\begin{document}

\begin{abstract}
We prove that every Kirchberg algebra in the UCT class has nuclear dimension~1. We first
show that Kirchberg $2$-graph algebras with trivial $K_0$ and finite $K_1$ have nuclear
dimension~1 by adapting a technique developed by Winter and Zacharias for Cuntz algebras.
We then prove that every Kirchberg algebra in the UCT class is a direct limit of
$2$-graph algebras to obtain our main theorem.
\end{abstract}

\maketitle

\section{Introduction}

Nuclear dimension for $C^*$-algebras, introduced by Winter and Zacharias in
\cite{WinterZacharias:Adv10}, is a noncommutative notion of rank based on covering
dimension for topological spaces. It has been shown \cite{SatoWhiteWinter:xx14,
TomsWinter:GAFA13, Winter:IM12} to be closely related to $\mathcal{Z}$-stability and
hence to the classification program for simple nuclear $C^*$-algebras. Winter and
Zacharias showed that all UCT-Kirchberg algebras (i.e., separable, nuclear, simple,
purely infinite $C^{*}$-algebras in the UCT class) have nuclear dimension at most 5 and
asked whether the precise value of their dimension is determined by algebraic properties
of their $K$-groups, such as torsion \cite[Problem~9.2]{WinterZacharias:Adv10}. Matui and
Sato \cite{MatuiSato} subsequently improved the estimate for simple Kirchberg algebras
from~5 to~3, and their result is valid for non-UCT Kirchberg algebras, if any exist; and
Barlak, Enders, Matui, Szab\'o and Winter have showed how to recover the Matui-Sato
estimate from a general relationship between the nuclear dimension of an
$\Oo_\infty$-stable $C^*$-algebra and its $\Oo_2$-stablization that implies, in
particular, that every $\Oo_\infty$-absorbing $C^*$-algebra with compact metrizable
primitive-ideal space has nuclear dimension at most~7
\cite{BarlakEndersEtAl:arXiv:1312.6289}. For UCT-Kirchberg algebras, a further
improvement due to Enders \cite{Enders:arXiv:1405.6538} shows that every UCT-Kirchberg
algebra with torsion-free $K_1$ has nuclear dimension~1. But the question remained open
whether torsion in $K_1$ precludes having nuclear dimension~1. Here we prove that every
UCT-Kirchberg algebra, regardless of its K-theory, has nuclear dimension~1. This settles
the question posed by Winter and Zacharias, and suggests another question: does every
simple separable non-AF $C^*$-algebra with finite nuclear dimension have nuclear
dimension~1? Indeed, the appearance of the present paper in preprint form motivated
Bosa--Brown--Sato--Tikuisis--White--Winter in their remarkable recent preprint
\cite{BBSTWW} to pursue the optimal bound of~1 for the nuclear dimension of very broad
classes of simple $C^*$-algebras (see \cite[page~4]{BBSTWW}).

We recall the definition of nuclear dimension. A completely positive map $\phi$ between
$C^*$-algebras is order zero if $ab = 0$ implies $\phi(a)\phi(b) = 0$ for positive $a,b$.
A separable $C^{*}$-algebra $A$ has \emph{nuclear dimension $r$}, denoted by
$\mathrm{dim}_{\mathrm{nuc}} ( A ) = r$, if $r$ is the least element in $\NN \cup \{
\infty \}$ for which there is a sequence of order-$r$ factorisations $\phi_n \circ
\psi_n$ through finite dimensional $C^*$-algebras $\mathcal{F}_n$ that pointwise
approximate the identity map on $A$. That is, there exist finite dimensional
$C^{*}$-algebras $( \mathcal{F}_{n} )_{ n \in \NN }$, completely positive, contractive
linear maps $( \ftn{\psi_{n}}{A}{ \mathcal{F}_{n} } )_{n \in \NN }$, and completely
positive linear maps $( \ftn{\phi_{n}}{ \mathcal{F}_{n} }{A} )_{n \in \NN }$ such that
\begin{enumerate}
\item $\lim_{n \to \infty } \| a - \phi_{n} \circ \psi_{n} (a) \| = 0$ for all $a \in
    A$ and
\item each $\mathcal{F}_{n}$ has a decomposition $\bigoplus_{i=0}^{r}
    \mathcal{F}_{n,i}$ such that $\phi_{n} \vert_{ \mathcal{F}_{n,i}}$ is an
    order-zero completely positive contraction for each $i$.
\end{enumerate}
Winter and Zacharias' calculation of nuclear dimension for Cuntz algebras in
\cite{WinterZacharias:Adv10} is related to a construction of Kribs and Solel
\cite{KribsSolel:JAMS07} which builds from a directed graph $E$ a sequence of directed
graphs $( E(n) )_{ n = 1}^{\infty}$ comprising a kind of generalised combinatorial
solenoid. The first two authors, with Tomforde, used the Kribs-Solel construction
explicitly to compute nuclear dimension of many purely infinite nonsimple graph algebras
in \cite{RuizSimsTomforde:arXiv:1312.0507}. The key feature of $E(n)$ used in
nuclear-dimension calculations is that there are inclusions of the Toeplitz algebras
$\ftn{\iota_n}{\Tt C^*(E)}{\Tt C^*(E(n))}$ that can be approximated modulo compacts by
order-1 factorisations through finite-dimensional $C^*$-algebras. These are parlayed into
an approximation of the identity on $C^*(E)$ using a completely positive splitting
$C^*(E) \to \Tt C^*(E)$ (which exists since every graph algebra is nuclear), a suitable
sequence of homomorphisms $j_n : C^*(E(n)) \to C^*(E) \otimes \Kk$, and classification
results for purely infinite $C^*$-algebras.

Here, we develop a version of this machinery for higher-rank graphs and their
$C^*$-algebras as introduced in \cite{KumjianPask:NYJM00}.  We use this, and the fact
that each Kirchberg algebra with trivial $K_0$ and finite $K_1$ is isomorphic to a tensor
product of purely infinite simple graph $C^*$-algebras, to show that such Kirchberg
algebras have nuclear dimension~1. This is, on the face of it, somewhat surprising: one
expects the tensor product of $C^*$-algebras of nuclear dimension $n$ and $m$ to have
nuclear dimension $(n+1)(m+1)-1$, as is the case for commutative $C^*$-algebras. Here the
underlying combinatorial model plays a key role. For graph $C^*$-algebras
\cite{RuizSimsTomforde:arXiv:1312.0507}, following
\cite[Section~7]{WinterZacharias:Adv10}, the $n$\textsuperscript{th} order-1
factorisation through finite-dimensional $C^*$-algebras was obtained by producing a pair
of pavings of the combinatorial quadrant $\NN \times \NN$ by copies of a completely
positive contractive $n \times n$ real matrix so that when the two pavings are
superimposed, entries at any fixed distance from the diagonal, and sufficiently far from
the origin, approach~1 as $n \to \infty$. For $2$-graphs, we must perform an analogous
decomposition in $\NN^2 \times \NN^2$, and the expected increase in dimension turns out
to be illusory: facing no topological constraints, we can just pick convenient bijections
of $\NN^2 \times \NN^2$ onto $\NN \times \NN$ that carry points close to the diagonal to
points close to the diagonal. In essence, this is the idea underlying the technical
argument of Theorem~\ref{thm: order 1 approximation}.

To deduce our main theorem, we apply the Kirchberg-Phillips theorem to show that every
stable UCT-Kirchberg algebra is a direct limit of purely infinite $2$-graph
$C^*$-algebras that are known to have nuclear dimension~1 either by Enders' results or by
the result of the preceding paragraph. We believe that this realisation of Kirchberg
algebras using $2$-graph $C^*$-algebras has independent interest: many important
$C^*$-algebraic properties are preserved under direct limits, and as our approach here
shows, the combinatorial structure of $2$-graphs can provide a good line of attack in
establishing structural properties of the associated $C^*$-algebras. Related
combinatorial models have been used to great effect to study weak semiprojectivity
\cite{Spielberg:CMB07} and prime-order automorphisms \cite{Spielberg:JraM07} of
UCT-Kirchberg algebras. Our techniques also have potential applications to calculations
of nuclear dimension for large classes of nonsimple Kirchberg algebras along the lines of
\cite{RuizSimsTomforde:arXiv:1312.0507} (see Remark~\ref{rmk:SpecSeq}).


\smallskip


We start with some background on higher-rank graphs in Section~\ref{sec:k-graphs and
repns}. In Section~\ref{sec:k-graphKS}, we show how to generalise the Kribs-Solel
construction to higher-rank graphs, and produce analogues of the homomorphisms $\iota_n$
and $j_n$ discussed in the preceding paragraph. In Section~\ref{sec:CP1-graphKS}, we
investigate how our construction behaves with respect to the cartesian-product
construction for $k$-graphs \cite{KumjianPask:NYJM00}; this allows us to relate the
results of the preceding section to tensor products of graph $C^*$-algebras. In
Section~\ref{sec:order1factorisations}, we show that for $2$-graphs, the maps
$\ftn{\tilde\iota_n}{C^*( \Lambda )}{C^*( \Lambda(n))}$ induced by the $\iota_n$ can be
asymptotically approximated by sums of two order-zero maps through AF-algebras. In
Section~\ref{sec:main}, we prove our main result. We first show that if $E$ and $F$ are
1-graphs whose $C^*$-algebras are Kirchberg algebras with $K$-theory $(T, 0)$ and
$(0,\ZZ)$ respectively, where $T$ is a finite abelian group, then for the $2$-graph
$\Lambda = E \times F$, the composition $j_{(n_{1} , n_{2} )} \circ \tilde\iota_{(n_{1},
n_{2} )}$ implements multiplication by $n_1n_2$ in $K_*(C^*(\Lambda)) \cong (0, T)$. By
choosing increasing $(n_{1}, n_{2} )$ for which multiplication by $n_1n_2$ is the
identity on $T$, and applying classification machinery, we deduce that UCT-Kirchberg
algebras with trivial $K_0$ and finite $K_1$ have nuclear dimension~1. We then prove our
main result by combining this with Enders' results and a direct-limit argument.


\section{Higher rank graphs and their $C^{*}$-algebras} \label{sec:k-graphs and repns}

We recall the standard conventions for $k$-graphs and their $C^*$-algebras introduced in
\cite{KumjianPask:NYJM00}. We regard $\NN^k$ as an additive semigroup with identity 0.
For $m,n \in \NN^k$, we write $m \vee n$ for their coordinatewise maximum. We write $n
\leq m$ if $n_{i} \leq m_{i}$ for all $i$. We also write $n < m$ to mean $n_{i} < m_{i}$
for all $i$. Warning: this convention means that $n < m$ and $n \lneq m$ mean different
things.

\begin{dfn}[{See~\cite{KumjianPask:NYJM00}}] \label{dfn:k-graph}
Let $k \in \NN \setminus \{0\}$.  A \emph{graph of rank $k$}, or \emph{$k$-graph}, is a
pair $(\Lambda,d)$ where $\Lambda$ is a countable category and $d$ is a functor from
$\Lambda$ to $\NN^k$ that satisfies the \emph{factorisation property}:  for all $\lambda
\in \Mor(\Lambda)$ and all $m,n \in \NN^k$ such that $d(\lambda) = m+n$, there exist
unique morphisms $\mu$ and $\nu$ in $\Mor(\Lambda)$ such that $d(\mu) = m$, $d(\nu) = n$
and $\lambda = \mu\nu$.
\end{dfn}

Since we are regarding $k$-graphs as generalised directed graphs, we refer to elements of
$\Mor(\Lambda)$ as \emph{paths}.  The factorisation property implies that $\setof{\id_o}{
o \in \Obj(\Lambda) } = \setof{\lambda \in \Mor(\Lambda)}{ d(\lambda) = 0}$. So the
codomain and domain maps $\ftn{\operatorname{cod},
\operatorname{dom}}{\Mor(\Lambda)}{\Obj(\Lambda)}$ determine maps $r : \lambda \mapsto
\id_{\operatorname{cod}(\lambda)}$ and $s : \lambda \mapsto
\id_{\operatorname{dom}(\lambda)}$ from $\Mor(\Lambda)$ to $d^{-1}(0)$. We refer to the
elements of $d^{-1}(0)$ as \emph{vertices}, and call $r(\lambda)$ and $s(\lambda)$ the
range and source of $\lambda$. We have $r(\lambda)\lambda = \lambda = \lambda
s(\lambda)$. We write $\lambda \in \Lambda$ to mean $\lambda \in \Mor(\Lambda)$.

We use the following notation from \cite{PaskQuiggEtAl:JA05}: given $\lambda \in \Lambda$
and $E \subseteq \Lambda$, we define
\[
\lambda E :=\setof{\lambda\mu}{\mu \in E, r(\mu) = s(\lambda)} \  \text{ and } \ E\lambda := \setof{\mu\lambda }{\mu \in E, s(\mu) =
r(\lambda) }.
\]
In particular if $d(v) = 0$, then $vE = \setof{\lambda \in E}{ r(\lambda) = v }$, and $Ev
= \setof{\lambda \in E}{ s(\lambda) = v }$.

For $n \in \NN^{k}$, we let $\Lambda^{n} = d^{-1} (n)$.  For $n < m$, we set $\Lambda^{[n
, m)} = \setof{\lambda \in \Lambda}{n \leq d(\mu) < m}$.  We use the convention that for
$m \le n \le d(\lambda)$, the path $\lambda(m,n)$ is the unique element of
$\Lambda^{n-m}$ such that $\lambda = \lambda'\lambda(m,n)\lambda''$ for some $\lambda'
\in \Lambda^m$. An application of the factorisation property shows that for $m \le
d(\lambda)$ we have $\lambda = \lambda(0,m)\lambda(m,d(\lambda))$.

As in \cite{KumjianPask:NYJM00}, we say that $\Lambda$ is \emph{row-finite} if $v
\Lambda^n$ is finite for each $v \in \Lambda^0$ and $n \in \NN^k$.  We say that $\Lambda$
has \emph{no sources} if each $v \Lambda^n \not= \emptyset$. All $k$-graphs in this paper
will be row-finite with no sources.

\begin{dfn}[See~{\cite{RaeburnSimsEtAl:JFA04}}] \label{dfn:common extensions}
Let $(\Lambda,d)$ be a $k$-graph. Given $\mu,\nu \in \Lambda$, we say that $\lambda$ is a
\emph{minimal common extension} of $\mu$ and $\nu$ if $\lambda \in \mu\Lambda \cap
\nu\Lambda$ and $d(\lambda) = d(\mu) \vee d(\nu)$. We denote the collection $\mu\Lambda
\cap \nu\Lambda \cap \Lambda^{d(\mu) \vee d(\nu)}$ of all minimal common extensions of
$\mu$ and $\nu$ by $\MCE(\mu,\nu)$. We define
\[\Lambda^{\mathrm{min}}(\mu,\nu) := \setof{(\alpha,\beta) \in \Lambda \times \Lambda }{ \mu\alpha =  \nu\beta \in \MCE(\mu,\nu) }.
\]
\end{dfn}
For a row-finite $k$-graph $\Lambda$, the set $\MCE(\mu,\nu)$ is finite for all $\mu, \nu \in \Lambda$, since each $\MCE(\mu,\nu) \subseteq r(\mu)\Lambda^{d(\mu) \vee d(\nu)}$.  The factorisation property
ensures that $(\alpha,\beta) \mapsto \mu\alpha$ is a bijection from
$\Lambda^{\min}(\mu,\nu)$ to $\MCE(\mu,\nu)$.

