# Real and complex $\displaystyle K$-theory for higher rank graph algebras
arising from cube complexes

Jeffrey L. Boersema  Seattle University
Department of Mathematics
Seattle, Washington 98133, USA<EMAIL_ADDRESS>and  Alina Vdovina  The
City College of New York and Graduate Center, CUNY
Department of Mathematics
New York, 10031, USA<EMAIL_ADDRESS>
###### Abstract.

Using the Evans spectral sequence and its counter-part for real $\displaystyle
K$-theory, we compute both the real and complex $\displaystyle K$-theory of
several infinite families of $\displaystyle C^{*}$-algebras based on higher-
rank graphs of rank $\displaystyle 3$ and $\displaystyle 4$. The higher-rank
graphs we consider arise from double-covers of cube complexes. By considering
the real and complex $\displaystyle K$-theory together, we are able to carry
these computations much further than might be possible considering complex
$\displaystyle K$-theory alone. As these algebras are classified by
$\displaystyle K$-theory, we are able to characterize the isomorphism classes
of the graph algebras in terms of the combinatorial and number-theoretic
properties of the construction ingredients.

##  1\. Introduction

Higher rank graphs were defined in [9] and their further theory was developed
in [15] and [7]. The main motivation was a systematic study of a large class
of $\displaystyle C^{*}$-algebras classifiable by their $\displaystyle
K$-theory. Despite of the vast literature on the subject, explicit
computations of the K-theory of the higher-rank $\displaystyle C^{*}$-algebras
is very rare, especially in rank three and higher. The first rank three
example was done in [5], and the first infinite series of rank three and
higher examples were described in [12]. Nevertheless, in both [5] and [12],
there were open questions on exact order of certain abelian subgroups in
K-theory. In this paper we present several infinite series of $\displaystyle
C^{*}$-algebras associated to rank-3 and rank-4 graphs and we compute their
K-theory completely and explicitly.

Not only are we considering the (complex) $\displaystyle C^{*}$-algebras
associated to these higher rank graphs, we are also considering real
$\displaystyle C^{*}$-algebras for these graphs and we are computing the real
$\displaystyle K$-theory. Real $\displaystyle C^{*}$-algebras associated to a
higher rank graph, and more generally real $\displaystyle C^{*}$-algebras
associated to a higher rank graph with an involution, were introduced in [2].
The analog of Evans’ spectral sequence was also developed in [2] to compute
the real $\displaystyle K$-theory of such algebras. The examples that we
consider in this paper are rank-3 and rank-4 graphs with two vertices and a
non-trivial involution that swaps the two vertices. We will be calculating the
$\displaystyle K$-theory of both real $\displaystyle C^{*}$-algebras: the one
associated with the graph with the trivial involution and the one associated
with the graph with the non-trivial involution. Previous calculations of
$\displaystyle K$-theory for real $\displaystyle C^{*}$-algebras of higher
rank graphs have been conducted in [2] and [3]. However, in those cases the
graphs either had rank no more than 2, or the graphs could be factored as a
product of graphs with rank no more than 2. Remarkably, we find that the
consideration of the real $\displaystyle K$-theory also allows us to compute
the (complex) $\displaystyle K$-theory in some cases where that was otherwise
intractable. In particular, we are able to resolve the open question in [12]
mentioned above.

The complex $\displaystyle C^{*}$-algebras associated to our higher rank
graphs fall in the category of purely infinite simple $\displaystyle
C^{*}$-algebras classified by $\displaystyle K$-theory in [8] and [13].
Similarly, the real $\displaystyle C^{*}$-algebras in this paper fall in the
category of purely infinite simple $\displaystyle C^{*}$-algebras classified
in [4]. Based on this, we will be able to characterize the isomorphism classes
of the resulting algebras in terms of the combinatoric and number-theory
properties of the construction ingredients.

The higher-rank graphs that we consider in this paper arise from cube
complexes, and their double covers, as in Section 6 of [12]. We will review
this construction in the next section. We will also review the key preliminary
notions, including the definition of the real and complex $\displaystyle
C^{*}$-algebras based on higher rank graphs, real $\displaystyle K$-theory,
and the spectral sequence technology to calculate the $\displaystyle K$-theroy
in the real case.

The geometric core of higher-rank graphs was introduced in [10]. Initially the
higher-rank graphs were defined as small categories in [9]. Connections of
higher rank graphs with geometry and combinatorics was known before, see, for
example, [7], the combinatorial analogue without reference to category theory
was done only in [10]. The higher-dimensional digraphs introduced in [10]
provide a bridge between cube complexes and higher-rank graphs. The
automorphism groups of cube complexes covered by products of trees induce
automorphism groups of higher-rank graphs. If the fundamental group of such a
cube complex has a subgroup of index two, then the double-cover of the cube
complex has an involution.

