies
$\mathrm{dim}_{\mathrm{diag}}(D_{A}\subset A)=0$ and is unique (up to
approximately inner automorphisms of $A$) with this property.

It was shown in [34] that there is a simple separable unital monotracial AF
algebra $A$ containing a Cartan subalgebra $D$ which (as an abelian
$\mathrm{C}^{*}$-algebra) is isomorphic to a regular canonical masa $D_{A}$,
but which does not have the unique extension property, hence is not a
diagonal, hence the pairs $(D\subset A)$ and $(D_{A}\subset A)$ are not
isomorphic.

##  5\. Topological dynamical systems

In this section we link diagonal dimension to Kerr’s notion of (fine) tower
dimension for topological dynamical systems.

######  Definition 5.1.

[26, Definition 4.1] Let $G\curvearrowright X$ be an action of a countable
discrete group on a compact Hausdorff space $X$.

A _tower_ is a pair $(V,S)$ consisting of a subset $V$ of $X$ and a finite
subset $S$ of $G$ such that the sets $sV$, called the _levels_ , for $s\in S$
are pairwise disjoint. $V$ is called the _base_ and $S$ is called the _shape_
of the tower. We write $SV:=\\{sv\mid s\in S,\,v\in V\\}$ for the union of the
levels.

A _castle_ is a family $((V_{j},S_{j}))_{j\in J}$ of towers such that all the
levels are pairwise disjoint.

A tower $(V,S)$ is _open_ if $V$ is an open subset of $X$. A castle whose
towers are open will be referred to as an _open castle_.

######  Definition 5.2.

[26, Definition 4.3, Definition 4.10] Let $G\curvearrowright X$ be an action
of a countable discrete group on a compact Hausdorff space $X$. The _tower
dimension_ of the action $G\curvearrowright X$ is said to be at most $d$,
written $\mathrm{dim}_{\mathrm{tow}}(X,G)\leq d$, if for every finite subset
$E\subset G$ there exist a finite family of open towers $((V_{j},S_{j}))_{j\in
J}$ and a partition $J=J^{(0)}\sqcup\ldots\sqcup J^{(d)}$ such that

  1. (1)

$\bigcup_{j\in J}S_{j}V_{j}=X$,

  2. (2)

for every $x\in X$ there are $j\in J$ and $t\in S_{j}$ such that $x\in tV_{j}$
and $Et\subset S_{j}$ (this is called $E$-Lebesgueness in [26]),

  3. (3)

for each $i\in\\{0,\ldots,d\\}$ the sets $S_{j}V_{j}$ are pairwise disjoint
for $j\in J^{(i)}$ (in other words, for each $i\in\\{0,\ldots,d\\}$ the family
$((V_{j},S_{j}))_{j\in J^{(i)}}$ forms a castle).

If, in addition, for any given open cover $\mathcal{U}$ of $X$ one can choose
the family of open towers $((V_{j},S_{j}))_{j\in J}$ so that the open cover
$(sV_{j})_{j\in J,s\in S_{j}}$ refines $\mathcal{U}$, then the action is said
to have _fine tower dimension_ at most $d$, and in this case we write
$\mathrm{dim}_{\mathrm{ftow}}(X,G)\leq d$. If no such $d$ exists, then the
tower dimension (or the fine tower dimension, respectively) is said to be
infinite.

######  Remarks 5.3.

(i) The definitions above insist on $G$ being countable discrete, since non-
discrete topological groups will require a different setup; cf. [23] vs. [24].

(ii) We have stated the definition of fine tower dimension in a version
slightly different from [26, Definition 4.10] in order to avoid asking $X$ to
carry a metric (but in the metrisable case the two formulations are
equivalent).

This additional generality will not cause technical difficulties, and will
prove useful when we look at uniform Roe algebras in Section 7. The only
_caveat_ is that for non-metrisable spaces the various notions of covering
dimension are no longer equivalent, so that one has to make a choice. The
appropriate version for our purposes is the one characterised in [28,
Proposition 1.5]: every finite open cover of $X$ has a finite open refinement
with chromatic number $d+1$.