The following definition of a Toeplitz-Cuntz-Krieger family for a higher-rank graph is
essentially \cite[Definition~7.1]{RaeburnSims:JOT05}, with the appropriate changes of
conventions to translate from product-systems of graphs to $k$-graphs (see also
\cite[Section~2.2]{anHuefLacaEtAl:JFA14}).

\begin{dfn}
Let $\Lambda$ be a row-finite $k$-graph with no sources.  A \emph{Toeplitz-Cuntz-Krieger
$\Lambda$-family} is a collection $\{t_{\lambda} \}_{\lambda \in \Lambda}$ of partial
isometries in a $C^{*}$-algebra satisfying
\begin{itemize}
\item[(TCK1)] $\{t_{v}\}_{v \in \Lambda^{0}}$ is a collection of mutually
    orthogonal projections;
\item[(TCK2)] $t_{\lambda} t_{\mu} = \delta_{s(\lambda), r(\mu)} t_{\lambda \mu}$ for
    all $\lambda, \mu \in \Lambda$;
\item[(TCK3)] $t_{\lambda}^{*} t_{\lambda} = t_{s(\lambda)}$ for all $\lambda \in
    \Lambda$; and
\item[(TCK4)] $t_{\lambda}^{*} t_{\mu} =  \sum_{(\alpha, \beta) \in
    \Lambda^{\mathrm{min}} (\lambda, \mu)} t_{\alpha} t_{\beta}^{*}$ for all
    $\lambda, \mu \in \Lambda$.
\end{itemize}

\begin{rmk}
In previous treatments (see, for example, \cite{anHuefLacaEtAl:JFA14}), the definition of
a Toeplitz-Cuntz-Krieger $\Lambda$-family for a row-finite $k$-graph $\Lambda$ with no
sources has included the additional relation
\begin{equation}\label{eq:TCK5}
\sum_{\lambda \in v\Lambda^n} t_\lambda t^*_\lambda \le t_v\qquad\text{ for all $v \in \Lambda^0$ and $n \in \NN^k$.}
\end{equation}
But in fact this relation follows from the other four. To see this, observe that if
$\mu,\nu \in \Lambda^n$, then $\Lambda^{\mathrm{min}}(\mu,\nu) = \{s(\mu),s(\mu)\}$ if
$\mu = \nu$ and is empty otherwise. So~(TCK4) shows that the $t_\lambda t^*_\lambda$ for
$\lambda \in \Lambda^n$ are mutually orthogonal, and so the sum on the left-hand side
of~\eqref{eq:TCK5} is a projection. For each $\Lambda$, we have
$\Lambda^{\mathrm{min}}(\lambda, r(\lambda)) = \{(s(\lambda), \lambda)\}$. Hence
\[
\Big(\sum_{\lambda \in v\Lambda^n} t_\lambda t^*_\lambda\Big) t_v
    = \sum_{\lambda \in v\Lambda^n} \sum_{(\alpha,\beta) \in \Lambda^{\mathrm{min}}(\lambda, r(\lambda))} t_\lambda t_\alpha t^*_\beta
    = \sum_{\lambda \in v\Lambda^n} t_\lambda t_{s(\lambda)} t^*_\lambda
    = \sum_{\lambda \in v\Lambda^n} t_\lambda t^*_\lambda.
\]
\end{rmk}

As in \cite{KumjianPask:NYJM00}, a \emph{Cuntz-Krieger $\Lambda$-family} is a collection
$\{s_{\lambda}\}_{\lambda \in \Lambda}$ of partial isometries in a $C^{*}$-algebra
satisfying (TCK1), (TCK2), (TCK3), and
\begin{itemize}
\item[(CK)] $s_{v} = \sum_{\lambda \in v \Lambda^{n}} s_{\lambda} s_{\lambda}^{*}$
    for each $v \in \Lambda^{0}$ and $n \in \NN^{k}$.
\end{itemize}

Let $\Lambda$ be a row-finite $k$-graph with no sources.  There is a universal
$C^{*}$-algebra $\Tt C^{*} (\Lambda)$ generated by a universal Toeplitz-Cuntz-Krieger
$\Lambda$-family $\{t_{\lambda}\}_{\lambda \in \Lambda}$. We call this $C^{*}$-algebra
the \emph{Toeplitz algebra of $\Lambda$}.  There is also a universal $C^{*}$-algebra
$C^{*} (\Lambda)$ generated by a universal Cuntz-Krieger $\Lambda$-family
$\{s_{\lambda}\}_{\lambda \in \Lambda}$.  We call this $C^{*}$-algebra the
\emph{Cuntz-Krieger algebra of $\Lambda$}, or just the $C^*$-algebra of $\Lambda$.
\end{dfn}


\section{The Kribs-Solel construction for $k$-graphs}\label{sec:k-graphKS}

For the duration of this section, we fix a row-finite $k$-graph $\Lambda$ with no
sources. The key tool for understanding nuclear dimension of graph algebras in
\cite{RuizSimsTomforde:arXiv:1312.0507} was a construction due to Kribs and Solel
\cite{KribsSolel:JAMS07}. The first step in our analysis here is to adapt this
construction to $k$-graphs.

Choose $n = (n_1, \dots, n_k) \in \NN^k$ with each $n_i \ge 1$. Let
\[
H_{n} := \setof{a n}{ a \in \ZZ^k } = \setof{m \in \ZZ^k}{ m_i/n_i \in \ZZ \text{ for all $i$} }.
\]
We will often just write $H$ for $H_{n}$. For $m \in \NN^k$, we write $[m]$ for $m + H
\in \ZZ^k/H$. We often identify $\ZZ^k/H$ as a set with $\setof{m \in \NN^k}{ m < n }$.

For $\lambda \in \Lambda$, we define
\[[\lambda]_H := \lambda(0, [d(\lambda)]),
\]
and we usually write $[\lambda]$ for $[\lambda]_H$. So $[\lambda]$ is the unique element
of $\Lambda$ such that $d([\lambda]) < n$ and $\lambda = [\lambda]\lambda'$ with
$d(\lambda') \in H$. The factorisation property implies that if $d(\mu) \in H$, then
$[\lambda\mu] = [\lambda]$.

Following \cite{RuizSimsTomforde:arXiv:1312.0507}, we write $\Lambda^{<n} := \setof{\lambda
\in \Lambda }{d(\lambda) <n }$.  Let
\[\Lambda(n) := \setof{(\lambda, \lambda') \in \Lambda \times \Lambda^{<n} }{s(\lambda) = r(\lambda')}.
\]
We aim to make this set into a $k$-graph. For $(\lambda, \lambda') \in \Lambda(n)$,
define
\[d((\lambda, \lambda')) := d(\lambda).
\]
So $\Lambda(n)^0 = \setof{(r(\lambda),\lambda) }{ \lambda \in \Lambda^{<n} }$. Define
$\ftn{r,s}{\Lambda(n)}{\Lambda(n)^0}$ by
\begin{align*}
s((\lambda, \lambda')) &:= (s(\lambda), \lambda'), \quad\text{and}\\
r((\lambda, \lambda')) &:= (r(\lambda), [\lambda\lambda']).
\end{align*}
Identify $\Lambda(n)^0$ with $\Lambda^{<n}$ via $(r(\lambda),\lambda) \mapsto \lambda$.
Then $s((\lambda,\lambda')) = \lambda'$ and $r((\lambda, \lambda')) = [\lambda\lambda']$.
Suppose that $s((\lambda,\lambda')) = r((\mu,\mu'))$; that is, $\lambda' = [\mu\mu']$.
Then we define
\[
(\lambda,\lambda')(\mu,\mu') := \big(\lambda\mu,\mu'\big).
\]

\begin{lem}
Under the operations just described, $\Lambda(n)$ is a row-finite $k$-graph with no
sources.
\end{lem}
\begin{proof}
We show that $\Lambda(n)$ is a category. We first check that $s$ and $r$ are compatible
with composition. Suppose that $s((\lambda,\lambda')) = r((\mu,\mu'))$. Then
\[s((\lambda,\lambda')(\mu,\mu'))
    = s((\lambda\mu, \mu'))
    = \mu' = s((\mu,\mu')).
\]
Writing $\mu\mu' = [\mu\mu']\tau = \lambda'\tau$, we have
\[r\big((\lambda,\lambda')(\mu,\mu')\big)
    = r((\lambda\mu, \mu'))
    = \big[\lambda\mu\mu'\big]
    = \big[\lambda[\mu\mu']\tau\big]
    = [\lambda\lambda'\tau];
\]
and since $d(\tau) \in H$, we have $r((\lambda,\lambda')(\mu,\mu')) =
[\lambda\lambda'\tau] = [\lambda\lambda'] = r((\lambda,\lambda'))$.

We now check that $r((\lambda,\lambda'))$ and $s((\lambda,\lambda'))$ act as left- and
right identities for $(\lambda, \lambda')$:
\[r((\lambda,\lambda'))(\lambda,\lambda')
    = (r(\lambda), [\lambda\lambda'])(\lambda,\lambda')
    = (r(\lambda)\lambda, \lambda')
    = (\lambda,\lambda'),
\]
and
\[(\lambda,\lambda')s((\lambda,\lambda'))
    = (\lambda, \lambda')\big(s(\lambda), \lambda'\big)
    = (\lambda s(\lambda), \lambda')
    = (\lambda, \lambda').
\]
To check associativity, suppose that $s((\lambda,\lambda')) = r((\mu,\mu'))$ and
$s((\mu,\mu')) = r((\nu,\nu'))$.  Then
\begin{align*}
((\lambda, \lambda')(\mu,\mu'))(\nu,\nu')
    &= (\lambda\mu, \mu')(\nu, \nu')
    = (\lambda\mu\nu, \nu'),\\
    &= (\lambda,\lambda')(\mu\nu, \nu')
    = (\lambda,\lambda')((\mu,\mu')(\nu,\nu')).
\end{align*}
So $\Lambda(n)$ is a category.

We check that $d$ is a functor:
\[d((\lambda,\lambda')(\mu,\mu'))
    = d((\lambda\mu, \mu'))
    = d(\lambda\mu) = d(\lambda) + d(\mu)
    = d((\lambda,\lambda')) + d((\mu, \mu')).
\]
Now we check the factorisation property. Suppose that $d((\lambda,\lambda')) = p+q$. Then
$d(\lambda) = p+q$, and the factorisation property in $\Lambda$ gives $\mu \in \Lambda^p$
and $\nu \in \Lambda^q$ such that $\lambda = \mu\nu$. Now $(\nu,\lambda') \in
\Lambda(n)^q$ and has range $r((\nu,\lambda')) = [\nu\lambda']$. Hence $(\mu, [\nu\lambda'])
\in \Lambda(n)^p r((\nu,\lambda'))$, and $(\mu,[\nu\lambda'])(\nu,\lambda') = (\mu\nu,
\lambda') = (\lambda, \lambda')$. For uniqueness, suppose that $(\alpha, \alpha') \in
\Lambda(n)^p$ and $(\beta,\beta') \in \Lambda(n)^q$ satisfy
$(\alpha,\alpha')(\beta,\beta') = (\lambda, \lambda')$. By definition of composition, we
have $(\alpha\beta, \beta') = (\lambda,\lambda')$. This forces $\beta' = \lambda'$ and
$\alpha\beta = \lambda$. Since $d(\alpha) = d((\alpha, \alpha')) = p$ and $d(\beta) =
d((\beta,\beta')) = q$, the factorisation property in $\Lambda$ forces $\alpha = \mu$ and
$\beta = \nu$. Since $(\alpha, \alpha')$ and $(\beta, \beta')$ are composable, we have
$\alpha' = s((\mu, \alpha')) = r ((\nu, \lambda')) = [\nu\lambda']$. Hence $\Lambda(n)$
is a $k$-graph.

To see that $\Lambda(n)$ is row-finite with no sources, take $(r(\lambda), \lambda) \in
\Lambda(n)^0$ and $m \in \NN^k$. Then
\begin{align*}
(r(\lambda),\lambda)\Lambda(n)^m
    &= \setof{(\mu, \mu')}{ \mu \in \Lambda^m, \mu' \in s(\mu)\Lambda^{<n}, [\mu\mu'] = \lambda } \\
    &= \setof{(\mu, \mu')}{ \mu \in \Lambda^m, \mu' \in s(\mu)\Lambda^{[d(\lambda) - m]}, [\mu\mu'] = \lambda }.
\end{align*}
Since $m + [d(\lambda) - m]$ is positive and congruent to $d(\lambda)$ mod $H$, we have
$m + [d(\lambda) - m] \ge d(\lambda)$. Let $p := m + [d(\lambda) - m] - d(\lambda) \in
H$. Then
\[(r(\lambda),\lambda)\Lambda(n)^m
    = \big\{\big((\lambda\nu)(0,m), (\lambda\nu)(m, p+d(\lambda))\big) \mathbin{\big|}  \nu \in s(\lambda)\Lambda^p\big\},
\]
which is finite and nonempty because $s(\lambda)\Lambda^p$ is finite and nonempty.
\end{proof}

To work with Toeplitz-Cuntz-Krieger $\Lambda(n)$-families we first compute
$\Lambda(n)^{\min}((\lambda,\lambda'),(\mu,\mu'))$.

\begin{lem}\label{lem:Lambda(n)min}
For $(\lambda,\lambda'), (\mu,\mu') \in \Lambda(n)$, if $[\lambda\lambda'] \not=
[\mu\mu']$, then $\Lambda(n)^{\min}((\lambda,\lambda'), (\mu,\mu')) = \emptyset$;
otherwise,
\begin{equation}\label{eq:Lmin desc}
\begin{split}
\Lambda(n)^{\min}((\lambda,\lambda'),(\mu,\mu'))
    = \big\{((\alpha,\tau), (\beta,\tau)) \mathbin{\big|} {}&(\alpha,\beta) \in \Lambda^{\min}(\lambda,\mu),\\
                &\quad\tau \in s(\alpha)\Lambda^{<n}, [\alpha\tau] = \lambda' \text{ and } [\beta\tau] = \mu'\big\}.
\end{split}
\end{equation}
\end{lem}
\begin{proof}
If $[\lambda\lambda'] \not= [\mu\mu']$, then $r((\lambda, \lambda') ) \not= r((\mu,
\mu'))$, and so $\Lambda(n)^{\min}((\lambda,\lambda'), (\mu,\mu')) = \emptyset$.

Suppose that $[\lambda\lambda'] = [\mu\mu']$.  Suppose further that $(\alpha,\beta) \in
\Lambda^{\min}(\lambda,\mu)$, that $\tau \in s(\alpha)\Lambda^{<n}$, and that
$[\alpha\tau] = \lambda'$ and $[\beta\tau] = \mu'$. Then $(\alpha, \tau), (\beta, \tau)
\in \Lambda(n)$, and $r((\alpha,\tau)) = \lambda' = s((\lambda,\lambda'))$ and
$r((\beta,\tau)) = \mu' = s((\mu,\mu'))$. We have
\[(\lambda,\lambda')(\alpha,\tau) = (\lambda\alpha, \tau)
    = (\mu\beta, \tau) = (\mu,\mu')(\beta,\tau).
\]
Since $d((\lambda\alpha,\tau)) = d(\lambda\alpha) = d(\lambda) \vee d(\mu) = d( ( \lambda, \lambda') ) \vee d( (\mu, \mu') )$, we have
$((\alpha,\tau), (\beta,\tau)) \in \Lambda(n)^{\min}((\lambda,\lambda'), (\mu,\mu'))$.

Conversely, suppose that $(\alpha, \tau), (\beta,\rho) \in
\Lambda(n)^{\min}((\lambda,\lambda'), (\mu,\mu'))$. Then
\begin{equation}\label{eq:CE calc}
(\lambda\alpha,\tau)
    = (\lambda,\lambda')(\alpha,\tau)
    = (\mu,\mu')(\beta,\rho)
    = (\mu\beta,\rho).
\end{equation}
So $\lambda\alpha = \mu\beta$, and
\[d(\lambda\alpha)
    = d((\lambda\alpha, \tau))
    = d((\lambda,\lambda')(\alpha,\tau))
    = d((\lambda,\lambda')) \vee d((\mu,\mu'))
    = d(\lambda) \vee d(\mu),
\]
so $(\alpha,\beta) \in \Lambda^{\min}(\lambda,\mu)$. By~\eqref{eq:CE calc}, $\tau =
s((\lambda\alpha, \tau)) = s((\mu\beta,\rho)) = \rho$. Since $[\alpha\tau] = r((\alpha,
\tau)) = s((\lambda,\lambda')) = \lambda'$ and $[\beta\tau] = r((\beta,\tau)) =
s((\mu,\mu')) = \mu'$, we deduce that $((\alpha,\tau), (\beta, \rho)) = ((\alpha, \tau),
(\beta, \tau))$ belongs to the right-hand side of~\eqref{eq:Lmin desc}.
\end{proof}

For each $n$ we now construct a homomorphism from $C^{*}(\Lambda)$ to $C^{*}(\Lambda(n))$
analogous to those for directed graphs described in
\cite[Lemma~2.5]{RuizSimsTomforde:arXiv:1312.0507}.

\begin{lem}\label{l:homomorphism KribSolel}
Let $\{t_{\lambda} \}_{\lambda \in \Lambda} \subseteq \Tt C^*(\Lambda)$ and
$\{t_{(\lambda, \lambda')} \}_{(\lambda, \lambda') \in \Lambda(n)} \subseteq \Tt
C^*(\Lambda(n))$ be the generating Toeplitz-Cuntz-Krieger families and let $\{s_{\lambda}
\}_{\lambda \in \Lambda} \subseteq C^*(\Lambda)$ and $\{s_{(\lambda, \lambda')}
\}_{(\lambda, \lambda') \in \Lambda(n)} \subseteq C^*(\Lambda(n))$ be the generating
Cuntz-Krieger families.  For $n \in \NN^{k}$, there are homomorphisms $\ftn{\iota_n}{\Tt
C^*(\Lambda)}{\Tt C^*(\Lambda(n))}$ and
$\ftn{\tilde\iota_n}{C^*(\Lambda)}{C^*(\Lambda(n))}$ such that
\begin{equation}\label{eq:iota formula}
\iota_n(t_\lambda) = \sum_{\lambda' \in s(\lambda)\Lambda^{<n}} t_{(\lambda, \lambda')} \quad\text{and}\quad
\tilde\iota_n(s_\lambda) = \sum_{\lambda' \in s(\lambda)\Lambda^{<n}} s_{(\lambda, \lambda')}.
\end{equation}
The homomorphism $\iota_{n}$ descends to the homomorphism $\tilde{\iota}_{n}$ under the
canonical quotient maps from Toeplitz algebras to Cuntz-Krieger algebras.
\end{lem}
\begin{proof}
For $\lambda \in \Lambda$, define $T_\lambda := \sum_{\lambda' \in
s(\lambda)\Lambda^{<n}} t_{(\lambda, \lambda')} \in \Tt C^*(\Lambda(n))$. We check that
$\{T_\lambda \}_{\lambda \in \Lambda}$ is a Toeplitz-Cuntz-Krieger $\Lambda$-family. Take
$v,w \in \Lambda^0$. Since $\{t_{(r(\nu),\nu)}\}_{ \nu \in\Lambda^{<n} }$ is a set of
mutually orthogonal projections,
\[T_v^* T_w
    = \sum_{\lambda \in v\Lambda^{<n}} t_{(v,\lambda)} \sum_{\mu \in w\Lambda^{<n}} t_{(w, \mu)}
    = \sum_{\lambda \in v\Lambda^{<n}, \mu \in w\Lambda^{<n}} \delta_{(v,\lambda), (w,\mu)} t_{( v, \lambda )}
    = \delta_{v,w} T_v,
\]
and so $\{T_v \}_{v \in \Lambda^{0}}$ are mutually orthogonal projections,
giving~(TCK1).