We would like to mention a couple of connections of the C*-algebras we
consider with other topics in mathematics. In [11], K-theory of C*-algebras is
used as an invariant of higher-dimensional Thompson groups which are otherwise
very hard to distinguish. In [6], K-theory was studies in connection with
Matui’s HK-conjecture. The further development of both real and complex
$\displaystyle K$-theory will serve to increase the possibility of such
positive connections.

##  2\. Preliminaries

###  2.1. Higher rank graphs

We recall the definition of a $\displaystyle k$-graph due to Kumjian and Pask
[9]. For an integer $\displaystyle k\geq 1$, we view
$\displaystyle{\mathbb{N}}^{k}$ as a monoid under pointwise addition. A
$\displaystyle k$-graph is a countable small category $\displaystyle\Lambda$
together with an assignment of a _degree_ $\displaystyle
d(\mu)\in{\mathbb{N}}^{k}$ to every morphism $\displaystyle\mu\in\Lambda$ such
that for all $\displaystyle\mu,\nu,\pi\in\Lambda$ the following hold

  1. (1)

$\displaystyle d(\mu\nu)=d(\mu)+d(\nu)$; and

  2. (2)

whenever $\displaystyle d(\pi)=m+n$ for $\displaystyle
m,n\in{\mathbb{N}}^{k}$, there is a unique factorisation
$\displaystyle\pi=\mu\nu$ such that $\displaystyle d(\mu)=m$ and
$\displaystyle d(\nu)=n$.

Condition (2) is known as the _factorisation property_ in the $\displaystyle
k$-graph. The composition in $\displaystyle\mu\nu$ is understood in the sense
of morphisms, thus the source $\displaystyle s(\mu)$ of $\displaystyle\mu$
equals the range $\displaystyle r(\nu)$ of $\displaystyle\nu$. Note that the
morphisms of degree $\displaystyle 0$ (in $\displaystyle{\mathbb{N}}^{k}$) are
necessarily the identity morphisms in the category. Denote this set by
$\displaystyle\Lambda^{0}$, and refer to its elements as _vertices_ of
$\displaystyle\Lambda$. With $\displaystyle e_{1},\dots,e_{k}$ denoting the
generators of $\displaystyle{\mathbb{N}}^{k}$, the set
$\displaystyle\Lambda^{e_{i}}=\\{\lambda\in\Lambda\mid d(\lambda)=e_{i}\\}$
consists of edges (or morphisms) of degree $\displaystyle e_{i}$, for
$\displaystyle i=1,\dots,k$. We write $\displaystyle v\Lambda^{n}$ for the set
of morphisms of degree $\displaystyle n\in{\mathbb{N}}^{k}$ with range
$\displaystyle v$.

Throughout this paper we are concerned with $\displaystyle k$-graphs where
$\displaystyle\Lambda^{0}$ and all $\displaystyle\Lambda^{e_{i}}$,
$\displaystyle i=1,\dots,k$, are finite. A $\displaystyle k$-graph
$\displaystyle\Lambda$ so that $\displaystyle 0<\\#v\Lambda^{n}<\infty$ for
all $\displaystyle v\in\Lambda^{0}$ and all $\displaystyle
n\in{\mathbb{N}}^{k}$ is source free and row-finite. The _adjacency matrices_
$\displaystyle
M_{1},\dots,M_{k}\in\operatorname{Mat}_{\Lambda^{0}}({\mathbb{N}})$ of
$\displaystyle\Lambda$ are $\displaystyle\Lambda^{0}\times\Lambda^{0}$
matrices with

$\displaystyle M_{i}(v,w)=|v\Lambda^{e_{i}}w|.$

By the factorisation property, the matrices $\displaystyle M_{i}$ pairwise
commute for $\displaystyle i=1,\dots,k$.

###  2.2. Rank-$\displaystyle k$ graphs with two vertices

We now review the specific construction of two-vertex $\displaystyle k$-graphs
involving cube complexes discussed in Section 6 of [12]. The construction
consists of two steps: First, we construct a family of cube complexes with two
vertices, covered by products of $\displaystyle k$ trees, and second, we
explain how to get a $\displaystyle k$-graph from each complex. These
$\displaystyle k$-graphs happen to have a natural non-trivial involution
$\displaystyle\gamma$, which will be important later on. For the background on
cube complexes covered by products of $\displaystyle k$ trees, see [10], [12],
and [16].