(iii) Unlike [26, Definition 4.10] we explicitly ask the index set $J$ to be
finite to begin with, whereas in [26] this is derived from compactness of $X$
whenever necessary.

(iv) Finiteness of tower dimension requires the action to be _free_. Indeed,
let $g\in G\setminus\\{e\\}$, let $x\in X$, and let $((V_{j},S_{j}))_{j\in J}$
be a family witnessing the tower dimension for the finite subset
$E:=\\{g,g^{-1},e\\}$. By the $E$-Lebesgueness condition 5.2(2) there are
$j\in J$ and $t\in S_{j}$ such that $x\in tV_{j}$ and $Et\subset S_{j}$. Since
$gtV_{j}$ and $tV_{j}$ are disjoint, $x$ cannot be fixed by $g$.

It will therefore in particular follow from the theorem below that finite
diagonal dimension of $(C(X)\subset C(X)\rtimes_{\mathrm{r}}G)$ can occur only
for free actions. In upcoming work, the third named author will introduce a
variant of diagonal dimension which will, at least to some extent, also allow
isotropy.

From [26, Proposition 4.11] and the definition of fine tower dimension we know
that

$\displaystyle\max\\{\mathrm{dim}_{\mathrm{tow}}^{+1}(X,G),\,\dim^{+1}(X)\\}$
$\displaystyle\qquad\qquad\leq\mathrm{dim}_{\mathrm{ftow}}^{+1}(X,G)$
$\displaystyle\qquad\qquad\qquad\qquad\leq\mathrm{dim}_{\mathrm{tow}}^{+1}(X,G)\cdot\dim^{+1}(X).$
(5.1)

The main result of this section fits diagonal dimension into this chain of
estimates.

######  Theorem 5.4.

Let $\alpha:G\curvearrowright X$ be an action of a countable discrete group on
a compact Hausdorff space. Then

$\displaystyle\mathrm{dim}_{\mathrm{ftow}}^{+1}(X,G)$
$\displaystyle\qquad\qquad\leq\mathrm{dim}_{\mathrm{diag}}^{+1}(C(X)\subset
C(X)\rtimes_{\mathrm{r}}G)$ (5.2)
$\displaystyle\qquad\qquad\qquad\qquad\leq\mathrm{dim}_{\mathrm{tow}}^{+1}(X,G)\cdot\dim^{+1}(X).$
(5.3)

In particular, if $X$ is totally disconnected333Recall that a compact
Hausdorff space is totally disconnected if and only if it has covering
dimension zero. then

$\mathrm{dim}_{\mathrm{tow}}(X,G)=\mathrm{dim}_{\mathrm{ftow}}(X,G)=\mathrm{dim}_{\mathrm{diag}}(C(X)\subset
C(X)\rtimes_{\mathrm{r}}G).$ (5.4)

The proof is divided into two parts. To implement the upper bound (5.3), we
only need to carefully follow Kerr’s proof of [26, Theorem 6.2], which
establishes the respective bound for the nuclear dimension of the crossed
product, but in fact keeps track of the diagonal by design.

The key new feature in this section is the lower bound (5.2), which extracts
purely dynamical information from $\mathrm{C}^{*}$-algebraic data.

The first equality in (5.4) is already contained in (5.1), and the second one
follows upon combining the latter with (5.2) and (5.3).

We start by isolating the lemma below from the proof of [26, Theorem 6.2].
Roughly speaking, it expresses tower dimension in terms of partitions of unity
(as opposed to open covers) which are approximately compatible with the group
action. We spell out the proof partly for the convenience of the reader and
partly because [26, Theorem 6.2] is stated and proven under the slightly more
special assumption that $X$ is metrisable.