For $(\lambda,\lambda'), (\mu,\mu') \in \Lambda(n)$,
we have
\[t_{(\lambda,\lambda')}t_{(\mu,\mu')}
    =  \delta_{s((\lambda,\lambda')), r((\mu,\mu'))} t_{(\lambda,\lambda')(\mu,\mu')}
    = \delta_{s(\lambda), r(\mu)} \delta_{\lambda', [\mu\mu']} t_{(\lambda\mu, \mu')}.
\]
Hence, for $\lambda, \mu \in \Lambda$,
\begin{align*}
T_\lambda T_\mu
    &= \sum_{\lambda' \in s(\lambda)\Lambda^{<n}, \mu' \in s(\mu)\Lambda^{<n}} t_{(\lambda,\lambda')}t_{(\mu,\mu')} \\
    &= \sum_{\lambda' \in s(\lambda)\Lambda^{<n}, \mu' \in s(\mu)\Lambda^{<n}} \delta_{s(\lambda), r(\mu)} \delta_{\lambda', [\mu\mu']} t_{(\lambda\mu,\mu')} \\
    &= \sum_{\mu' \in s(\mu)\Lambda^{<n}} \delta_{s(\lambda), r(\mu)} t_{(\lambda\mu,\mu')}
    = \delta_{s(\lambda), r(\mu)} T_{\lambda\mu}.
\end{align*}
So $\{ T_{\lambda} \}_{ \lambda \in \Lambda}$ satisfies~(TCK2).

For (TCK3)~and~(TCK4), fix $\lambda,\mu \in \Lambda$. We calculate:
\begin{align*}
T^*_\lambda T_\mu
    &= \sum_{\lambda' \in s(\lambda)\Lambda^{<n}, \mu' \in s(\mu)\Lambda^{<n}} t^*_{(\lambda,\lambda')} t_{(\mu,\mu')} \\
    &= \sum_{\substack{\lambda' \in s(\lambda)\Lambda^{<n} \\ \mu' \in s(\mu)\Lambda^{<n}}}
        \Big(\sum_{((\alpha,\alpha'), (\beta,\beta')) \in \Lambda(n)^{\min}((\lambda,\lambda'),(\mu,\mu'))} t_{(\alpha,\alpha')} t^*_{(\beta,\beta')}\Big)\\
    &= \sum_{\substack{\lambda' \in s(\lambda)\Lambda^{<n} \\ \mu' \in s(\mu)\Lambda^{[d(\lambda\lambda') - d(\mu)]}}}
        \sum_{\substack{(\alpha, \beta) \in \Lambda^{\min}(\lambda,\mu), \tau \in s(\alpha)\Lambda^{<n}, \\
                        [\alpha\tau] = \lambda', [\beta\tau] = \mu'}}
            t_{(\alpha,\tau)} t^*_{(\beta,\tau)} \quad\text{(by Lemma~\ref{lem:Lambda(n)min})}\\
    &= \sum_{(\alpha, \beta) \in \Lambda^{\min}(\lambda,\mu)} \Big(\sum_{\tau \in s(\alpha)\Lambda^{<n}}
            t_{(\alpha,\tau)} t^*_{(\beta,\tau)}\Big).
\end{align*}
If $(\alpha,\beta) \in \Lambda^{\min}(\lambda, \mu)$ and $\tau, \rho \in \Lambda^{<n}$,
then $t_{(\alpha,\tau)} t^*_{(\beta,\rho)} \not= 0$ forces $\tau = s((\alpha,\tau)) =
s((\beta,\rho)) = \rho$. So summing over two variables $\tau \in s(\alpha)\Lambda^{<n}$
and $\rho \in s(\alpha)\Lambda^{<n} = s(\beta)\Lambda^{<n}$ adds no new nonzero terms to
the final line of the preceding calculation. Hence
\[T^*_\lambda T_\mu
    = \sum_{(\alpha, \beta) \in \Lambda^{\min}(\lambda,\mu)} \Big(\sum_{\tau \in s(\alpha)\Lambda^{<n}, \rho \in s(\beta)\Lambda^{<n}}
            t_{(\alpha,\tau)} t^*_{(\beta,\rho)}\Big)
    = \sum_{(\alpha,\beta) \in \Lambda^{\min}(\lambda,\mu)} T_\alpha T^*_\beta.
\]
This gives~(TCK4); and~(TCK3) then follows from~(TCK1) because
$\Lambda^{\min}(\lambda,\lambda) = \{(s(\lambda), s(\lambda))\}$. Hence
$\{T_\lambda\}_{\lambda \in \Lambda}$ is a Toeplitz-Cuntz-Krieger $\Lambda$-family.

The universal property of $\Tt C^*(\Lambda)$ gives a homomorphism $\iota_n : \Tt
C^*(\Lambda) \to \Tt C^*(\Lambda(n))$ such that
\[
\iota_n(t_\lambda) = T_\lambda = \sum_{\lambda' \in s(\lambda)\Lambda^{<n}} t_{(\lambda, \lambda')}
\]
for all $\lambda$. To see that $\iota_{n}$ descends to the desired homomorphism
$\ftn{\tilde\iota_{n}}{C^*(\Lambda)}{C^*(\Lambda(n))}$, let $\ftn{q_n}{\Tt
C^*(\Lambda(n))}{C^*(\Lambda(n))}$ denote the quotient map. We check that the family
$S_\lambda := q_n(T_\lambda)$ satisfies~(CK). For $v \in \Lambda^0$ and $m \in \NN^k$,
\[\sum_{\lambda \in v\Lambda^m} S_\lambda S^*_\lambda
    = \sum_{\lambda \in v\Lambda^m} \sum_{\mu,\nu \in s(\lambda)\Lambda^{<n}} s_{(\lambda,\mu)}s^*_{(\lambda,\nu)}.
\]
As above, $s_{(\lambda,\mu)}s^*_{(\lambda,\nu)} \not= 0$ forces $s((\lambda,\mu)) =
s((\lambda,\nu))$, and so $\mu = \nu$. Using this at the first equality and relation~(CK)
in $C^*(\Lambda(n))$ at the second-last equality, we calculate:
\begin{align*}
\sum_{\lambda \in v\Lambda^m} S_\lambda S^*_\lambda
    &= \sum_{\lambda \in v\Lambda^m} \sum_{\lambda' \in s(\lambda)\Lambda^{<n}} s_{(\lambda,\lambda')}s^*_{(\lambda,\lambda')}
    = \sum_{(\lambda,\lambda') \in \Lambda(n)^m, r(\lambda) = v} s_{(\lambda,\lambda')}s^*_{(\lambda,\lambda')} \\
    &= \sum_{\alpha \in v\Lambda^{<n}} \sum_{(\lambda,\lambda') \in \Lambda(n)^m, [\lambda\lambda'] = \alpha} s_{(\lambda,\lambda')}s^*_{(\lambda,\lambda')}
    = \sum_{\alpha \in v\Lambda^{<n}} \sum_{(\lambda,\lambda') \in (v,\alpha)\Lambda(n)^m} s_{(\lambda,\lambda')}s^*_{(\lambda,\lambda')} \\
    &= \sum_{\alpha \in v\Lambda^{<n}} s_{(v,\alpha)}
    = S_v.
\end{align*}
So $\{ S_\lambda \}_{ \lambda \in \Lambda }$ is a Cuntz-Krieger $\Lambda$-family. The
universal property of $C^*(\Lambda)$ now gives a homomorphism
$\ftn{\tilde{\iota}_{n}}{C^*(\Lambda)}{C^*(\Lambda(n))}$ such that
$\tilde{\iota}_{n}(s_\lambda) = S_\lambda = q_n(\iota_{n}(t_\lambda))$. The quotient maps
$\ftn{q}{\Tt C^*(\Lambda)}{C^*(\Lambda)}$ and $\ftn{q_n}{\Tt
C^*(\Lambda(n))}{C^*(\Lambda(n))}$ satisfy $\tilde{\iota}_{n} \circ q = q_n \circ
\iota_{n}$, and the formula for $\tilde{\iota}_n$ in~\eqref{eq:iota formula} follows.
\end{proof}

Now we construct an analogue of the map of
\cite[Proposition~3.1]{RuizSimsTomforde:arXiv:1312.0507}. For $\lambda \in \Lambda$, we
write $T(\lambda)$ for the unique path such that $\lambda = [\lambda]T(\lambda)$. Note
that $d( T( \lambda ) ) = d(\lambda) - [d(\lambda)] \in H_{n}$.

For a set $X$, we write $\Kk_X$ for the $C^*$-algebra of compact operators on
$\ell^2(X)$, with canonical matrix units $\setof{\theta_{x, y}}{x,y \in X}$.

\begin{lem}\label{lem:return homomorphism}
Let $\{s_{\lambda} \}_{\lambda \in \Lambda} \subseteq C^*(\Lambda)$ and $\{s_{(\lambda,
\lambda')} \}_{(\lambda, \lambda') \in \Lambda(n)} \subseteq C^*(\Lambda(n))$ be the
generating Cuntz-Krieger families. There is a homomorphism
$\ftn{j_n}{C^*(\Lambda(n))}{C^*(\Lambda) \otimes \Kk_{\Lambda^{<n}}}$ such that
\[j_n(s_{(\lambda,\lambda')})
    = s_{T(\lambda\lambda')} \otimes \theta_{[\lambda\lambda'], \lambda'}
        \quad\text{for all $(\lambda,\lambda') \in \Lambda(n)$}.
\]
\end{lem}
\begin{proof}
We just have to check the Cuntz-Krieger relations for the elements
$S_{(\lambda,\lambda')} := s_{T(\lambda\lambda')} \otimes \theta_{[\lambda\lambda'],
\lambda'}$. For $\lambda \in \Lambda^{<n}$, we have $T(\lambda) = s(\lambda)$ and
$[\lambda] = \lambda$. Thus $\{ S_{(r(\lambda),\lambda)} = s_{s(\lambda)}
\otimes \theta_{\lambda, \lambda} \}_{ ( r(\lambda), \lambda) \in \Lambda(n)^{0}}$ is a collection of mutually orthogonal projections.

Let $(\lambda, \lambda')$ and $( \mu, \mu' )$ be elements in $\Lambda(n)$.  Then
\begin{align*}S_{(\lambda,\lambda')}S_{(\mu,\mu')}
    &= (s_{T(\lambda\lambda')} \otimes \theta_{[\lambda\lambda'], \lambda'})
        (s_{T(\mu\mu')} \otimes \theta_{[\mu\mu'], \mu'}) \\
    &= \delta_{ \lambda', [ \mu \mu' ] } s_{T(\lambda\lambda')}s_{T(\mu\mu')} \otimes \theta_{[\lambda\lambda'], \mu'}\\
    &= \delta_{ s( (\lambda, \lambda') ), r( (\mu, \mu' )) } s_{T(\lambda\lambda')}s_{T(\mu\mu')} \otimes \theta_{[\lambda\lambda'], \mu'}.
\end{align*}
Suppose that $\lambda' = s((\lambda,\lambda')) = r((\mu,\mu'))  = [\mu\mu']$.  Then
$r(T(\mu\mu')) = s([\mu\mu']) = s(\lambda') = s(T(\lambda\lambda'))$. Moreover,
$\lambda\lambda'T(\mu\mu') = \lambda\mu\mu'$ because $[\mu\mu'] = \lambda'$. So
$T(\lambda\lambda'T(\mu\mu')) = T(\lambda\mu\mu') = T(\lambda\lambda')T(\mu\mu')$. Since
$d(T(\mu\mu')) \in H$, we also have $[\lambda\lambda'] = [\lambda\lambda'T(\mu\mu')]$,
and hence $[\lambda\mu\mu'] = [\lambda\lambda'T(\mu\mu')] = [\lambda\lambda']$. Putting
these two observations together, we deduce that
\begin{align*}
S_{(\lambda,\lambda')}S_{(\mu,\mu')}
    &= \delta_{ s( (\lambda, \lambda') ), r( (\mu, \mu' )) } s_{T(\lambda\mu\mu')} \otimes \theta_{[\lambda\mu\mu'], \mu'} \\
    &= \delta_{ s( (\lambda, \lambda') ), r( (\mu, \mu' )) } S_{(\lambda\mu,\mu')} \\
    &= \delta_{ s( (\lambda, \lambda') ), r( (\mu, \mu' )) } S_{(\lambda,\lambda')(\mu,\mu')},
\end{align*}
establishing~(TCK2).

For~(TCK3), fix $(\lambda,\lambda') \in \Lambda(n)$. We have
\begin{align*}
S^*_{(\lambda,\lambda')} S_{(\lambda,\lambda')}
    &= \big(s^*_{T(\lambda\lambda')} \otimes \theta_{\lambda', [\lambda\lambda']}\big)
            \big(s_{T(\lambda\lambda')} \otimes \theta_{[\lambda\lambda'], \lambda'}\big)\\
    &= s_{s(T(\lambda\lambda'))} \otimes \theta_{\lambda', \lambda'}
    = s_{s(\lambda')} \otimes \theta_{\lambda', \lambda'}
    = S_{(r(\lambda'),\lambda')}
    = S_{s((\lambda,\lambda'))}.
\end{align*}

Finally for~(CK), fix $(v,\lambda) \in \Lambda(n)^0$ and $m \in \NN^k$. Then
\begin{align*}
\sum_{(\mu,\mu') \in (v,\lambda)\Lambda(n)^m} S_{(\mu,\mu')} S^*_{(\mu,\mu')}
    &= \sum_{\mu \in v\Lambda^m, \mu' \in s(\mu)\Lambda^{<n}, [\mu\mu'] = \lambda}
        S_{(\mu,\mu')} S^*_{(\mu,\mu')} \\
    &= \sum_{\mu \in v\Lambda^m, \mu' \in s(\mu)\Lambda^{<n}, [\mu\mu'] = \lambda}
        s_{T(\mu\mu')} s^*_{T(\mu\mu')} \otimes \theta_{[\mu\mu'], [\mu\mu']}\\
    &= \sum_{\mu \in v\Lambda^m, \mu' \in s(\mu)\Lambda^{<n}, [\mu\mu'] = \lambda}
        s_{T(\mu\mu')} s^*_{T(\mu\mu')} \otimes \theta_{\lambda, \lambda}.
\end{align*}
Let $p := m + [d(\lambda) - m]$. Then $p \ge 0$ and $[p] = d(\lambda)$, so $p \ge
d(\lambda)$. The factorisation property implies that $\setof{\mu\mu'}{\mu \in v\Lambda^m,
\mu' \in s(\mu)\Lambda^{<n}, [\mu\mu'] = \lambda } = \setof{\lambda\nu}{ \nu \in
s(\lambda)\Lambda^{p - d(\lambda)} }$. For $\nu \in s(\lambda)\Lambda^{p - d(\lambda)}$,
we have $T(\lambda\nu) = \nu$. We deduce that
\[\sum_{(\mu,\mu') \in (v,\lambda)\Lambda(n)^m} S_{(\mu,\mu')} S^*_{(\mu,\mu')}
    = \sum_{\nu \in s(\lambda)\Lambda^{p - d(\lambda)}} s_\nu s^*_\nu \otimes \theta_{\lambda,\lambda}
    = s_{s(\lambda)} \otimes \theta_{\lambda,\lambda}
    = S_{(v,\lambda)}
\]
as required. Now the universal property of $C^*(\Lambda(n))$ gives the desired
homomorphism $j_n$.
\end{proof}


\section{Cartesian products, $1$-graphs, and the Kribs-Solel construction}\label{sec:CP1-graphKS}

Kumjian and Pask show that a cartesian product $\Lambda \times \Gamma$ of higher-rank
graphs is itself a higher-rank graph with $C^*(\Lambda \times \Gamma) \cong C^*(\Lambda)
\otimes C^*(\Gamma)$ (\cite[Corollary~3.5(iv)]{KumjianPask:NYJM00}). In this section we
show that the construction of the preceding section is compatible with the
cartesian-product operation, and also that the construction of \cite{KribsSolel:JAMS07}
and that of the preceding section are compatible via the passage from directed graphs to
$1$-graphs. We will use these results to compute the map $\ftn{K_*(j_n \circ
\tilde\iota_n)}{K_*(C^*(\Lambda))}{K_*(C^*(\Lambda) \otimes \Kk_{\Lambda^{<n}} )} \cong
K_*(C^*(\Lambda))$ for a particular class of $2$-graphs $\Lambda$.