Step 1. Let $\displaystyle X_{1},...,X_{k}$ be distinct alphabets, such that
$\displaystyle\left|X_{i}\right|=m_{i}$ and

$\displaystyle X_{i}=\\{x_{1}^{i},x_{2}^{i},...,x_{m_{i}}^{i}\\}.$

Let $\displaystyle F_{i}$ be the free group generated by $\displaystyle
X_{i}$. Then the direct product

$\displaystyle G=F_{1}\times F_{2}\times\ldots\times F_{k}$

of $\displaystyle k$ free groups $\displaystyle F_{1}$,$\displaystyle
F_{2}$,…,$\displaystyle F_{k}$ has a presentation

$\displaystyle G=\langle
X_{1},X_{2},...,X_{k}\mid[x^{i}_{s},x^{j}_{l}]=1,i\neq
j=1,...,k;s=1,...,m_{i};l=1,...,m_{j}\rangle,$

where $\displaystyle[x,y]$ means commutator $\displaystyle xyx^{-1}y^{-1}$.
The group $\displaystyle G$ acts simply transitively on a Cartesian product
$\displaystyle\Delta$ of $\displaystyle k$ trees $\displaystyle
T_{1},T_{2},..,T_{k}$ of valencies $\displaystyle 2m_{1},2m_{2},...,2m_{k}$
respectively. The quotient of this action is a cube complex $\displaystyle P$
with one vertex such that the universal cover of $\displaystyle P$ is
$\displaystyle\Delta$. The edges of the cube complex $\displaystyle P$ are
naturally equipped with orientations and labellings by elements of
$\displaystyle X=X_{1}\cup X_{2}...\cup X_{k}$ and the $\displaystyle
1$-skeleton of $\displaystyle P$ is a wedge of
$\displaystyle\sum_{i=1}^{k}m_{i}$ circles. We construct a family of double
covers of $\displaystyle P$ in the following way. A double cover
$\displaystyle P^{2}$ of $\displaystyle P$ has two vertices, say
$\displaystyle v_{1}$ and $\displaystyle v_{2}$. For each edge $\displaystyle
x$ there are two edges, say $\displaystyle x_{1}$ and $\displaystyle x_{2}$,
in the $\displaystyle 1$-skeleton of $\displaystyle P^{2}$. In fact there are
two choices for the structure of these edges: either both $\displaystyle
x_{1}$ and $\displaystyle x_{2}$ are loops, one at $\displaystyle v_{1}$ and
the other at $\displaystyle v_{2}$; or one of the edges $\displaystyle
x_{1},x_{2}$ points from $\displaystyle v_{1}$ to $\displaystyle v_{2}$ and
the other points from $\displaystyle v_{2}$ to $\displaystyle v_{1}$. We will
say that the edge pair $\displaystyle x_{1},x_{2}$ has type one in the first
case, and has type two otherwise.

For example, in Figure 1, the edge pair $\displaystyle b_{1},b_{2}$ is type 1
and the edge pair $\displaystyle a_{1},a_{2}$ is type 2. Figures 1,2,3,4,5
show that our double covers are well defined.

Fig. 1

In the double covers we consider, we stipulate that at least one edge pair has
type two (so the graph is connected) and that all of the edge pairs with
labels in the same set $\displaystyle X_{i}$ will have the same type (just for
convenience).

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Step 2. We explain now how to construct a $\displaystyle k$-graph
$\displaystyle C$ from the cube complex $\displaystyle P^{2}$. The graph
$\displaystyle C$ will have the same set of vertices as $\displaystyle P^{2}$,
but the number of edges will double. Specifically, for each edge
$\displaystyle x$ in $\displaystyle P^{2}$, we obtain two edges $\displaystyle
x$ and $\displaystyle x^{\prime}$ where $\displaystyle s(x)=r(x^{\prime})$ and
$\displaystyle s(x^{\prime})=r(x)$. Furthermore, the degree of $\displaystyle
x$ and $\displaystyle x^{\prime}$ is $\displaystyle e_{i}$, descending from
the labels associated from the edges of $\displaystyle\Delta$ (colloquially we
say that the edges $\displaystyle x$ and $\displaystyle x^{\prime}$ have color
$\displaystyle i$). Each geometric square $\displaystyle abcd$ in
$\displaystyle P^{2}$ will give rise to four squares (or commutativity
relations) in $\displaystyle C$: namely,

$\displaystyle
ab=d^{\prime}c^{\prime},bc=a^{\prime}d^{\prime},cd=b^{\prime}a^{\prime},da=c^{\prime}b^{\prime}.$

For example, the square $\displaystyle a_{1}b_{2}a_{2}^{-1}b_{1}^{-1}$ (the
front face of the cube in Fig. 5) will give rise to four squares in
$\displaystyle C$: namely,

$\displaystyle
a_{1}b_{2}=b_{1}a_{2},~{}b_{2}a_{2}^{\prime}=a_{1}^{\prime}b_{1},~{}a_{2}^{\prime}b_{1}^{\prime}=b_{2}^{\prime}a_{1}^{\prime},~{}b_{1}^{\prime}a_{1}=a_{2}b_{2}^{\prime}.$

This completes the description of the construction of a large collection of
examples of rank-$\displaystyle k$ graphs. In the rest of this paper we will
consider graphs that arise from this construction, restricting our attention
to the ones in which $\displaystyle P$ has one vertex, so that the
rank-$\displaystyle k$ graph $\displaystyle C$ has two vertices (and the
number of edges is a multiple of 4).