######  Lemma 5.5.

Let $\alpha:G\curvearrowright X$ be as before. Suppose
$\mathrm{dim}_{\mathrm{tow}}(X,G)\leq d<\infty$. Then for every finite subset
$e\in E\subset G$ and $\eta>0$, there exist a finite family of open towers
$((V_{j},S_{j}))_{j\in J}$, a partition $J=J^{(0)}\sqcup\ldots\sqcup J^{(d)}$,
and a partition of unity $(h_{j})_{j\in J}$ for $X$ such that

  1. (1)

$(S_{j}V_{j})_{j\in J}$ forms an open cover for $X$ to which the partition of
unity $(h_{j})_{j\in J}$ is subordinate,

  2. (2)

for each $i\in\\{0,\ldots,d\\}$, $j\neq j^{\prime}\in J^{(i)}$, and for
$g,g^{\prime}\in E$, $\alpha_{g}(h_{j})\alpha_{g^{\prime}}(h_{j^{\prime}})=0$
(we abuse notation and write $\alpha$ also for the induced action on $C(X)$
here),

  3. (3)

for each $i\in\\{0,\ldots,d\\}$ the family $((V_{j},S_{j}))_{j\in J^{(i)}}$
forms a castle,

  4. (4)

$|h_{j}(x)-h_{j}(gx)|<\eta$ for all $j\in J$, $x\in X$ and $g\in E$.

###### Proof (cf. that of Theorem 6.2 in [26]).

We may assume $E$ to be symmetric by replacing it with $E\cup E^{-1}$, if
necessary. Take $n\in\mathbb{N}$ such that $(d+2)/n<\eta$.

Choose a finite family of open towers $((V_{j},S_{j}))_{j\in J}$ and a
partition $J=J^{(0)}\sqcup\ldots\sqcup J^{(d)}$ satisfying conditions 5.2(1)
and 5.2(3) (so that in particular 5.5(3) holds), and such that 5.2(2) holds
with $E$ replaced by $E^{n}$. For each $j\in J$ define the $E^{n}$-core of
$S_{j}$ as

$B_{j,n}:=\\{s\in S_{j}\mid E^{n}s\subset S_{j}\\}.$

Upon shifting the $S_{j}$ from the right and the $V_{j}$ from the left, if
necessary, we may assume each of the nonempty $B_{j,n}$ to contain $e$.

By 5.2(2), the family $(sV_{j})_{j\in J,s\in B_{j,n}}$ is an open cover of
$X$, and since $X$ is normal we can find a partition of unity
$(\hat{f}_{j,s})_{j\in J,s\in B_{j,n}}$ subordinate to it. For every $j\in J$
define $f_{j}\in C(X)^{1}_{+}$ by

$f_{j}:=\max_{s\in B_{j,n}}\alpha_{s^{-1}}(\hat{f}_{j,s})$ (5.5)

provided $B_{j,n}\neq\emptyset$; otherwise set $f_{j}:=0$. Note that for every
$j\in J$ and $s\in B_{j,n}$ we have
$0\leq\hat{f}_{j,s}\leq\alpha_{s}(f_{j})\in C_{0}(sV_{j})$, whence

$\sum\nolimits_{j\in J}\sum\nolimits_{s\in
B_{j,n}}\alpha_{s}(f_{j})\geq\mathbf{1}_{X}.$

Next, for each $j$ define further subsets $B_{j,k}$ of $S_{j}$ by

$\displaystyle B_{j,0}$ $\displaystyle:=S_{j}\setminus\bigcap\nolimits_{g\in
E}gS_{j}\mbox{ and }$ (5.6) $\displaystyle B_{j,k}$
$\displaystyle:=\Big{(}\bigcap\nolimits_{g\in
E^{k}}gS_{j}\Big{)}\setminus\Big{(}\bigcap\nolimits_{g\in
E^{k+1}}gS_{j}\Big{)}\mbox{ for }k=1,\ldots,n-1$

and note that

$B_{j,n}=\bigcap\nolimits_{g\in E^{n}}gS_{j}.$

This in particular means that for each $j$ the family
$(B_{j,k})_{k=0,\ldots,n}$ forms a partition of $S_{j}$. Moreover, by
construction we have for all $j\in J$ and $g\in E$

$\displaystyle gB_{j,k}$ $\displaystyle\subset B_{j,k-1}\cup B_{j,k}\cup
B_{j,k+1}\mbox{ for }k=1,\ldots,n-1,\mbox{ and }$ $\displaystyle gB_{j,n}$