For $i = 1,2$, let $(\Lambda_{i}, d_{i})$ be a $k_{i}$-graph.  The product category
$(\Lambda_{1} \times \Lambda_{2}, d_{1} \times d_{2})$ is a $(k_{1} + k_{2})$-graph with
degree map $(d_{1} \times d_{2})((\mu_{1}, \mu_{2})) = (d_{1} (\mu_{1}), d (\mu_{2}))$.
If each $(\Lambda_{i}, d_{i})$ is row-finite with no sources, then so is $(\Lambda_{1}
\times \Lambda_{2}, d_{1} \times d_{2})$. By
\cite[Corollary~3.5(iv)]{KumjianPask:NYJM00}, there exists an isomorphism
$\ftn{\Theta_{\Lambda_{1} \times \Lambda_{2}}}{C^{*} (\Lambda_{1} \times
\Lambda_{2})}{C^{*} (\Lambda_{1}) \otimes C^{*} (\Lambda_{2})}$ such that
$\Theta_{\Lambda_{1} \times \Lambda_{2}} (s_{(\mu_{1}, \mu_{2})}) = s_{\mu_{1}} \otimes
s_{\mu_{2}}$.

\begin{rmk}\label{rmk:pathcat}
Let $(\Lambda_{i}, d_{i})$ be a row-finite $k_{i}$-graph with no sources for $i = 1,2$.
For $n_1 \in \NN^{k_1}$ and $n_2 \in \NN^{k_2}$, the functor that sends $((\lambda_{1},
\lambda_{2}), (\lambda_{1}', \lambda_{2}')) \in (\Lambda_{1} \times \Lambda_{2})((n_{1},
n_{2}))$ to $((\lambda_{1}, \lambda_{1}'), (\lambda_{2}, \lambda_{2}')) \in  \Lambda_{1}
(n_{1}) \times \Lambda_{2} (n_{2})$ is an isomorphism of $(k_{1} + k_{2})$-graphs. So
there is an isomorphism $C^{*} ((\Lambda_{1} \times \Lambda_{2})((n_{1}, n_{2})) ) \cong
C^{*}(\Lambda_{1} (n_{1}) \times \Lambda_{2} (n_{2}) )$ sending $s_{((\lambda_{1},
\lambda_{2}), (\lambda_{1}', \lambda_{2}'))}$ to $s_{((\lambda_{1}, \lambda_{1}'),
(\lambda_{2}, \lambda_{2}'))}$.
\end{rmk}

We show that the homomorphism in Lemma~\ref{l:homomorphism KribSolel} is compatible with
the isomorphism $C^{*}((\Lambda_{1} \times \Lambda_{2})((n_{1}, n_{2})) ) \cong
C^{*}(\Lambda_{1} (n_{1}) \times \Lambda_{2} (n_{2}))$ just described.

\begin{lem}\label{l:prodgraph}
For $i = 1,2$, let $(\Lambda_{i}, d_{i})$ be a row-finite $k_{i}$-graph with no sources.
For $(n_{1}, n_{2})  \in\NN^{k_{1}} \times \NN^{k_{2}}$, we have $(\tilde{\iota}_{n_{1}} \otimes \tilde{\iota}_{n_{2}}) \circ \Theta_{\Lambda_{1} \times \Lambda_{2}}
    = (\Theta_{\Lambda_{1}(n_{1}) \times \Lambda_{2}(n_{2})}) \circ \tilde{\iota}_{(n_{1}, n_{2})}$.
\end{lem}
\begin{proof}
Let $(\mu_{1}, \mu_{2}) \in  \Lambda_{1} \times \Lambda_{2}$.  Then
\[(\tilde{\iota}_{n_{1}} \otimes \tilde{\iota}_{n_{2}}) \circ \Theta_{\Lambda_{1} \times \Lambda_{2}} (s_{(\mu_{1}, \mu_{2})})
    = \tilde{\iota}_{n_{1}} \otimes \tilde{\iota}_{n_{2}}  (s_{\mu_{1}} \otimes s_{\mu_{2}})
    = \sum_{\substack{\nu \in s(\mu_{1}) \Lambda_{1}^{<n_{1}}  \\ \nu' \in s(\mu_{2}) \Lambda_{2}^{<n_{2}}}} s_{(\mu_{1}, \nu)} \otimes s_{(\mu_{2}, \nu')}.
\]
Identifying $((\Lambda_{1} \times \Lambda_{2})((n_{1}, n_{2})), d_{1} \times d_{2})$ with
$(\Lambda_{1} (n_{1}), d_{1}) \times (\Lambda_{2} (n_{2}) , d_{2})$ as in
Remark~\ref{rmk:pathcat}, we have
\begin{align*}
\Theta_{\Lambda_{1}(n_{1}) \times \Lambda_{2}(n_{2})} \circ \tilde{\iota}_{(n_{1}, n_{2})} (s_{(\mu_{1}, \mu_{2})})
    &= \Theta_{\Lambda_{1}(n_{1}) \times \Lambda_{2}(n_{2})}
         \Big(\sum_{(\alpha, \beta) \in s((\mu_{1}, \mu_{2})) (\Lambda_{1} \times \Lambda_{2})^{< (n_{1}, n_{2})}}
                    s_{((\mu_{1}, \alpha), (\mu_{2}, \beta))}
         \Big) \\
	&= \sum_{\substack{\alpha \in s(\mu_{1}) \Lambda^{<n_{1}} \\ \beta \in s(\mu_{2}) \Lambda^{< n_{2}}}}
            \Theta_{\Lambda_{1}(n_{1}) \times \Lambda_{2}(n_{2})} (s_{((\mu_{1}, \alpha), (\mu_{2}, \beta))}) \\
	&= \sum_{\substack{\alpha \in s(\mu_{1}) \Lambda^{<n_{1}} \\ \beta \in s(\mu_{2}) \Lambda^{< n_{2}}}}
            s_{(\mu_{1}, \alpha)} \otimes s_{(\mu_{2}, \beta)}.
\end{align*}
Therefore, $\tilde{\iota}_{n_{1}} \otimes \tilde{\iota}_{n_{2}} \circ \Theta_{\Lambda_{1}
\times \Lambda_{2}} = \Theta_{\Lambda_{1}(n_{1}) \times \Lambda_{2}(n_{2})} \circ
\tilde{\iota}_{(n_{1}, n_{2})}$.
\end{proof}

We will need to apply Lemma~\ref{l:prodgraph} where $\Lambda_1$ and $\Lambda_2$ are the
$1$-graphs associated to directed graphs $E$ and $F$, and relate this to
\cite[Lemma~2.5]{RuizSimsTomforde:arXiv:1312.0507} for $C^*(E)$ and $C^*(F)$. We
therefore find ourselves in an unfortunate clash of conventions. The convention used in
\cite{RuizSimsTomforde:arXiv:1312.0507} is that of \cite{KPRR, KPR} where,
for historical reasons, the partial
isometries in a Cuntz-Krieger family point in the opposite direction to the edges in the
graph. This is at odds with the $k$-graph convention where the partial isometries go in
the same direction as the morphisms in the $k$-graph. To deal with this, we take the
approach that the range and source maps are interchanged when passing from a directed
graph $E$ to its path category $E^*$.

We recall the definition of the Toeplitz algebra $\Tt C^*(E)$ and the Cuntz-Krieger
algebra $C^*(E)$ of a directed graph $E$ as used in
\cite{RuizSimsTomforde:arXiv:1312.0507}. Let $E = (E^{0}, E^{1}, r_{E}, s_{E})$ be a
row-finite directed graph with no sinks (so $0 < |\setof{e \in E^{1}}{s_{E} (e) = v}| <
\infty$  for $v \in E^{0}$). Then $\Tt C^*(E)$ is the universal $C^*$-algebra generated
by mutually orthogonal projections $\{q_v\}_{v \in E^0 }$ and elements $\{t_e\}_{ e
\in E^1 }$ such that
\begin{itemize}
\item[1)] $t^*_e t_e = q_{r_{E}(e)}$ for all $e \in E^1$, and
\item[2)] $q_v \ge \sum_{e \in E^1, s_{E}(e) = v} t_e t^*_e$ for each $v \in E^0$.
\end{itemize}
The graph $C^{*}$-algebra $C^*(E)$ is the universal $C^*$-algebra generated by mutually
orthogonal projections $\{p_v\}_{v \in E^0 }$ and elements $\{s_e\}_{ e \in E^1 }$
such that
\begin{itemize}
\item[3)] $s^*_e s_e = p_{r_{E}(e)}$ for all $e \in E^1$, and
\item[4)] $p_v = \sum_{e \in E^1, s_{E}(e)= v} t_e t^*_e$ for each $v \in E^0$.
\end{itemize}

We recall the construction described in \cite[Section~4]{KribsSolel:JAMS07}.  Given $m
\in \NN$ and a directed graph $E = (E^{0}, E^{1}, r_{E}, s_{E})$, we define $E(m)$ to be
the directed graph with
\begin{align*}
E(m)^0 &= E^{<m},&
E(m)^1 &= \setof{(e, \mu)}{ e \in E^1, \mu \in E^{<m}, r_{E}(e) = s_{E}(\mu) },\\
r_{E(m)}((e,\mu)) &= \mu,&
s_{E(m)}((e,\mu)) &= \begin{cases}
		e\mu &\text{if $|\mu| < m-1$}\\
		s_{E}(e) &\text{if $|\mu| = m-1$.}
	\end{cases}
\end{align*}

The next lemma is due to James Rout, and will appear in his PhD thesis. We thank James
for providing us with the details (a proof appears in
\cite[Lemma~2.5]{RuizSimsTomforde:arXiv:1312.0507}).

\begin{lem}[Rout]\label{l:Rout}
Let $E$ be a row-finite directed graph and take $m \ge 1$.  There is an injective homomorphism $\ftn{\iota_{m, E}}{\Tt
C^*(E)}{\Tt C^*(E(m))}$ such that
\[\iota_{m, E}(q_v) = \sum_{\substack{\mu \in E^{<m} \\ s_{E} (\mu) = v}} q^m_\mu\quad\text{and}\quad
\iota_{m, E}(t_e) = \sum_{(e,\mu)  \in E(m)^{1}} t^m_{(e,\mu)},
\]
where $\{q^{m}_{\mu} , t_{(e, \nu)}^{m} \}_{ \mu \in E(m)^{0} , (e, \nu ) \in E(m)^{1}}$
are the universal generators of $\Tt C^{*} (E)$.  The map $\iota_{m, E}$ descends to an
injective homomorphism $\ftn{\tilde{\iota}_{m, E}}{C^*(E)}{C^*(E(m))}$.
\end{lem}

We describe canonical isomorphisms $C^{*}(E) \cong C^{*} (E^{*})$ and $C^{*} (E(m)) \cong
C^{*} (E^{*} (m))$ and show that these isomorphisms intertwine the homomorphism
$\tilde{\iota}_{m, E}$ of Lemma~\ref{l:Rout} and the homomorphism $\tilde{\iota}_{m}$ of
Lemma~\ref{l:homomorphism KribSolel}.

\begin{rmk}\label{rmk:C*(E)congC*(E*)}
Let $E$ be a row-finite directed graph with no sinks, and let $E^*$ be its path-category regarded
as a row-finite $1$-graph with no sources. Let $\{p_v, s_e\}_{v \in E^0, e \in E^1}$ be
the universal generators of $C^*(E)$ and let $\{S_\lambda \}_{ \lambda \in E^* }$ be the
universal generators of $C^*(E^*)$. By~\cite[Examples~1.7]{KumjianPask:NYJM00}, there is
an isomorphism $\ftn{\psi_E}{C^*(E)}{C^*(E^*)}$ such that $\psi_E(p_v) = S_v$ and
$\psi_E(s_e) = S_e$ for all $v \in E^0$ and $e \in E^1$.
\end{rmk}

\begin{lem}\label{lem:E*(m)congE(m)*}
Let $E$ be a row-finite directed graph with no sinks, and let $E^*$ be its path-category
regarded as a row-finite $1$-graph with no sources. There is an isomorphism of $1$-graphs
$E(m)^* \cong E^*(m)$ extending the identity map on $(E(m)^*)^1 = E^*(m)^1$. There is an
isomorphism $C^*(E(m)^*) \cong C^*(E^*(m))$ satisfying $s_{(e,\mu)} \mapsto s_{(e,\mu)}$ for
$(e,\mu) \in E^*(m)^1 = (E(m)^*)^1$.
\end{lem}
\begin{proof}
Example~1.3 of \cite{KumjianPask:NYJM00} says that $1$-graphs $\Lambda$ and $\Gamma$ are
isomorphic if and only if there is a bijection $\Lambda^1 \to \Gamma^1$ that intertwines
range maps and source maps. Since $(e,\mu) \mapsto ( e, \mu )$ is such a bijection between $(E(m)^*)^1$ and
$E^*(m)^1$, there is an isomorphism $E^*(m) \cong E(m)^*$ as claimed. Since isomorphic
1-graphs have canonically isomorphic $C^*$-algebras, the result follows.
\end{proof}

\begin{lem}\label{l:equivalence construction}
Let $E$ be a row-finite directed graph with no sinks, and fix $m \in \NN \setminus\{0\}$.
Identify $C^*(E(m)^*)$ with $C^*(E^*(m))$ using Lemma~\ref{lem:E*(m)congE(m)*}. Then the
isomorphisms $\ftn{\psi_{E}}{C^{*} (E)}{C^{*} (E^{*})}$ and $\ftn{\psi_{E(m)}}{C^{*}
(E(m))}{C^{*} (E^{*}(m))}$ of Remark~\ref{rmk:C*(E)congC*(E*)} satisfy $\tilde{\iota}_{m}
\circ \psi_{E} = \psi_{E(m)} \circ \tilde{\iota}_{m,E}$.
\end{lem}
\begin{proof}
Let $v \in E^{0}$.  Then
\[\tilde{\iota}_{m} \circ \psi_{E}  (p_{v})
    = \tilde{\iota}_{m} (S_{v})
    = \sum_{\lambda \in v (E^{*})^{< m}} S_{(v, \lambda)}
    = \sum_{\substack{\lambda \in E^{< m} \\  s_{E}(\lambda) = v}} S_{(v, \lambda)},
\]
and
\[\psi_{E(m)} \circ \tilde{\iota}_{m,E} (p_{v})
    = \psi_{E(m)} \Big(\sum_{\substack{\lambda \in E^{<m} \\ s_{E} (\lambda) = v}} p_{\lambda} \Big)
    = \sum_{\substack{\lambda \in E^{<m} \\ s_{E} (\lambda) = v}} S_{(s_{E} (\lambda), \lambda)}
    = \sum_{\substack{\lambda \in E^{<m} \\ s_{E} (\lambda) = v}} S_{(v, \lambda)}.
\]
Thus, $\tilde{\iota}_{m} \circ \psi_{E}  (p_{v}) = \psi_{E(m)} \circ \tilde{\iota}_{m,E}
(p_{v})$ for all $v \in E^{0}$. For $e \in E^{1}$,
\[
\tilde{\iota}_{m} \circ \psi_{E}  (s_{e})
    = \tilde{\iota}_{m} (S_{e})
    = \sum_{\lambda \in s(e) (E^{*})^{<m}} S_{(e, \lambda)}
    = \sum_{\substack{\lambda \in E^{<m}  \\ s_{E} (\lambda)  = r_{E} (e)}} S_{(e, \lambda)}
    = \sum_{(e, \lambda) \in E(m)^{1}} S_{(e, \lambda)},
\]
and
\[\psi_{E(m)} \circ \tilde{\iota}_{m,E} (s_{e})
    = \psi_{E(m)} \Big(\sum_{(e, \lambda) \in E(m)^{1}} s_{(e, \lambda)} \Big)
    = \sum_{(e, \lambda) \in E(m)^{1}} S_{(e, \lambda)}.
\]
So $\tilde{\iota}_{m} \circ \psi_{E}  (s_{e}) = \psi_{E(m)} \circ \tilde{\iota}_{m,E}
(s_{e})$ for all $e \in E^{1}$.  Since $C^{*} (E)$ is generated by $\{p_{v}, s_{e} \}_{ v
\in E^{0} , e \in E^{1}}$, we see that $\tilde{\iota}_{m} \circ \psi_{E}  = \psi_{E(m)}
\circ \tilde{\iota}_{m,E}$.
\end{proof}


\section{Asymptotic order-1 approximations}\label{sec:order1factorisations}

In this section, we show that given a row-finite $2$-graph with no sources, the family of
homomorphisms $(\tilde{\iota}_{n})_{n \in \NN^k}$ has an asymptotic order-1 approximation
through AF-algebras.  Thus, the family $( j_{n} \circ \tilde{\iota}_{n} )_{n \in
\NN^{k}}$ has an asymptotic order-1 approximation through AF-algebras.  We will use this
family of homomorphisms in the next section to prove that the nuclear dimension of a
UCT-Kirchberg algebra with trivial $K_{0}$ and finite $K_{1}$ is~1.