To proceed, we need to define two special kinds of matrices,

$\displaystyle D_{i}=\begin{bmatrix}2m_{i}&0\\\
0&2m_{i}\end{bmatrix}\text{~{}and~{}}T_{i}=\begin{bmatrix}0&2m_{i}\\\
2m_{i}&0\end{bmatrix}.$

If the edges in $\displaystyle P$ associated with color $\displaystyle i$ have
lifts in $\displaystyle P^{2}$ that are type 1, then the adjacency matrix of
the rank-$\displaystyle k$ graph $\displaystyle C$ will have the form
$\displaystyle T_{i}$ for $\displaystyle i\in\\{1,2,\dots,k\\}$. If the lifts
are of type 2, then the adjacency matrix will have the form $\displaystyle
D_{i}$.

###  2.3. Real and Complex $\displaystyle C^{*}$-algebras

From these higher-rank graphs, we construct real and complex $\displaystyle
C^{*}$-algebras, following [2] and [9], as follows. For any row-finite source-
free k-graph $\displaystyle\Lambda$, $\displaystyle C^{*}(\Lambda)$ is the
universal complex $\displaystyle C^{*}$-algebra generated by partial
isometries $\displaystyle s_{\lambda}$, for $\displaystyle\lambda\in\Lambda$,
subject to the relations

  1. (1)

For each $\displaystyle v\in\Lambda^{0}$, $\displaystyle s_{v}$ is a
projection, and $\displaystyle s_{v}s_{w}=\delta_{v,w}s_{v}$.

  2. (2)

For each $\displaystyle\lambda\in\Lambda,\
s_{\lambda}^{*}s_{\lambda}=s_{s(\lambda)}$.

  3. (3)

For each $\displaystyle\lambda,\mu\in\Lambda,\
s_{\lambda}s_{\mu}=s_{\lambda\mu}$.

  4. (4)

For each $\displaystyle v\in\Lambda^{0}$ and each $\displaystyle
n\in{\mathbb{N}}^{k}$, $\displaystyle\displaystyle s_{v}=\sum_{\lambda\in
v\Lambda^{n}}s_{\lambda}s_{\lambda}^{*}.$

The real $\displaystyle C^{*}$-algebra $\displaystyle
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda)$ is the universal real
$\displaystyle C^{*}$-algebra generated by the same partial isometries
$\displaystyle s_{\lambda}$ as above subject to the same relations. We can and
do typically represent $\displaystyle
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda)$ as the real subalgebra of
$\displaystyle C^{*}(\Lambda)$ generated by $\displaystyle s_{\lambda}$. Thus
$\displaystyle C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda)$ is the closure
of the set of all linear combinations of products of $\displaystyle
s_{\lambda}$ and $\displaystyle s_{\lambda}^{*}$.

In addition, there is a obvious involution $\displaystyle\gamma$ on the graph
$\displaystyle\Lambda$ that interchanges the two vertices and interchanges
pairs of edges in a way consistent with the action on the vertices. In this
situation, there is a different real $\displaystyle C^{*}$-algebra
$\displaystyle C_{\scriptscriptstyle{\mathbb{R}}}^{*}(\Lambda,\gamma)$,
associated to this graph with involution, as constructed in [2], which is
represented as the real $\displaystyle C^{*}$-algebra in $\displaystyle
C^{*}(\Lambda)$ generated by the elements of the form $\displaystyle
zs_{\lambda}+\overline{z}s_{\gamma(\lambda)}$ for $\displaystyle
z\in{\mathbb{C}}$.

The two real $\displaystyle C^{*}$-algebras $\displaystyle
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda)$ and $\displaystyle
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma)$ are both real
structures associated with $\displaystyle C^{*}(\Lambda)$, in the sense that
the complexification of each one is isomorphic to the complex $\displaystyle
C^{*}$-algebra $\displaystyle C^{*}(\Lambda)$. A typical problem in the theory
of real $\displaystyle C^{*}$-algebras is to identify up to isomorphism all of
the real structures associated with a given complex $\displaystyle
C^{*}$-algebra. The constructions of these real $\displaystyle C^{*}$-algebras
depend on the integer values of $\displaystyle m_{i}$ (for $\displaystyle
i\in\\{1,2,\dots,k\\}$), on the choices of the type of lifts for each
$\displaystyle i$ (that is the form of the adjacency matrices $\displaystyle
M_{i}$), and the choice of whether we are consider the real $\displaystyle
C^{*}$-algebra $\displaystyle C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda)$
or $\displaystyle C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma)$.