If $\ftn{f}{\NN^k}{\RR}$ is a function, then we write $\lim_{n \to \infty} f(n) = 0$ if
for every $\varepsilon > 0$ there exists $N \in \NN^k$ such that $|f(n)| < \varepsilon$
whenever $n \ge N$ in $\NN^k$.

Recall that a completely positive map $\ftn{\phi}{A}{B}$ has \emph{order-zero} if for $a,
b \in A_{+}$ with $ab = 0$, we have $\phi(a)\phi(b) = 0$. Suppose that $(\beta_n)_{n\in
\NN^k}$ is a family of homomorphisms $\ftn{\beta_n}{A}{B_n}$, and let $\mathcal{C}$ be a
class of $C^*$-algebras. Following
\cite[Definition~2.8]{RuizSimsTomforde:arXiv:1312.0507}\footnote{In the preprint version
of \cite{RuizSimsTomforde:arXiv:1312.0507} the authors mistakenly require just that each
$F_n$, rather than each $F_n^{(i)}$, belonged to $\mathcal{C}$; the intention was that
$\mathcal{C}$ should be closed under hereditary subalgebras and direct sums.}, a family
$(F_n, \phi_n, \psi_n)_{n\in \NN^k}$ is an \emph{asymptotic order-$r$ factorisation of
the family $(\beta_n)$ through elements of $\mathcal{C}$} if each $F_n$ is a direct sum
$F_n = \bigoplus^r_{i=0} F_n^{(i)}$ of $C^*$-algebras $F_n^{(i)} \in \mathcal{C}$, each
$\ftn{\psi_n}{A}{F_n}$ is a completely positive contraction, each
$\ftn{\phi_n}{F_n}{B_n}$ restricts to an order-zero completely positive contraction on
each $F_n^{(i)}$, and $\lim_{n \to \infty} \|\phi_n \circ \psi_n(a) - \beta_n(a)\| = 0$
for each $a \in A$. We say that $(F_n, \phi_n, \psi_n)_{n \in \NN^k}$ is an
\emph{asymptotic order-$r$ factorisation of $\ftn{\beta}{A}{B}$} if it is an asymptotic
order-$r$ factorisation of $(\beta)_{n \in \NN^k}$.

\begin{rmk}\label{rmk:approximation subseq}
Suppose that $(\ftn{\beta_n}{A}{B_n})_{n \in \NN^k}$ has an asymptotic order-$r$
factorisation through elements of $\mathcal{C}$. Then for any strictly increasing
sequence $(n^m)_{m \in \NN}$ in $\NN^k$ such that $n^m_j \to \infty$ as $m \to \infty$
for each $j \le k$, the sequence $(\beta_{n^m})_{m \in \NN}$ has an asymptotic order-$r$
factorisation through elements of $\mathcal{C}$ in the sense of
\cite[Definition~2.8]{RuizSimsTomforde:arXiv:1312.0507}.
\end{rmk}

Throughout this section, we use the following notation.  Let $\Lambda$ be a row-finite
$k$-graph with no sources and let $n \in \NN^{k}$. Then $\{t_{\lambda} \}_{\lambda \in
\Lambda} \subseteq \Tt C^{*} (\Lambda)$ and $\{T_{(\lambda,\lambda')}
\}_{(\lambda,\lambda') \in \Lambda(n)} \subseteq \Tt C^{*} (\Lambda(n))$ will be the
universal generating Toeplitz-Cuntz-Krieger families, and $\{s_{\lambda} \}_{\lambda \in
\Lambda } \subseteq C^{*} (\Lambda)$ and $\{S_{(\lambda, \lambda')} \}_{(\lambda,
\lambda') \in \Lambda(n)} \subseteq C^{*} (\Lambda(n))$ will be the universal generating
Cuntz-Krieger families. We will regard $\Tt C^{*} (\Lambda)$ as a sub-$C^{*}$-algebra of
$B(\ell^{2} (\Lambda))$. When $s(\mu) = s(\nu)$, we have
\begin{equation}\label{eq:Toeplitz series}
    t_{\mu} t_{\nu}^{*} = \sum_{\tau \in s(\mu)\Lambda} \theta_{\mu \tau, \nu \tau},
\end{equation}
where the series converges in the strict topology.

First we construct a homomorphism that we will use to define the maps $\phi_n$ in our
asymptotic factorisation.

\begin{lem}\label{lem:Lambda homomorphism}
Let $\Lambda$ be a row-finite $k$-graph with no sources. For $p,n \in \NN^{k}$, there is
a homomorphism $\ftn{\Gamma_p^{p+n}}{\bigoplus_{v \in \Lambda^0} \Kk_{\Lambda^{[p, p+n)}
v}}{\Tt C^*(\Lambda(n))}$ such that
\[\Gamma_p^{p+n}(\theta_{\mu,\nu}) = T_{(\mu, s(\mu))} T^*_{(\nu, s(\nu))}
\]
for all $\mu,\nu \in \Lambda^{[p, p+n)}$ with $s(\mu) = s(\nu)$.
\end{lem}
\begin{proof}
We just have to check that the elements $\{T_{(\mu, s(\mu))} T^*_{(\nu, s(\nu))}\}_{\mu,
\nu \in \Lambda^{[p, p+n)} , s(\mu) = s(\nu) }$ are nonzero and are matrix units in the
sense that $(T_{(\mu, s(\mu))} T^*_{(\nu, s(\nu))})^* = T_{(\nu, s(\nu))} T^*_{(\mu,
s(\mu))}$ and
\[
T_{(\mu, s(\mu))} T^*_{(\nu, s(\nu))}T_{(\mu', s(\mu'))} T^*_{(\nu', s(\nu'))}
    = \delta_{\nu,\mu'} T_{(\mu, s(\mu))} T^*_{(\nu', s(\nu'))}.
\]
(It follows from the displayed equation that $\lsp\{(T_{(\mu, s(\mu))} T^*_{(\nu,
s(\nu))})^* : s(\mu) = s(\nu) = v\} \perp \lsp\{(T_{(\mu, s(\mu))} T^*_{(\nu, s(\nu))})^*
: s(\mu) = s(\nu) = w\}$ for distinct $v,w$.)

The $T_{(\mu, s(\mu))} T^*_{(\nu, s(\nu))}$ are nonzero by~\eqref{eq:Toeplitz series}.
Let $\mu, \nu \in \Lambda^{[p, p+n)}$. By Lemma~\ref{lem:Lambda(n)min},
\begin{align*}
&\Lambda(n)^{\mathrm{min}} ((\nu, s(\nu)), (\mu, s(\mu))) \\
    &\ = \begin{cases}
    \setof{((\alpha, \tau), (\beta, \tau))}
             {(\alpha, \beta) \in \Lambda^{\mathrm{min}} (\nu, \mu), \tau \in s(\alpha) \Lambda^{<n},
                [\alpha \tau] = s(\nu), [\beta \tau] = s(\mu)} &\text{ if $[ \mu ] = [ \nu ]$} \\
               \emptyset &\text{ otherwise.}
               \end{cases}
\end{align*}
We claim that
\[
\Lambda(n)^{\mathrm{min}} ((\nu, s(\nu)), (\mu, s(\mu)))
    = \begin{cases}
        \big\{\big((s(\nu), s (\nu)), (s(\nu ), s (\nu))\big)\big\} &\text{ if $\mu = \nu$}\\
        \emptyset &\text{ otherwise.}
    \end{cases}
\]
Indeed, if $\Lambda(n)^{\mathrm{min}} ((\nu, s(\nu)), (\mu, s(\mu))) \neq \emptyset$, say
$((\alpha, \tau), (\beta, \tau)) \in \Lambda(n)^{\mathrm{min}}((\nu, s(\nu)), (\mu,
s(\mu)))$, then $[ \mu ] = [ \nu ]$.  In particular, $[ d( \mu ) ] = d( [ \mu ] ) = d( [ \nu ] ) = [ d( \nu ) ]$.  Since $p \leq d(\nu), d(\mu) < p + n$, we have that $d(\nu) = d(\mu)$. Since
$\mu\alpha = \nu\beta$, the factorisation property forces $\mu = \nu$. We then have
$\Lambda^{\mathrm{min}} (\nu, \mu) = \Lambda^{\mathrm{min}} (\nu, \nu) = \{(s(\nu),
s(\nu)) \}$, giving
\[
\Lambda(n)^{\mathrm{min}} ((\nu, s(\nu)), (\mu, s(\mu))) = \{((s(\nu), s(\nu)), (s(\nu), s(\nu))) \}
\]
as claimed.

We now show that $\{ T_{(\mu, s(\mu))}T^*_{(\nu, s(\nu))} \}_{\mu, \nu \in \Lambda^{[p, p+n)} , s( \mu
) = s(\nu) }$ form a system of matrix units, so that
the formula given for $\Gamma_p^{p+n}$ indeed defines a homomorphism. For $\mu, \nu,
\mu', \nu' \in \Lambda^{[p, p+n)}$ with $s(\mu) = s(\nu)$ and $s (\mu') = s(\nu')$,
\begin{align*}
T_{(\mu, s(\mu))}T^*_{(\nu, s(\nu))}&  T_{(\mu', s(\mu'))} T_{(\nu', s(\nu'))}^{*} \\
    &= T_{(\mu, s(\mu))} \Big(\sum_{((\alpha, \gamma), (\beta, \delta))  \in  \Lambda(n)^{\mathrm{min}} ((\nu, s(\nu)), (\mu', s(\mu')))}
                                        T_{(\alpha, \gamma)} T_{(\beta, \delta)}^{*} \Big) T_{(\nu', s(\nu'))}^{*} \\
    &= \delta_{\nu, \mu'} T_{(\mu, s(\mu))} T_{(s(\nu), s(\nu))} T_{(\nu', s(\nu'))}^{*} \\
    &= \delta_{\nu, \mu'} T_{(\mu, s(\mu))} T_{(s(\mu), s(\mu))} T_{(s(\nu'), s(\nu'))} T_{(\nu', s(\nu'))}^{*} \\
    &= \delta_{\nu, \mu'} \delta_{s(\mu), s(\nu')} T_{(\mu, s(\mu))}T_{(\nu', s(\nu'))}^{*}.\qedhere
\end{align*}
\end{proof}

Next we provide a technical lemma and a proposition that summarises what we require to construct an
approximate order-1 factorisation of the family $(\tilde\iota_n)_{n \in \NN^k}$ obtained
from Lemma~\ref{l:homomorphism KribSolel}.

\begin{lem}\label{lem:equiv-exprs}
Let $\Lambda$ be a row-finite $k$-graph with no sources and let $n \in \NN^k$.  For each
$\mu \in \Lambda$
\begin{equation}\label{eq:iota_n relation}
    \iota_{n} (t_{\mu}) T_{(s(\mu), s(\mu))} = T_{(\mu, s(\mu))} =  T_{(r(\mu), [\mu])} \iota_{n} (t_{\mu}).
\end{equation}
For $\mu, \nu, \tau \in \Lambda$ with $s(\mu) = s(\nu) = r(\tau)$,
\[
T_{(\mu \tau, s(\mu \tau))} T_{(\nu \tau, s(\nu \tau))}^{*} = \iota_{n} (t_{\mu}) \iota_{n} (t_{\tau} t_{\tau}^{*}) T_{(r(\tau), [\tau])} \iota_{n} (t_\nu^{*})
    \quad \text{and} \quad T_{(\mu, s(\mu))} T_{(\mu, s(\mu))}^{*} = \iota_{n}(t_{\mu} t_{\mu}^{*}) T_{(r(\mu), [\mu])}.
\]
\end{lem}
\begin{proof}
Recall that $\iota_{n} (t_{\mu}) = \sum_{\lambda \in s(\mu) \Lambda^{<n}} T_{(\mu,
\lambda)}$.  So
\begin{align*}
\iota_{n} (t_{\mu}) T_{(s(\mu), s(\mu))}
    &=  \Big(\sum_{\lambda \in s(\mu) \Lambda^{<n}} T_{(\mu, \lambda)} \Big) T_{(s(\mu), s(\mu))}\\
    &=  \Big(\sum_{\lambda \in s(\mu) \Lambda^{<n}} T_{(\mu, \lambda)} T_{(s(\mu), \lambda)} \Big) T_{(s(\mu), s(\mu))}
	= T_{(\mu, s(\mu))}.
\end{align*}

We now prove that $ T_{(\mu, s(\mu))} = T_{(r(\mu), [\mu])} \iota_{n} (t_{\mu} )$. We
have
\[T_{(r(\mu), [\mu])} \iota_{n} (t_{\mu})
    = T_{(r(\mu), [\mu])} \sum_{\lambda \in s(\mu) \Lambda^{< n}} T_{(\mu, \lambda)}
    =  T_{(r(\mu), [\mu])} \sum_{\lambda \in s(\mu) \Lambda^{< n}} T_{(r(\mu), [\mu \lambda])} T_{(\mu, \lambda)}.
\]
Note that $T_{(r(\mu), [\mu])} T_{(r(\mu), [\mu \lambda])} \neq 0$ if and only if $[\mu]
= [\mu \lambda]$.  Let $\lambda \in s(\mu) \Lambda^{<n}$ with $[\mu] = [\mu \lambda]$.
Since $[\mu] = \mu (0,  [d(\mu)])$ and $[\mu \lambda] = (\mu \lambda)(0, [d(\mu
\lambda)])$, we see that $d(\lambda) = d(\mu\lambda) - d(\mu) \in H_n$. Since $d(\lambda)
< n$, we deduce that $d(\lambda) = 0$, giving $\lambda = r(\lambda) = s(\mu)$. Hence,
\[T_{(r(\mu), [\mu])} \iota_{n} (t_{\mu})
    =  T_{(r(\mu), [\mu])} \sum_{\lambda \in s(\mu) \Lambda^{< n}} T_{(r(\mu), [\mu \lambda])} T_{(\mu, \lambda)}
    = T_{(\mu, s(\mu))}.
\]
This proves~\eqref{eq:iota_n relation}.

For the second assertion, take $\mu, \nu, \tau \in \Lambda$ with $s(\mu) = s(\nu) = r
(\tau)$. Then~\eqref{eq:iota_n relation} gives
\begin{align*}
T_{(\mu \tau, s(\mu \tau))} T_{(\nu \tau, s(\nu \tau))}^{*}
    &= \iota_{n}(t_{\mu\tau}) T_{(s(\mu \tau), s(\mu \tau))} T_{(s(\nu \tau), s(\nu \tau))} \iota_{n} (t_{\nu \tau}^{*})
	= \iota_{n} (t_{\mu}) \iota_{n} (t_{\tau}) T_{(s(\tau), s(\tau))} \iota_{n} (t_{\tau}^{*}) \iota_{n} (t_{\nu}^{*}) \\
	&= \iota_{n} (t_{\mu}) \iota_{n} (t_{\tau}) \iota_{n} (t_{\tau}^{*}) T_{(r(\tau), [\tau])} \iota_{n} (t_{\nu}^{*})
	= \iota_{n} (t_{\mu}) \iota_{n} (t_{\tau} t_{\tau}^{*}) T_{(r(\tau), [\tau])} \iota_{n} (t_{\nu}^{*}),
\end{align*}
and
\begin{align*}
T_{(\mu, s(\mu))} T_{(\mu, s(\mu))}^{*}
	&= \iota_{n} (t_{\mu}) T_{(s (\mu), s(\mu))} \big(\iota_{n} (t_{\mu})  T_{(s (\mu), s(\mu))} \big)^{*}\\
    &= \iota_{n} (t_{\mu}) T_{(s (\mu), s(\mu))} \iota_{n} (t_{\mu}^{*})
	= \iota_{n}(t_{\mu} t_{\mu}^{*}) T_{(r(\mu), [\mu])}.\qedhere
\end{align*}
\end{proof}

Recall that for $n \in \NN^k$ with each $n_i
\ge 1$, the group $H_n$ is the subgroup
\[
	\setof{p \in \ZZ^k}{ n_i \text{ divides } p_i\text{ for each }i \le k}.
\]
For $x \in \RR^{k}$, let $\lceil x \rceil = \big(\lceil x_{1} \rceil, \dots, \lceil x_{k}
\rceil  \big) \in \ZZ^k$, and for $a \in \RR \setminus\{0\}$, put $\frac{x}{a} =
\big(\frac{{x}_{1}}{a}, \dots, \frac{x_{k}}{a} \big)$.

\begin{prp}\label{p: order 1 approximation}
Let $\Lambda$ be a row-finite $k$-graph with no sources. For each $n \in \NN^k$ such that
each $n_j > 0$, each $p < n$, and each $\mu \in \Lambda$, let $h_{n,\mu}(p)$ and
$g_{n,\mu}(p)$ be the unique elements in $H_n$ such that
\[\textstyle
n \leq d(\mu) + p + h_{n, \mu}(p) < 2n
    \quad \text{and} \quad \Big \lceil \frac{3n}{2} \Big \rceil \leq d(\mu) + p + g_{n, \mu}(p) <  \Big \lceil \frac{5n}{2} \Big \rceil.
\]
For each $n \in \NN^k$, let $\Delta_n$ be a function $\ftn{\Delta_n}{\NN^{k} \times
\NN^{k}}{[0,1)}$. For $i = 1,2$, define $\Delta_{n,i}^{\mu, \nu} : \NN^k  \to [0,1)$ by
$\Delta_{n,1}^{\mu, \nu}(p) := \Delta_{n} \big(d(\mu) + p + h_{n, \mu}(p) - n , d (\nu) +
p + h_{n,\mu}(p) - n\big)$ and $\Delta_{n,2}^{\mu, \nu}(p) := \Delta_{n} \big(d(\mu) + p
+ g_{n, \mu}(p) - \lceil \frac{3n}{2} \rceil, d (\nu) + p + g_{n, \mu}(p) - \lceil
\frac{3n}{2} \rceil\big)$. Suppose that for each $\mu, \nu \in \Lambda$,
\[
\lim_{n \to \infty} \max \big\{|\Delta_{n,1}^{\mu, \nu}(p) + \Delta_{n,2}^{\mu, \nu}(p) - 1 | \mathbin{\big|} p < n\big\} = 0.
\]
Suppose that there exist completely positive, contractive linear maps $\ftn{P_n}{\Tt C^{*}
(\Lambda)}{\mathcal{K}_{\Lambda^{[n, 2n)}}}$ and $\ftn{Q_n}{\Tt C^{*}
(\Lambda)}{\mathcal{K}_{\Lambda^{[\lceil 3n/2 \rceil, \lceil 5n/2 \rceil)}}}$ such that
\[
P_n(t_{\mu} t_{\nu}^{*})
    = \sum_{\substack{\tau \in s(\mu) \Lambda  \\ n \leq d(\mu \tau), d(\nu \tau)  < 2n}}
        \Delta_{n} (d(\mu \tau) - n, d (\nu \tau) - n) \theta_{\mu\tau, \nu \tau}
\]
and
\[
Q_n (t_{\mu} t_{\nu}^{*})
    = \sum_{\substack{\tau \in s(\mu) \Lambda  \\
            \left \lceil \frac{3n}{2} \right \rceil
                \leq d(\mu \tau), d(\nu \tau)
                < \left \lceil \frac{5n}{2} \right \rceil}}
                    {\textstyle\Delta_{n} \left(d(\mu \tau) - \left \lceil \frac{3n}{2} \right \rceil, d (\nu \tau)
                        - \left \lceil \frac{3n}{2} \right \rceil  \right) \theta_{\mu\tau, \nu \tau}}
\]
for all $\mu, \nu \in \Lambda$ with $s(\mu) = s(\nu)$. Then the family
$(\tilde{\iota}_{n})_{n \in \NN^k}$ has an order-1 approximation through AF-algebras.
\end{prp}
\begin{proof}
For each $n \in \NN^k$, let $\ftn{\pi_n}{\Tt C^{*} (\Lambda(n))}{C^{*} (\Lambda(n))}$ be
the quotient homomorphism.  We first show that for all $\mu, \nu \in \Lambda$ with
$s(\mu) = s(\nu)$,
\begin{equation}\label{eq:approx iota}
\lim_{n \to \infty} \Big\| \pi_{n}\Big(\Big(\big(\Gamma_{n}^{2n} \circ P_n
                                + \Gamma_{\lceil \frac{3n}{2} \rceil}^{\lceil \frac{5n}{2} \rceil} \circ Q_n\big) - \iota_{n}\Big)
                                    (t_{\mu} t_{\nu}^{*})\Big)\Big\| = 0,
\end{equation}
where the $\Gamma$'s are the homomorphisms constructed in Lemma~\ref{lem:Lambda
homomorphism}. For this, let $\mu, \nu \in \Lambda$ with $s(\mu) = s(\nu)$, and fix $n
\in \NN^k$. Lemma~\ref{lem:Lambda homomorphism} gives
\[
\Gamma_{n}^{2n} \circ P_{n} (t_{\mu} t_{\nu}^{*})
    = \sum_{\substack{\tau \in s(\mu) \Lambda  \\ n \leq d(\mu \tau), d(\nu \tau)  < 2n}}
        \Delta_{n} (d(\mu \tau) - n, d (\nu \tau) - n) T_{(\mu \tau, s(\mu \tau))} T_{(\nu \tau, s(\nu \tau))}^{*}.
\]
Lemma~\ref{lem:equiv-exprs} shows that $T_{(\mu \tau, s(\mu \tau))} T_{(\nu \tau, s(\nu
\tau))}^{*} = \iota_{n} (t_{\mu}) \iota_{n} (t_{\tau} t_{\tau}^{*}) T_{(r(\tau), [\tau])}
\iota_{n} (t_\nu^{*})$. So,
\begin{align*}
\Gamma_{n}^{2n} \circ P_{n} (t_{\mu} t_{\nu}^{*})
    &= \iota_{n} (t_{\mu})  \Big(\sum_{\substack{\tau \in s(\mu) \Lambda  \\ n \leq d(\mu \tau), d(\nu \tau)  < 2n}}
        \Delta_{n} (d(\mu \tau) - n, d (\nu \tau) - n) \iota_{n} (t_{\tau} t_{\tau}^{*}) T_{(r(\tau), [\tau])} \Big) \iota_{n} (t_\nu^{*})  \\
    &=\iota_{n} (t_{\mu}) \Big(\sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}} \sum_{\rho \in s(\alpha) \Lambda^{h_{n,\mu}(p)}}
        \Delta_{n,1}^{\mu, \nu}(p)  \iota_{n} (t_{\alpha \rho} t_{\alpha \rho}^{*}) T_{(r(\alpha \rho), [\alpha \rho])} \Big) \iota_{n} (t_\nu^{*}) \\
    &= \iota_{n} (t_{\mu}) \Big(\sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}} \sum_{\rho \in s(\alpha) \Lambda^{h_{n,\mu}(p)}}
        \Delta_{n,1}^{\mu, \nu}(p)  \iota_{n} (t_{\alpha}) \iota_{n}(t_{\rho} t_{\rho}^{*}) \iota_{n} (t_{\alpha}^{*})
            T_{(s(\mu), \alpha)} \Big) \iota_{n} (t_\nu^{*}) \\
    &=  \iota_{n} (t_{\mu}) \Big(\sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}}
        \Delta_{n,1}^{\mu, \nu}(p)  \iota_{n} (t_{\alpha})
            \Big(\sum_{\rho \in s(\alpha) \Lambda^{h_{n,\mu}(p)}} \iota_{n}(t_{\rho} t_{\rho}^{*}) \Big)
            \iota_{n} (t_{\alpha}^{*})  T_{(s(\mu), \alpha)} \Big) \iota_{n} (t_\nu^{*}).
\end{align*}
Relation~(CK) for $\{s_\lambda\}_{\lambda \in \Lambda}$ gives
\begin{align*}
\pi_{n} &{}\circ \Gamma_{n}^{2n} \circ P_n (t_{\mu} t_{\nu}^{*}) \\
    &= \widetilde{\iota}_{n} (s_{\mu}) \Big(\sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}}
        \Delta_{n,1}^{\mu, \nu}(p)  \widetilde{\iota}_{n} (s_{\alpha}) \widetilde{\iota}_{n} (s_{s(\alpha)}) \widetilde{\iota}_{n} (s_{\alpha}^{*})
            S_{(s(\mu), \alpha)} \Big) \widetilde{\iota}_{n} (s_\nu^{*}) \\
    &= \widetilde{\iota}_{n} (s_{\mu}) \Big(\sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}}
        \Delta_{n,1}^{\mu, \nu}(p) \widetilde{\iota}_{n} (s_{\alpha}) \widetilde{\iota}_{n} (s_{\alpha}^{*})
            S_{(s(\mu), \alpha)} \Big) \widetilde{\iota}_{n} (s_\nu^{*})
\end{align*}
By Lemma~\ref{lem:equiv-exprs}, $S_{(\alpha, s(\alpha))} S_{(\alpha, s(\alpha))}^{*} =
\widetilde{\iota}_{n}(s_{\alpha} s_{\alpha}^{*}) S_{(r(\alpha), [\alpha])} =
\widetilde{\iota}_{n} (s_{\alpha}) \widetilde{\iota}_{n} (s_{\alpha}^{*}) S_{(s(\mu),
\alpha)}$ for all $\alpha \in s(\mu) \Lambda^{<n}$, and hence
\begin{equation}\label{eq:piGammaP sum}
\pi_n \circ \Gamma^{2n}_n \circ P_n(t_\mu t^*_\nu)
    = \widetilde{\iota}_{n} (s_{\mu}) \Big(\sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}}
        \Delta_{n,1}^{\mu, \nu}(p) S_{(\alpha, s(\alpha))} S_{(\alpha, s(\alpha))}^{*} \Big) \widetilde{\iota}_{n} (s_\nu^{*}).
\end{equation}

Take $p < n$ and $\alpha \in s(\mu) \Lambda^{p}$. Then
\begin{align*}
\{(\lambda, \lambda') \in \Lambda(n)^{p}  \mid  {}&r((\lambda, \lambda')) = (s(\mu), \alpha)\} \\
    &= \setof{(\lambda, \lambda') \in \Lambda \times \Lambda^{< n}}{s(\lambda) = r(\lambda'), d(\lambda) = p,
        (r(\lambda), [\lambda \lambda']) = (s(\mu), \alpha)}.
\end{align*}
Suppose that $(\lambda, \lambda') \in \Lambda(n)^{p}$ with $r ((\lambda, \lambda')) =
(s(\mu), \alpha)$.  Then $[\lambda\lambda'] = \alpha$, so $p = d(\alpha) = d([\lambda
\lambda']) = [d(\lambda \lambda')] = [p + d(\lambda')]$. Hence $[p] = [p + d(\lambda')]$,
and since $d(\lambda') < n$, this forces $d(\lambda') = 0$.  Therefore, $\alpha = [ \lambda \lambda' ] = [ \lambda ] = \lambda$ since $d( \lambda ) = p < n$ and $\lambda' = r( \lambda' ) = s( \lambda ) = s( \alpha )$.  Hence
\[
\setof{(\lambda, \lambda') \in \Lambda(n)^{p}}{ r((\lambda, \lambda')) = (s(\mu), \alpha)} = \{(\alpha, s(\alpha))\},
\]
which implies that each $S_{(\alpha, s(\alpha))} S_{(\alpha, s(\alpha))}^{*} =
S_{(s(\mu), \alpha)}$ by~(CK) in $C^*(\Lambda(n))$. Combining this
with~\eqref{eq:piGammaP sum} gives
\[\pi_{n} \circ \Gamma_{n}^{2n} \circ P_n (t_{\mu} t_{\nu}^{*})
    = \widetilde{\iota}_{n} (s_{\mu}) \Big(\sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}}
        \Delta_{n,1}^{\mu, \nu}(p) S_{(s(\mu), \alpha)}  \Big) \widetilde{\iota}_{n} (s_\nu^{*}).
\]
A similar computation gives
\[\pi_{n} \circ \Gamma_{\lceil \frac{3n}{2}  \rceil}^{\lceil \frac{5n}{2} \rceil} \circ Q_n (t_{\mu} t_{\nu}^{*})
    =  \widetilde{\iota}_{n} (s_{\mu}) \Big(\sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}}
        \Delta_{n,2}^{\mu, \nu}(p)  S_{(s(\mu), \alpha)}  \Big) \widetilde{\iota}_{n} (s_\nu^{*}).
\]

Since $\{S_{(s(\mu), \alpha)} \}_{ \alpha \in s(\mu) \Lambda^{<n}}$ is a collection of mutually orthogonal projections,
\begin{align*}
\big\| \big(\pi_{n} \circ \Gamma_{n}^{2n}& \circ P_n (t_{\mu} t_{\nu}^{*})
            + \pi_{n} \circ \Gamma_{\lceil \frac{3n}{2} \rceil}^{\lceil \frac{5n}{2} \rceil} \circ Q_n (t_{\mu} t_{\nu}^{*})\big)
            - \pi_{n} \circ \iota_{n} (t_{\mu} t_{\nu}^{*}) \big\| \\
    &\leq \Big\|\sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}}
            (\Delta_{n,1}^{\mu, \nu}(p)  + \Delta_{n,2}^{\mu, \nu}(p)) S_{(s(\mu), \alpha)}
                - \widetilde{\iota}_{n} (s_{s(\mu)}) \Big\| \\
    &= \Big\|\sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}}
            (\Delta_{n,1}^{\mu, \nu}(p)  + \Delta_{n,2}^{\mu, \nu}(p)) S_{(s(\mu), \alpha)}
            - \sum_{p < n} \sum_{\alpha \in s(\mu) \Lambda^{p}} S_{(s(\mu), \alpha)} \Big\| \\
    &= \max_{p < n} | \Delta_{n,1}^{\mu, \nu}(p)  + \Delta_{n,2}^{\mu, \nu}(p)  - 1|.
\end{align*}
By assumption, $\lim_{n \to \infty} \max_{p < n} | \Delta_{n,1}^{\mu, \nu}(p) +
\Delta_{n,2}^{\mu, \nu}(p)  - 1 | = 0$.  This proves~\eqref{eq:approx iota}.

Since $k$-graph algebras are nuclear \cite[Theorem~5.5]{KumjianPask:NYJM00}, we may apply
\cite[Theorem~3.10]{ChoiEffros:ANMATH} to obtain a contractive completely positive
splitting  $\ftn{\sigma}{C^{*} (\Lambda)}{\Tt C^{*} (\Lambda)}$ for the quotient map. For
each $n$, define $\ftn{\psi_{n}}{C^{*}(\Lambda)}{\Kk_{\Lambda^{[n, 2n)}} \oplus
\Kk_{\Lambda^{[\lceil 3n/2 \rceil, \lceil 5n/2 \rceil)}}}$ by $\psi_n(a) :=
\big(P_n(\sigma(a)), Q_n(\sigma(a))\big)$ and $\ftn{\phi_{n}}{\Kk_{\Lambda^{[n, 2n)}}
\oplus \Kk_{\Lambda^{[\lceil 3n/2 \rceil, \lceil 5n/2 \rceil)}}}{C^{*}(\Lambda(n))}$ by
$\phi_{n} ( ( a, b ) ) = \pi_{n} (\Gamma_{n}^{2n}(a) +\Gamma^{\lceil \frac{5n}{2}
\rceil}_{\lceil \frac{3n}{2} \rceil}(b))$. By Lemma~\ref{lem:Lambda homomorphism},
$\phi_{n}$ restricts to a homomorphism (and in particular an order-zero map) on each of
$\Kk_{\Lambda^{[n, 2n)}}$ and $\Kk_{\Lambda^{[\lceil 3n/2 \rceil, \lceil 5n/2 \rceil)}}$.
Since $\Tt C^{*} ( \Lambda ) = \overline{ \mathrm{span} } \setof{ t_{\mu} t_{\nu}^{*} }{
\mu, \nu \in \Lambda, s( \mu ) = s( \nu )}$ and since
\[\lim_{n \to \infty} \big\| \pi_{n} \circ \Gamma_{n}^{2n} \circ P_n (t_{\mu} t_{\nu}^{*})
        + \pi_{n} \circ \Gamma_{\lceil \frac{3n}{2} \rceil}^{\lceil \frac{5n}{2} \rceil} \circ Q_n (t_{\mu} t_{\nu}^{*})
        -  \pi_{n} \circ \iota_{n} (t_{\mu} t_{\nu}^{*}) \big\| = 0
\]
for all $\mu, \nu \in \Lambda$ with $s(\mu) = s(\nu)$, the family $\left(\Kk_{\Lambda^{[n, 2n)}} \oplus \Kk_{\Lambda^{[\lceil 3n/2 \rceil,
\lceil 5n/2 \rceil)}}, \psi_{n}, \phi_{n} \right)$ is an asymptotic order-1 approximation
of $(\tilde{\iota}_{n})_{n \in \NN^k}$ through AF-algebras.
\end{proof}

\begin{ntn}
Following \cite{WinterZacharias:Adv10}, for each $m \in \NN$, define $\kappa_{m} \in
M_{\{0, \dots, m-1\}}\big([0,1)\big)$ as follows: put $l := \lceil\frac{m}{2}\rceil$, and
define
\begin{gather*}
\kappa_{m} = \frac{1}{l+1}
    \left(
    \begin{matrix}
        1 & 1 & \dots & 1 & 1 &\dots & 1 & 1 \\
        1 & 2 & \dots & 2 & 2 &\dots & 2 & 1 \\
        \vdots & \vdots & \ddots  &\vdots &\vdots & \iddots & \vdots &\vdots  \\
        1 & 2 & \dots & l & l & \dots & 2 & 1 \\
        1 & 2 & \dots & l & l & \dots & 2 & 1 \\
        \vdots & \vdots & \iddots &\vdots & \vdots & \ddots &\vdots &\vdots  \\
        1 & 2 & \dots & 2 & 2 &\dots & 2 & 1 \\
        1 & 1 & \dots & 1 & 1 &\dots & 1 & 1 \\
    \end{matrix}
    \right) \qquad\text{ if $m$ is even} \\
\kappa_{m} = \frac{1}{l+2}
    \left(
    \begin{matrix}
        1 & 1 & \dots & 1 &\dots & 1 & 1 \\
        1 & 2 & \dots & 2 &\dots & 2 & 1 \\
        \vdots & \vdots & \ddots &\vdots & \iddots & \vdots &\vdots  \\
        1 & 2 & \dots & l+1 & \dots & 2 & 1 \\
        \vdots & \vdots & \iddots &\vdots & \ddots &\vdots &\vdots  \\
        1 & 2 & \dots & 2 &\dots & 2 & 1 \\
        1 & 1 & \dots & 1 &\dots & 1 & 1 \\
    \end{matrix}
    \right) \qquad\text{ if $m$ is odd.}
\end{gather*}
Define $\kappa_{m}(i,j) = 0$ for $(i,j) \in \ZZ^2 \setminus (\{0, \dots, m-1\} \times
\{0, \dots, m-1\})$.
\end{ntn}