###  2.4. K-theory

In our work, we will use the abbreviated version of united $\displaystyle
K$-theory $\displaystyle K^{\scriptscriptstyle{\it CR}}(A)$ that was
introduced in [1] for real $\displaystyle C^{*}$-algebras. From Theorem 10.2
of [4], this invariant classifies the category of real purely infinite simple
$\displaystyle C^{*}$-algebras consisting of exactly those real $\displaystyle
C^{*}$-aglebras whose complexifications fall under the classification theorem
for complex purely infinite simple $\displaystyle C^{*}$-algebras, by
Kirchberg and Phillips in [8] and [13]. This category includes all of the real
graph algebras we will consider in this paper. Specifically, for a real
$\displaystyle C^{*}$-algebra $\displaystyle A$ we define

$\displaystyle K^{\scriptscriptstyle{\it CR}}(A)=\\{KO_{*}(A),KU_{*}(A)\\}$

where $\displaystyle KO_{*}(A)$ is the standard 8-periodic real $\displaystyle
K$-theory for a real $\displaystyle C^{*}$-algebra and $\displaystyle
KU_{*}(A)=K_{*}({\mathbb{C}}\otimes_{\scriptscriptstyle{\mathbb{C}}}A)$ is the
standard 2-periodic $\displaystyle K$-theory of the complexification of
$\displaystyle A$. The invariant $\displaystyle K^{\scriptscriptstyle{\it
CR}}(A)$ also includes the natural transformations

$\displaystyle\displaystyle r_{i}$ $\displaystyle\displaystyle\colon
KU_{i}(A)\rightarrow KO_{i}(A)$ $\displaystyle\displaystyle\text{induced by
the standard inclusion }\mathbb{C}\rightarrow M_{2}(\mathbb{R})$
$\displaystyle\displaystyle c_{i}$ $\displaystyle\displaystyle\colon
KO_{i}(A)\rightarrow KU_{i}(A)$ $\displaystyle\displaystyle\text{induced by
the standard inclusion }\mathbb{R}\rightarrow\mathbb{C}$
$\displaystyle\displaystyle\psi_{i}$ $\displaystyle\displaystyle\colon
KU_{i}(A)\rightarrow KU_{i}(A)$ $\displaystyle\displaystyle\text{induced by
conjugation }\mathbb{C}\rightarrow\mathbb{C}$
$\displaystyle\displaystyle\eta_{i}$ $\displaystyle\displaystyle\colon
KO_{i}(A)\rightarrow KO_{i+1}(A)$ induced by multiplication by
$\displaystyle\eta\in KO_{1}({\mathbb{R}})={\mathbb{Z}}_{2}$
$\displaystyle\displaystyle\xi_{i}$ $\displaystyle\displaystyle\colon
KO_{i}(A)\rightarrow KO_{i+4}(A)$ induced by multiplication by
$\displaystyle\xi\in KO_{4}({\mathbb{R}})={\mathbb{Z}}$.

The additional structure tends to aid in the computations of $\displaystyle
KO_{*}(A)$ because the natural transformations satisfy the relations

$\displaystyle\displaystyle rc=2$ $\displaystyle\displaystyle cr=1+\psi$
$\displaystyle\displaystyle 2\eta=0$ $\displaystyle\displaystyle r\psi=r$
$\displaystyle\displaystyle\psi^{2}=\mathrm{id}$
$\displaystyle\displaystyle\eta^{3}=0$ $\displaystyle\displaystyle\psi c=c$
$\displaystyle\displaystyle\psi\beta_{\scriptscriptstyle
U}=-\beta_{\scriptscriptstyle U}\psi$
$\displaystyle\displaystyle\xi=r\beta_{\scriptscriptstyle U}^{2}c$

and they fit into a long exact sequence

$\displaystyle\cdots\xrightarrow{r\beta_{\scriptscriptstyle
U}^{-1}}KO_{i}(A)\xrightarrow{\eta}KO_{i+1}(A)\xrightarrow{c}KU_{i+1}(A)\xrightarrow{r\beta_{\scriptscriptstyle
U}^{-1}}KO_{i-1}(A)\xrightarrow{\eta}\cdots$

The target category of this functor is the category of all
$\displaystyle\mathcal{CR}$-modules.

To compute $\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$, we will use the
spectral sequence of [2, Theorem 3.13], which generalizes the spectral
sequence of [5] for complex $\displaystyle K$-theory. The $\displaystyle
E^{2}$ page of the spectral sequence arises from the homology of a certain
chain complex $\displaystyle\mathcal{C}$, which is based on the
$\displaystyle\mathcal{CR}$-modules $\displaystyle K^{\scriptscriptstyle{\it
CR}}({\mathbb{R}})$ and $\displaystyle K^{\scriptscriptstyle{\it
CR}}({\mathbb{C}})$ and relies on the combinatorial data of
$\displaystyle\Lambda$ and $\displaystyle\gamma$. We will review the details
of the formation of this spectral sequence in our calculations in the
following sections. The spectral sequence converges to $\displaystyle
K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$ in the sense
that there is a filtration of $\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$, the subfactors
of which appear as the groups of the $\displaystyle E^{\infty}$ page.
Specifically, the groups $\displaystyle
KO_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$ and
$\displaystyle KU_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$
are obtained from the groups
$\displaystyle(E^{\infty}_{p,q})^{\scriptscriptstyle O}$ and
$\displaystyle(E^{\infty}_{p,q})^{\scriptscriptstyle U}$where $\displaystyle
p+q=n$.