\begin{thm}\label{thm: order 1 approximation}
Let $\Lambda$ be a row-finite $2$-graph with no sources. Then $(\tilde{\iota}_{n})_{n \in
\NN^k}$ has an asymptotic order-1 approximation through AF-algebras.
\end{thm}
\begin{proof}
For $m \in \NN$, let $\mathsf{A}_{m}$ denote the $m\times m$ matrix with all entries
equal to 1. For $n \in \NN^2$, define
\[\Delta_{n}
    := \frac{1}{2} \left(\kappa_{n_1} \otimes \mathsf{A}_{n_2} + \mathsf{A}_{n_1} \otimes \kappa_{n_2}\right)
    : \CC^{n_1} \otimes \CC^{n_2} \to \CC^{n_1} \otimes \CC^{n_2}.
\]
Since $\CC^{n_1} \otimes \CC^{n_2}$ is a finite-dimensional Hilbert space, $\Delta_{n}$
can be regarded as an $n_1n_2 \times n_1n_2$ matrix.  Since $\kappa_{n_1} \otimes
\mathsf{A}_{n_2}$ and $\mathsf{A}_{n_1} \otimes \kappa_{n_2}$ are positive elements in
the $C^{*}$-algebra $M_{n_1} \otimes M_{n_2}$, the matrix $\Delta_{n}$ is also positive.
Write $\{e_{i}\}$ for the canonical orthonormal basis elements of $\CC^{n_1}$ and of
$\CC^{n_2}$. Then
\begin{align*}
\Delta_{n} (i_{1}, i_{2}, j_{1}, j_{2}) &= \langle \Delta_{n} e_{i_{1}} \otimes e_{i_{2}}, e_{j_{1}} \otimes e_{j_{2}} \rangle \\
	&= \frac{1}{2} \big(\langle \kappa_{n_1} e_{i_{1}},  e_{j_{1}} \rangle \langle \mathsf{A}_{n_2} e_{i_{2}}, e_{j_{2}} \rangle
            + \langle \mathsf{A}_{n_1} e_{i_{1}},  e_{j_{1}} \rangle \langle \kappa_{n_2} e_{i_{2}}, e_{j_{2}} \rangle \big) \\
	&= \frac{1}{2} \big(\kappa_{n_1} (i_{1}, j_{1}) + \kappa_{n_2} (i_{2}, j_{2})\big).
\end{align*}
Define $M^{n,1} \in M_{\Lambda^{[n, 2n)}}$ by $M^{n,1}_{\mu, \nu} = \Delta_{n} (d(\mu) -
n, d(\nu) - n)$ and define $M^{n,2} \in M_{\Lambda^{[\lceil3n/2\rceil,
\lceil5n/2\rceil)}}$ by $M^{n,2}_{\mu, \nu} = \Delta_{n} (d(\mu) - \lceil \frac{3n}{2}
\rceil,  d(\nu) -\lceil \frac{3n}{2} \rceil)$. We claim that Schur multiplication by
$M^{n,i}$ is a completely positive contraction for $i = 1,2$. We just argue the case $i =
1$ and when $n_1$ and $n_2$ are even; the other cases are similar. For $1 \le j \le
n_1/2$, let $\Phi^{1,j}$ be the strong-operator sum $\sum_{|d(\lambda)_1 - (3n_1-1)/2| <
j} \theta_{\lambda,\lambda}$, and for $1 \le j \le n_2/2$, let $\Phi^{2,j} =
\sum_{|d(\lambda)_2 - (3n_2-1)/2| < j} \theta_{\lambda,\lambda}$, where $d( \lambda )_{i}$ denotes the $i$th coordinate of $d( \lambda )$. Each $\Phi^{i,j}$ is a
projection, and so $\Phi^i : a \mapsto \sum^{n_i/2}_{j=1} \frac{1}{n_i/2 + 1} \Phi^{i,j}
a \Phi^{i,j}$ is a completely positive contraction. Schur multiplication by $M^{n,1}$ is
equal to $\frac12(\Phi^1 + \Phi^2)$, and so is itself a completely positive contraction.

For $p < q \in \NN^2$, define $R^{q}_p \in \Bb(\ell^2(\Lambda))$ to be the
strong-operator sum
\[\textstyle
R^q_p
    = \sum_{\lambda \in \Lambda^{[p,q)}} \theta_{\lambda, \lambda}.
\]
Define $\ftn{P_{n}}{\Tt C^{*} (\Lambda)}{\Kk_{\Lambda^{[n, 2n)}}} \subseteq \Kk_\Lambda$
by $P_{n}(a)  =  M^{n,1}
* (R^{2n}_n a R^{2n}_n)$ and define $\ftn{Q_{n}}{\Tt C^{*} (\Lambda)}{\Kk_{\Lambda^{[\lceil
3n/2 \rceil, \lceil 5n/2 \rceil)}}} \subseteq \Kk_\Lambda$ by $Q_{n}(a) = M^{n,2} *
(R^{\lceil 5n/2 \rceil}_{\lceil 3n/2 \rceil} a R^{\lceil 5n/2 \rceil}_{\lceil 3n/2
\rceil})$. Since Schur multiplication by each $M^{n,i}$ is a completely positive,
contractive linear map and since $R^{2n}_n$ and $R^{\lceil 5n/2 \rceil}_{\lceil 3n/2
\rceil}$ are projections, $P_{n}$ and $Q_{n}$ are completely positive, contractive linear
maps.

We will show that $\Delta_{n}$, $P_{n}$, and $Q_{n}$ satisfy the hypotheses of
Proposition~\ref{p: order 1 approximation}.  Fix $\mu, \nu \in \Lambda$ with $s(\mu) =
s(\nu)$.  Recall that the series $\sum_{\tau \in \Lambda} \theta_{\mu \tau, \nu \tau}$
converges strictly to $t_\mu t^*_\nu$. Since $\theta_{\lambda, \lambda} \theta_{\mu \tau,
\nu \tau} \theta_{\beta, \beta} = \delta_{\lambda, \mu\tau} \delta_{\beta,\nu\tau}
\theta_{\mu \tau, \nu \tau}$,
\[
R^{2n}_n t_{\mu} t_{\nu}^{*} R^{2n}_n
    = \sum_{\substack{\tau \in s(\mu)\Lambda \\ n \leq d(\mu \tau), d(\nu \tau) < 2n}} \theta_{\mu\tau, \nu\tau}  \quad \text{and} \quad
R^{\lceil 5n/2 \rceil}_{\lceil 3n/2\rceil} t_{\mu} t_{\nu}^{*} R^{\lceil 5n/2 \rceil}_{\lceil 3n/2\rceil}
    = \sum_{\substack{\tau \in s(\mu)\Lambda \\
             \left\lceil \frac{3n}{2} \right\rceil \leq d(\mu \tau), d(\nu \tau) < \left \lceil \frac{5n}{2} \right\rceil}}
        \theta_{\mu\tau, \nu \tau}.
\]
Hence,
\[P_{n} (t_{\mu} t_{\nu}^{*})
    = \sum_{\substack{\tau \in s(\mu) \Lambda  \\ n \leq d(\mu \tau), d(\nu \tau)  < 2n}}
        \Delta_{n} (d(\mu \tau) - n, d (\nu \tau) - n) \theta_{\mu\tau, \nu \tau}
\]
and
\begin{align*}
Q_{n}(t_{\mu} t_{\nu}^{*})
    &= \sum_{\substack{\tau \in s(\mu) \Lambda  \\
        \left \lceil \frac{3n}{2} \right \rceil \leq d(\mu \tau), d(\nu \tau)  < \left \lceil \frac{5n}{2} \right \rceil}}
            {\textstyle\Delta_{n} \left(d(\mu \tau) - \left \lceil \frac{3n}{2} \right \rceil, d (\nu \tau)  - \left \lceil \frac{3n}{2} \right \rceil\right)
                \theta_{\mu\tau, \nu \tau}}.
\end{align*}

Let $p = (p_{1}, p_{2}) < n$, let $d(\mu) = (a_{1}, a_{2})$, and let $d(\nu) = (b_{1},
b_{2})$.  Let $h_{n, \mu}(p) = (h_{p_{1},n}^{\mu}, h_{p_{2}, n}^{\mu})$ be the unique
element in $H_{n}$ such that $n \leq d(\mu) + p + h_{n, \mu} (p) < 2n$ and let $g_{n,
\mu}(p) = (g_{p_{1}, n}^{\mu}, g_{p_{2}, n}^{\mu})$ be the unique element in $H_{n}$ such
that $\left\lceil \frac{3n}{2} \right\rceil \leq d(\mu) + p + g_{n, \mu} (p) <
\left\lceil \frac{5n}{2} \right\rceil$. Note that $h_{p_{j}, n}^{\mu}$ is the unique
element in $n_j\ZZ$ such that $n_j \leq a_j + p_{j} + h_{p_{j}, n}^{\mu} < 2n_j$ and
$g_{p_{j}, n}^{\mu}$ is the unique element in $n_j\ZZ$ such that $\big\lceil
\frac{3n_j}{2} \big\rceil \leq a_j + p_{j} + g_{p_{j}, n}^{\mu} < \big\lceil
\frac{5n_j}{2} \big\rceil$.

Set
\begin{align*}
\zeta_{n, \mu, p_{j}} := \kappa_{n_j} &(a_{j} + p_{j} + h_{p_{j}, n}^{\mu} - n_j, b_{j} + p_{j} + h_{p_{j}, n}^{\mu} - n_j) \\
    &\textstyle{} + \kappa_{n_j} \left(a_{j} + p_{j} + g_{p_{j}, n}^{\mu} - \big\lceil \frac{3n_j}{2} \big\rceil,
                b_{j} + p_{j} + g_{p_{j}, n}^{\mu} - \big\lceil \frac{3n_j}{2} \big\rceil \right).
\end{align*}
Using the definitions of the $h^\mu_{p_j, n}$ and $g^\mu_{p_j, n}$, one checks that
\[\textstyle
\big(a_{j} + p_{j} + h_{p_{j}, n}^{\mu} - n_j\big)
    - \big(a_{j} + p_{j} + g_{p_{j}, n}^{\mu} - \big\lceil \frac{3n_j}{2} \big\rceil\big)
    \in \big\{\big\lceil \frac{n_j}{2} \big\rceil, -\big\lfloor \frac{n_j}{2} \big\rfloor\big\}.
\]
For any integer $k$ and for any $x,y$, we have $\kappa_{k}(x,y) \ge \kappa_k(x,x) -
|x-y|/(\lceil\frac{k}{2}\rceil + 1)$, and $(\lceil k/2\rceil + 1)/(\lceil k/2\rceil + 2)
\le \kappa_k(x,x) + \kappa_k(x+\lceil\frac{k}{2}\rceil, x + \lceil\frac{k}{2}\rceil) \le
1$. Hence
\[
\frac{\lceil \frac{k}{2}\rceil + 1}{\lceil \frac{k}{2}\rceil + 2} - \frac{2|x-y|}{\lceil\frac{k}{2}\rceil + 1}
    \le \kappa_k(x,y) + \kappa_k\left({\textstyle x + \big\lceil\frac{k}{2}\big\rceil, y + \big\lceil\frac{k}{2}\big\rceil}\right) \le 1.
\]
Applying this inequality with $k = n_j$, $x = \min\big\{a_{j} + p_{j} + h_{p_{j},
n}^{\mu} - n_j, a_{j} + p_{j} + g_{p_{j}, n}^{\mu} - \big\lceil
\frac{3n_j}{2}\big\rceil\big\}$ and $y := \min\big\{b_{j} + p_{j} + h_{p_{j}, n}^{\mu} -
n_j, b_{j} + p_{j} + g_{p_{j}, n}^{\mu} - \big\lceil \frac{3n_j}{2} \big\rceil\big\}$, we
see that $|\zeta_{n, \mu, p_{j}} - 1| \le \big(1 + 2|a_j -
b_j|\big)/\big(\big\lceil\frac{n_j}{2}\big\rceil + 1\big)$.

For $p < n$, set $\Delta_{n,1}^{\mu, \nu}(p)  =  \Delta_{n} (d(\mu) + p + h_{n, \mu}(p) -
n, d(\nu) + p + h_{n,\mu}(p) - n)$ and $\Delta_{n,2}^{\mu, \nu}(p)  = \Delta_{n}(d(\mu) +
p + g_{n, \mu}(p) - \lceil \frac{3n}{2} \rceil, d(\nu) + p + g_{n, \mu}(p) - \lceil
\frac{3n}{2} \rceil)$.  Then
\[
\left|\Delta_{n,1}^{\mu, \nu}(p) + \Delta_{n,2}^{\mu, \nu}(p)  - 1 \right|
    = \left| \frac{1}{2} \left(\zeta_{n, \mu, p_{1}} - 1 \right)  + \frac{1}{2} \left(\zeta_{n, \mu,  p_{2}} - 1 \right) \right|
	\leq \frac{2(1 + |a_{1} - b_{1}| + |a_{2} - b_{2}|)}{2(\min\{\lceil \frac{n_1}{2} \rceil, \lceil \frac{n_2}{2} \rceil \} + 1)}.
\]
Hence
\[
\lim_{n \to \infty} \max \big\{|\Delta_{n,1}^{\mu, \nu}(p) + \Delta_{n,2}^{\mu, \nu}(p) - 1| \mathbin{\big|} p < n\big\} = 0.
\]

So $\Delta_{n}$, $P_{n}$ and $Q_{n}$ satisfy the hypotheses of Proposition~\ref{p: order
1 approximation}, which then says that $(\tilde{\iota}_{n})_{n \in \NN^k}$ has an
asymptotic order-1 approximation through AF-algebras.
\end{proof}

\begin{cor}\label{cor: order 1 approximation}
If $E$ and $F$ are row-finite directed graphs with no sinks, then $(\tilde{\iota}_{m, E}
\otimes \tilde{\iota}_{m, F})_{m \in \NN}$ has an asymptotic order-1 approximation
through AF-algebras.
\end{cor}
\begin{proof}
Let $\ftn{\tilde{\iota}_{m,1}}{C^{*} (E^{*})}{C^{*} (E^{*} (m))}$,
$\ftn{\tilde{\iota}_{m,2}}{C^{*} (F^{*})}{C^{*} (F^{*} (m))}$, $\ftn{\tilde{\iota}_{(m,
m)}}{C^{*} (E^{*} \times F^{*})}{C^{*} ((E^{*} \times F^{*})((m,m)))}$ be the
homomorphisms defined in Lemma~\ref{l:homomorphism KribSolel} for the 1-graphs $E^{*}$,
$F^{*}$, and the 2-graph $E^{*} \times F^{*}$ respectively.  By Lemma~\ref{l:prodgraph},
$\tilde{\iota}_{m,1} \otimes \tilde{\iota }_{m, 2} = \Theta_{E^{*} (m) \times F^{*} (m)}
\circ  \tilde{\iota}_{(m, m)} \circ \Theta_{E^{*} \times F^{*}}^{-1}$, where
$\Theta_{E^{*} \times F^{*}}$ and $\Theta_{E^{*} (m) \times F^{*} (m)}$ are isomorphisms.
By Theorem~\ref{thm: order 1 approximation} and Remark~\ref{rmk:approximation subseq},
the sequence $(\tilde{\iota}_{(m, m)})_{m \in \NN}$ has an asymptotic order-1
approximation through AF-algebras.  Hence, $(\tilde{\iota}_{m,1} \otimes
\tilde{\iota}_{m, 2})_{ m \in \NN}$ has an asymptotic order-1 approximation through AF-algebras.  By
Lemma~\ref{l:equivalence construction}, there exist isomorphisms $\ftn{\psi_{E}}{C^{*}
(E)}{C^{*} (E^{*})}$, $\ftn{\psi_{F}}{C^{*} (F)}{C^{*} (F^{*})}$,
$\ftn{\psi_{E(m)}}{C^{*} (E(m))}{C^{*} (E^{*}(m))}$, and $\ftn{\psi_{F(m)}}{C^{*}
(F(m))}{C^{*} (F^{*}(m))}$ such that $\tilde{\iota}_{m,E} = \psi_{E(m)}^{-1} \circ
\tilde{\iota}_{m,1} \circ \psi_{E}$ and $\tilde{\iota}_{m,F} = \psi_{F(m)}^{-1} \circ
\tilde{\iota}_{m,2} \circ \psi_{F}$. Hence,
\[\tilde{\iota}_{m, E} \otimes \tilde{\iota}_{m, F} = (\psi_{E(m)} \otimes \psi_{F(m)})^{-1}
    \circ  (\tilde{\iota}_{m,1} \otimes  \tilde{\iota}_{m,2}) \circ (\psi_{E} \otimes \psi_{F}).
\]
Thus $(\tilde{\iota}_{m, E} \otimes \tilde{\iota}_{m, F})_{m \in \NN}$ has an asymptotic
order-1 approximation through AF-algebras.
\end{proof}