This spectral sequence exists in the category of
$\displaystyle\mathcal{CR}$-modules, which means that it has both a real
component and a complex component, as alluded to above, and these components
are connected by the natural transformations including $\displaystyle r$ and
$\displaystyle c$. This is the case on each page of the spectral sequence
starting with the chain complex $\displaystyle\mathcal{C}$ and the natural
transformations commute with the differentials $\displaystyle d$. The complex
component of this spectral sequence coincides with the spectral sequence of
Evans in [5].

For reference, the groups of $\displaystyle K^{\scriptscriptstyle{\it
CR}}({\mathbb{R}})$ and $\displaystyle K^{\scriptscriptstyle{\it
CR}}({\mathbb{C}})$ are shown below from Tables 1 and 2 of [2].

Fig. 6 – $\displaystyle K^{\scriptscriptstyle{\it CR}}({\mathbb{R}})$

$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\hline\cr
n&\makebox[28.45274pt][c]{0}&\makebox[28.45274pt][c]{1}&\makebox[28.45274pt][c]{2}&\makebox[28.45274pt][c]{3}&\makebox[28.45274pt][c]{4}&\makebox[28.45274pt][c]{5}&\makebox[28.45274pt][c]{6}&\makebox[28.45274pt][c]{7}\\\
\hline\cr\hline\cr
KO_{n}({\mathbb{R}})&{\mathbb{Z}}&{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}&0&{\mathbb{Z}}&0&0&0\\\
\hline\cr
KU_{n}({\mathbb{R}})&{\mathbb{Z}}&0&{\mathbb{Z}}&0&{\mathbb{Z}}&0&{\mathbb{Z}}&0\\\
\hline\cr\hline\cr c_{n}&1&0&0&0&2&0&0&0\\\ \hline\cr r_{n}&2&0&1&0&1&0&0&0\\\
\hline\cr\psi_{n}&1&0&-1&0&1&0&-1&0\\\ \hline\cr\eta_{n}&1&1&0&0&0&0&0&0\\\
\hline\cr\hline\cr\end{array}$

Fig. 7 – $\displaystyle K^{\scriptscriptstyle{\it CR}}({\mathbb{C}})$

$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\hline\cr
n&\makebox[28.45274pt][c]{0}&\makebox[28.45274pt][c]{1}&\makebox[28.45274pt][c]{2}&\makebox[28.45274pt][c]{3}&\makebox[28.45274pt][c]{4}&\makebox[28.45274pt][c]{5}&\makebox[28.45274pt][c]{6}&\makebox[28.45274pt][c]{7}\\\
\hline\cr\hline\cr
KO_{n}({\mathbb{C}}))&{\mathbb{Z}}&0&{\mathbb{Z}}&0&{\mathbb{Z}}&0&{\mathbb{Z}}&0\\\
\hline\cr
KU_{n}({\mathbb{C}})&{\mathbb{Z}}^{2}&0&{\mathbb{Z}}^{2}&0&{\mathbb{Z}}^{2}&0&{\mathbb{Z}}^{2}&0\\\
\hline\cr\hline\cr c_{n}&\bigl{(}\begin{smallmatrix}{1}\\\
{1}\end{smallmatrix}\bigr{)}&0&\bigl{(}\begin{smallmatrix}{1}\\\
{-1}\end{smallmatrix}\bigr{)}&0&\bigl{(}\begin{smallmatrix}{1}\\\
{1}\end{smallmatrix}\bigr{)}&0&\bigl{(}\begin{smallmatrix}{1}\\\
{-1}\end{smallmatrix}\bigr{)}&0\\\ \hline\cr
r_{n}&\bigl{(}\begin{smallmatrix}{1}&{1}\end{smallmatrix}\bigr{)}&0&\bigl{(}\begin{smallmatrix}{1}&{-1}\end{smallmatrix}\bigr{)}&0&\bigl{(}\begin{smallmatrix}{1}&{1}\end{smallmatrix}\bigr{)}&0&\bigl{(}\begin{smallmatrix}{1}&{-1}\end{smallmatrix}\bigr{)}&0\\\
\hline\cr\psi_{n}&\bigl{(}\begin{smallmatrix}{0}&{1}\\\
{1}&{0}\end{smallmatrix}\bigr{)}&0&\bigl{(}\begin{smallmatrix}{0}&{-1}\\\
{-1}&{0}\end{smallmatrix}\bigr{)}&0&\bigl{(}\begin{smallmatrix}{0}&{1}\\\
{1}&{0}\end{smallmatrix}\bigr{)}&0&\bigl{(}\begin{smallmatrix}{0}&{-1}\\\
{-1}&{0}\end{smallmatrix}\bigr{)}&0\\\ \hline\cr\eta_{n}&0&0&0&0&0&0&0&0\\\
\hline\cr\hline\cr\end{array}$