\section{Nuclear dimension of UCT-Kirchberg algebras}\label{sec:main}

In this section, we show that all UCT-Kirchberg algebras have nuclear dimension~1.  We
already know from \cite{Enders:arXiv:1405.6538} that every UCT-Kirchberg algebra with
torsion free $K_{1}$-group has nuclear dimension~1. So we first show that each
UCT-Kirchberg algebra with trivial $K_{0}$-group and finite $K_{1}$-group has nuclear
dimension~1, and then prove our main theorem.


\begin{dfn}
A \emph{Kirchberg algebra} is a separable, nuclear, simple, purely infinite
$C^{*}$-algebra.  A \emph{UCT-Kirchberg algebra} is a Kirchberg algebra in the UCT class
of \cite{RosenbergSchochet:Duke87}.
\end{dfn}

For each finite abelian group $T$, let $E_{T}$ be an infinite, row-finite, strongly
connected graph such that $K_*(C^{*} (E_{T})) = ( T, \{0\} )$ and $C^{*} (E_{T})$ is a UCT-Kirchberg algebra (note that strongly connected implies that $E_{T}$ has no sinks and
sources).  Let $F_{\ZZ}$ be an infinite, row-finite, strongly connected graph such that
$K_{*} (C^{*} (F_{\ZZ}))  = (\{0\}, \ZZ)$ and $C^{*} (F_{\ZZ})$ is a UCT-Kirchberg
algebra.  Note that $E_{T}$ and $F_{\ZZ}$ exist by
\cite[Theorem~1.2]{Szymanski:IUMJ2002}.

\begin{lem}\label{l:nuclear dimension 2-graph}
Let $T$ be a finite abelian group.  Then the nuclear dimension of $C^{*} (E_{T}) \otimes
C^{*} (F_{\ZZ})$ is 1.  Consequently, every UCT-Kirchberg algebra with $K_{0}$ trivial
and $K_{1}$ finite has nuclear dimension 1.
\end{lem}
\begin{proof}
Consider the directed graphs $E_T$ and $F_\ZZ$. For $k \in \NN$, let
\[
\ftn{ \tilde{\iota}_{k, E_{T}} }{C^{*} ( E_{T} ) }{ C^{*} (E_{T}(k) ) }
    \quad\text{ and }\quad
\ftn{\tilde{\iota }_{k, F_{\ZZ}}}{ C^{*} (F_{\ZZ} ) }{ C^{*} ( F_{\ZZ} (k) ) }
\]
be the homomorphisms given in Lemma~\ref{l:Rout} for $E_{T}$ and $F_{\ZZ}$ respectively.
Let
\[
\ftn{ j_{k, E_{T}} }{C^{*} ( E_{T}(k) ) }{ C^{*} (E_{T}) \otimes \Kk }
    \quad\text{ and }\quad
\ftn{j_{k, F_{\ZZ}}}{ C^{*} ( F_{\ZZ}(k) ) }{ C^{*} ( F_{\ZZ} ) \otimes \Kk}
\]
be the homomorphisms given in \cite[Proposition~3.1]{RuizSimsTomforde:arXiv:1312.0507}
for $E_{T}$ and $F_{\ZZ}$ respectively.

By Corollary~\ref{cor: order 1 approximation}, there is an asymptotic order-1
approximation through AF-algebras for $(\tilde{\iota}_{k, E_{T}} \otimes \tilde{\iota
}_{k, F_{\ZZ}})_{k \in \NN}$. The composition of this sequence of homomorphisms with
$j_{k, E_{T}} \otimes j_{k, F_{\ZZ}}$ gives an asymptotic order-1 approximation through
AF-algebras for $((j_{k, E_{T}} \circ \tilde{\iota}_{k, E_{T}}) \otimes (j_{k, F_{\ZZ}}
\circ \tilde{\iota}_{k, F_{\ZZ}}))_{k \in \NN}$.

For $m \in \NN$, let $p_{m} = (|T|+1)^m$. Since the order of each element of $T$ divides
$|T|$, multiplication by each $p_{m}$ induces the identity map on $T$. Set $\gamma_{m} =
(j_{p_{m}, E_{T}} \circ \tilde{\iota}_{p_{m}, E_{T}}) \otimes (j_{p_{m}, F_{\ZZ}} \circ
\tilde{\iota}_{p_{m}, F_{\ZZ}})$.  By construction, $(\gamma_{m})_{m \in \NN}$ has an
asymptotic order-1 approximation through AF-algebras.  The K\"unneth formula in
\cite{RosenbergSchochet:Duke87} combined with
\cite[Lemma~3.2]{RuizSimsTomforde:arXiv:1312.0507} shows that $K_{1} (\gamma_{m})$ is
multiplication by $p_{m}^{2}$ on $K_{1} (C^{*} (E_{T}) \otimes C^{*} (F_{\ZZ})) = T$.
Thus, $K_{1} (\gamma_{m}) = \id_T$. Since $K_{0} (C^{*} (E_{T}) \otimes C^{*} (F_{\ZZ}))
= 0$ the map $K_{0} (\gamma_{m})$ is trivially the identity. Since $E_{T}$ and $F_{\ZZ}$
are infinite directed graphs, $C^{*} (E_{T})$ and $C^{*} (F_{\ZZ})$ are non-unital UCT-Kirchberg
algebras, and hence stable. The Universal Coefficient Theorem in
\cite{RosenbergSchochet:Duke87} and the Kirchberg-Phillips classification (cf.
\cite{kirchpureinf} and \cite{Phillips:Doc00}), show that there exist an isomorphism
$\ftn{\beta_{m}}{(C^{*} (E_{T}) \otimes \Kk) \otimes (C^{*} (F_{\ZZ}) \otimes \Kk)}{C^{*}
(E_{T}) \otimes C^{*} (F_{\ZZ})}$ and a unitary $ u_{m}$ in $\mathcal{M}
( C^{*} (E_{T}) \otimes C^{*} (F_{\ZZ}) )$ for each $m \in \NN$ such that
\begin{equation}\label{eq:order-1 approximation id}
\lim_{m \to \infty} \| u_{m} (\beta_{m} \circ \gamma_{m})(a) u_{m}^{*} - a \| = 0
    \quad\text{ for all $a \in C^{*} (E_{T}) \otimes C^{*} (F_{\ZZ})$.}
\end{equation}
Since $(\gamma_{m})_{m \in \NN}$ has an asymptotic order-1 approximation through
AF-algebras, so does $(\mathrm{Ad}(u_{m}) \circ \beta_{m} \circ \gamma_{m})_{m \in \NN}$.
So~(\ref{eq:order-1 approximation id}) implies that $\mathrm{id}_{C^{*} (E_{T}) \otimes
C^{*} (F_{\ZZ})}$ has an asymptotic order-1 approximation through AF-algebras. Hence
\cite[Lemma~2.9]{RuizSimsTomforde:arXiv:1312.0507} shows that $\ndim\big(C^{*} (E_{T})
\otimes C^{*} (F_{\ZZ })\big) = 1$.

Let $A$ be a UCT-Kirchberg algebra with $K_{0}$ trivial and $K_{1}$ finite. Then $K_{*}
(A) \cong K_{*} (C^{*} (E_{T}) \otimes C^{*} (F_{\ZZ }) )$ where $T = K_{1} (A)$.  By
Kirchberg-Phillips classification, $A \otimes \Kk \cong C^{*} (E_{T}) \otimes C^{*}
(F_{\ZZ })$.  Hence, $\ndim(A \otimes \Kk) = 1$; so
\cite[Corollary~2.8]{WinterZacharias:Adv10} gives $\ndim(A) = 1$.
\end{proof}

\begin{rmk}\label{rmk:SpecSeq}
A key step in the preceding proof was to show that the maps $j_n \circ \iota_n :
C^*(\Lambda) \to C^*(\Lambda) \otimes \Kk_{\Lambda^{<n}}$ induce multiplication by
$n_1n_2$ in $K$-theory. We were able to do this using the K\"unneth formula because
tensor products of $1$-graph $C^*$-algebras provide a large enough class of models to
cover all the UCT-Kirchberg algebras in question. It seems likely that our techniques
could be used to compute the exact value of nuclear dimension for a large class of
nonsimple Kirchberg algebras with torsion in $K_1$ along the lines of
\cite{RuizSimsTomforde:arXiv:1312.0507}, but to do this, we need to know that $j_n \circ
\iota_n$ induces multiplication by $n_1 n_2$ in the $K$-theory of every ideal of
$C^*(\Lambda)$ for general $2$-graphs $\Lambda$. This can be proved using Evans'
calculation of $K$-theory for $k$-graph algebras \cite{Evans:NYJM08} and naturality of
Kasparov's spectral sequence (see \cite[page~185]{KumjianPaskSims:DM08}). But this would
require introducing extraneous notation or digging into the proofs in
\cite{Evans:NYJM08}, so we have not pursued this approach here.
\end{rmk}

\begin{dfn}
A homomorphism $\ftn{\phi} {A}{B}$ is called \emph{full} if for all $a \in A$ with $a
\neq 0$, the closed two-sided ideal generated by $\phi(a)$ is equal to $B$.
\end{dfn}

\begin{lem}\label{l:directlimitsimple}
Let $(\phi_n : A_n \to A_{n+1})^\infty_{n=1}$ be a directed system of $C^*$-algebras, and
set $A = \varinjlim (A_{n}, \phi_n)$.  If there exists $N \in \NN$ such that $\phi_n$ is
full for all $n \geq N$, then $A$ is simple.  If, in addition, each $A_{n}$ is a finite
direct sum of UCT-Kirchberg algebras, then $A$ is a UCT-Kirchberg algebra.
\end{lem}
\begin{proof}
Let $I$ be a nonzero ideal of $A$.  Then $I_{n} = \phi_{n, \infty}^{-1}(I)$ is an ideal
of $A_{n}$ and $\phi_n (I_{n}) \subseteq I_{n+1}$.  Since $I$ is nonzero, there exists
$M$ such that $I_{n} \neq 0$ for all $n \geq M$.  So for $n \geq \max\{N, M \}$ the ideal
generated by $\phi_n(I_{n})$ is $A_{n+1}$.  Since $\phi_n (I_{n}) \subseteq I_{n+1}$, we
have $I_{n+1} = A_{n+1}$ for $n \geq \max \{N, M \}$, and so $I = A$.

Suppose each $A_{n}$ is a finite direct sum of UCT-Kirchberg algebras.  Then every
nonzero projection of any $A_{n}$ is properly infinite.  Since $\phi_n$ is injective for
$n \geq N$ (because the maps are full), $\phi_n$ takes properly infinite projections to
properly infinite projections for all $n \geq N$.  Thus, every nonzero projection of $A$
is properly infinite.  Hence, $A$ is a purely infinite simple $C^{*}$-algebra.  Since
each $A_{n}$ is separable, nuclear, and in the UCT class, $A$ is too. Thus, $A$ is a
UCT-Kirchberg algebra.
\end{proof}

\begin{lem}\label{l:direct limit decomposition}
Let $A$ be a stable UCT-Kirchberg algebra. Then there exist sequences
$(\Lambda_n)_{n\in \NN}$ and $(\Gamma_n)_{n \in \NN}$ of row-finite $2$-graphs with no
sources, and homomorphisms $\phi_n : C^*(\Lambda_n) \oplus C^*(\Gamma_n) \to
C^*(\Lambda_{n+1}) \oplus C^*(\Gamma_{n+1})$ such that: each $C^*(\Lambda_n)$ and each
$C^*(\Gamma_n)$ is a stable UCT-Kirchberg algebra with finitely-generated $K$-theory;
each $K_1(C^*(\Lambda_n))$ is free abelian; each $K_0(C^*(\Gamma_n))$ is trivial and each
$K_1(C^*(\Gamma_n))$ is finite; and $A \cong \varinjlim(C^*(\Lambda_n) \oplus C^*(\Gamma_n),
\phi_n)$.
\end{lem}
\begin{proof}
Let $( G_{n,0} )_{n \in \NN}$ and $( G_{n,1} )_{ n \in \NN}$ be increasing families of finitely generated
abelian groups with $\bigcup_{n=1}^{\infty} G_{n, 0} = K_{0}(A)$ and $\bigcup_{n =
1}^{\infty} G_{n,1} = K_{1} (A)$.  Decompose each $G_{n,1}$ as $G_{n,1} = T( G_{n,1} )
\oplus F( G_{n,1} )$, where $T( G_{n,1} )$ is finite and $F( G_{n,1} )$ is free.

Let $E_{ \Kk }$ be the infinite row-finite graph $\cdots \to  \bullet \to \bullet \to \cdots$
(so $C^{*} ( E_{ \Kk} )  \cong \Kk$). For each $n$ apply
\cite[Theorem~1.2]{Szymanski:IUMJ2002} to obtain a row-finite strongly connected graph
$E_n$ such that $C^*(E_n)$ is a UCT Kirchberg algebra with $K_{*}(C^{*}( E_{n} ) ) =(
G_{n,0} , F( G_{n,1} ) )$.  Then each $\Lambda_n := E_{n}^{*} \times E_{\Kk}^{*}$ is a
row-finite 2-graph with no sources such that $C^*(\Lambda_n)$ is a stable UCT-Kirchberg
algebra with $K_{*} ( C^{*}(\Lambda_n)) = ( G_{n,0} , F( G_{n,1} ) )$. For each $n$, let
$E_{T(G_{n,1})}$ and $F_\ZZ$ be as in Lemma~\ref{l:nuclear dimension 2-graph} and the preceding
discussion. Let $\Gamma_n := E^*_{T(G_{n,1})} \times F^*_\ZZ$. Then $\Gamma_n$ is a
row-finite 2-graph with no sources, and $C^*(\Gamma_n)$ is a stable UCT-Kirchberg algebra with
$K_{*}(C^*(\Gamma_n)) = ( 0 , T( G_{n,1} ) )$.

Each $K_{*}(C^*(\Lambda_n) \oplus C^*(\Gamma_n)) = (G_{n,0} , G_{n,1} )$.  By
Kirchberg-Phillips (cf. \cite{kirchpureinf} and \cite{Phillips:Doc00}), for each $n \in
\NN$, there exists a full homomorphism $\ftn{\phi_n}{C^*(\Lambda_n) \oplus
C^*(\Gamma_n)}{C^*(\Lambda_{n+1}) \oplus C^*(\Gamma_{n+1})}$ which in $K$-theory induces
the inclusion map $G_{n,i} \hookrightarrow G_{n+1, i}$. Therefore, $K_{i} ( \varinjlim (
C^*(\Lambda_n) \oplus C^*(\Gamma_n) , \phi_{n} ) ) \cong K_{i} ( A )$.  So, by
Lemma~\ref{l:directlimitsimple} and the Kirchberg-Phillips classification, $A \cong
\varinjlim ( C^*(\Lambda_n) \oplus C^*(\Gamma_n), \phi_{n} )$.
\end{proof}


\begin{thm}
Every UCT-Kirchberg algebra has nuclear dimension~1.
\end{thm}
\begin{proof}
Let $A$ be a UCT-Kirchberg algebra. Since Kirchberg algebras are not AF,
\cite[Remarks~2.2(iii)]{WinterZacharias:Adv10} shows that $A$ has nuclear dimension at
least 1. Corollary~2.8 of \cite{WinterZacharias:Adv10} shows that the nuclear dimension
of $A \otimes \Kk$ is the same as that of $A$, so we may assume that $A$ is stable.

By Lemma~\ref{l:direct limit decomposition}, $A  \cong \varinjlim ( B_{n} \oplus C_{n} ,
\phi_n)$ where $B_{n}$ and $C_{n}$ are UCT-Kirchberg algebras such that $K_{1} ( B_{n} )$
is free, $K_{0} ( C_{n} ) = 0$, and $K_{1} ( C_{n} )$ is a finite abelian group. By
Lemma~\ref{l:nuclear dimension 2-graph}, $\ndim(C_{n}) = 1$.  By
\cite[Theorem~4.1]{Enders:arXiv:1405.6538}, $\ndim(B_{n}) = 1$. Proposition~2.3(i) of
\cite{WinterZacharias:Adv10} implies that each $B_{n} \oplus C_{n}$ has nuclear dimension
1.  It now follows from \cite[Proposition~2.3(iii)]{WinterZacharias:Adv10} that $A$ has
nuclear dimension 1.
\end{proof}
{"config": "arxiv", "file": "1406.2045.tex"}