##  3\. The rank-3 case, with no involution

Let $\displaystyle\Lambda$ be a rank-3 graph of the form discussed above.
Specifically, $\displaystyle\Lambda$ is a two-vertex graph and the incidence
matrices $\displaystyle M_{i}$ for $\displaystyle\Lambda$ each have the form

$\displaystyle T_{i}=\begin{bmatrix}0&2n_{i}\\\
2n_{i}&0\end{bmatrix}\quad\text{or}\quad D_{i}=\begin{bmatrix}2m_{i}&0\\\
0&2m_{i}\end{bmatrix}$

for $\displaystyle i=1,2,3$ (with the restriction that at least one of the
incidence matrices must have the form $\displaystyle T_{i}$). From [12], we
have the complex $\displaystyle K$-theory $\displaystyle
KU(C_{\scriptscriptstyle{\mathbb{R}}}^{*}(\Lambda)=K_{*}(C^{*}(\Lambda))$
given by

$\displaystyle\displaystyle K_{*}(C^{*}(\Lambda))$
$\displaystyle\displaystyle=0$ if $\displaystyle g=1$
$\displaystyle\displaystyle K_{0}(C^{*}(\Lambda))$
$\displaystyle\displaystyle=\text{some extension of
$\displaystyle{\mathbb{Z}}_{g}$ by $\displaystyle{\mathbb{Z}}_{g}$}$ if
$\displaystyle g\geq 3$ $\displaystyle\displaystyle K_{1}(C^{*}(\Lambda))$
$\displaystyle\displaystyle={\mathbb{Z}}_{g}^{2}$ if $\displaystyle g\geq 3$.

However, in [12] the nature of the extension was not determined. Furthermore,
in the cases where more than one of the matrices $\displaystyle M_{i}$ has the
off-diagonal form, the formula for $\displaystyle g$ in [12] is incorrect. In
this section, we will compute both $\displaystyle
KO_{*}(C_{\scriptscriptstyle{\mathbb{R}}}^{*}(\Lambda))$ and $\displaystyle
KU_{*}(C_{\scriptscriptstyle{\mathbb{R}}}^{*}(\Lambda))$ (thereby determining
the previously unknown extension), and we correct the formula for
$\displaystyle g$.

Here we define $\displaystyle g$ as follows:

  1. (1)

If $\displaystyle M_{1}=T_{1},M_{2}=D_{2},M_{3}=D_{3}$ then

$\displaystyle g=\gcd(1-4n_{1}^{2},1-2m_{2},1-2m_{3})\;.$

  2. (2)

If $\displaystyle M_{1}=T_{1},M_{2}=T_{2},M_{3}=D_{3}$ then

$\displaystyle g=\gcd(1-4n_{1}^{2},1-4n_{2}^{2},1-4n_{1}n_{2},1-2m_{3})\;.$

  3. (3)

If $\displaystyle M_{1}=T_{1},M_{2}=T_{2},M_{3}=T_{3}$ then

$\displaystyle
g=\gcd(1-4n_{1}^{2},1-4n_{2}^{2},1-4n_{3}^{2},1-4n_{1}n_{2},1-4n_{1}n_{3},1-4n_{2}n_{3})\;.$

As mentioned, this formula for $\displaystyle g$ agrees with [12] (Proposition
6.2) in case (a), but is a correction in cases (b) and (c).

######  Proposition 3.1.

For the rank-3 graph described above, $\displaystyle K^{\scriptscriptstyle{\it
CR}}(C_{\scriptscriptstyle{\mathbb{R}}}^{*}(\Lambda))$ is given by the table
below, for $\displaystyle g\geq 3$.

Note that if $\displaystyle g=1$, then $\displaystyle
K^{\scriptscriptstyle{\it
CR}}(C_{\scriptscriptstyle{\mathbb{R}}}^{*}(\Lambda))=0$ in all degrees.

$\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$
$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\hline\cr
n&\makebox[28.45274pt][c]{0}&\makebox[28.45274pt][c]{1}&\makebox[28.45274pt][c]{2}&\makebox[28.45274pt][c]{3}&\makebox[28.45274pt][c]{4}&\makebox[28.45274pt][c]{5}&\makebox[28.45274pt][c]{6}&\makebox[28.45274pt][c]{7}\\\
\hline\cr\hline\cr
KO_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
\hline\cr
KU_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}\\\
\hline\cr\hline\cr\end{array}$

###### Proof.

The graph $\displaystyle\Lambda$ has two vertices. So, following [2], we set
$\displaystyle\mathcal{A}=K^{\scriptscriptstyle{\it CR}}({\mathbb{R}})\oplus
K^{\scriptscriptstyle{\it CR}}({\mathbb{R}})$ and consider the chain complex

$\displaystyle
0\rightarrow\mathcal{A}\xrightarrow{\partial_{3}}\mathcal{A}^{3}\xrightarrow{\partial_{2}}\mathcal{A}^{3}\xrightarrow{\partial_{1}}\mathcal{A}\rightarrow
0$

the homology of which gives the $\displaystyle E^{2}$ page of a spectral
sequence which converges to $\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}(\Lambda))$.

The complex part of this chain complex in degree 0 is exactly the chain
complex analyzed in the proof of Proposition 6.3 of [12], specifically we have

$\displaystyle
0\rightarrow{\mathbb{Z}}^{2}\xrightarrow{\partial_{3}}{\mathbb{Z}}^{6}\xrightarrow{\partial_{2}}{\mathbb{Z}}^{6}\xrightarrow{\partial_{1}}{\mathbb{Z}}^{2}\rightarrow
0$

where

$\displaystyle\displaystyle\partial_{1}$
$\displaystyle\displaystyle=\begin{bmatrix}I-M_{1}^{T}&I-M_{2}^{T}&I-M_{3}^{T}\end{bmatrix}$
$\displaystyle\displaystyle\partial_{2}$
$\displaystyle\displaystyle=\begin{bmatrix}-(I-M_{2}^{T})&-(I-M_{3}^{T})&0\\\
I-M_{1}^{T}&0&-(I-M_{3}^{T})\\\ 0&I-M_{1}^{T}&I-M_{2}^{T}&\end{bmatrix}$
$\displaystyle\displaystyle\partial_{3}$
$\displaystyle\displaystyle=\begin{bmatrix}I-M_{3}^{T}\\\ -(I-M_{2}^{T})\\\
I-M_{1}^{T}\end{bmatrix}\;.$

We refer to Lemma 3.4 at the end of this section the calculation of the Smith
normal forms of these matrices, which come out to the following:

$\displaystyle\mathcal{S}(\partial_{1})=\mathcal{S}(\partial_{3})^{T}=\begin{bmatrix}1&0&0&0&0&0\\\
0&g&0&0&0&0\end{bmatrix}\\\
\quad\text{and}\quad\mathcal{S}(\partial_{2})=\mathrm{diag}(1,1,g,g,0,0)\;.$

Thus the homology of this chain complex in $\displaystyle
H_{*}(\mathcal{C})=({\mathbb{Z}}_{g},{\mathbb{Z}}_{g}^{2},{\mathbb{Z}}_{g},0)$
is degrees $\displaystyle p=0,1,2,3$.

The real part of this chain complex has period 8. In degrees 0 and 4, it is
identical to the complex part

$\displaystyle
0\rightarrow{\mathbb{Z}}^{2}\xrightarrow{\partial_{3}}{\mathbb{Z}}^{6}\xrightarrow{\partial_{2}}{\mathbb{Z}}^{6}\xrightarrow{\partial_{1}}{\mathbb{Z}}^{2}\rightarrow
0$

and with the same partial maps, so the homology is the same. The real part of
this chain complex in degrees 1 and 2 consists of $\displaystyle 2$-torsion
subgroups

$\displaystyle
0\rightarrow{\mathbb{Z}}_{2}^{2}\xrightarrow{\partial_{3}}{\mathbb{Z}}_{2}^{6}\xrightarrow{\partial_{2}}{\mathbb{Z}}_{2}^{6}\xrightarrow{\partial_{1}}{\mathbb{Z}}_{2}^{2}\rightarrow
0$

but the matrices describing the partials are the same as above, modulo 2.
Since $\displaystyle g$ is odd, the chain complex is exact and the homology
vanishes.

Therefore, the $\displaystyle E^{2}$ page of the spectral sequence of
$\displaystyle K^{\scriptscriptstyle{\it CR}}(C^{*}(\Lambda))$ looks like the
following, in the real and complex parts.

$\displaystyle E^{2}_{p,q}$ (for $\displaystyle g$ odd)
$\displaystyle\displaystyle\begin{array}[]{ ccccc
}\lx@intercol\hfil\underline{\text{real part}}\hfil\lx@intercol\\\
\vspace{.25cm}\hfil\\\ \vdots\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&\hskip
8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip
2.84544pt\vdots\\\ 7\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\
6\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 5\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 4\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
3\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 2\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 1\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 0\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
\hline\cr~{}\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&1&2&3\end{array}\hskip
85.35826pt\begin{array}[]{ ccccc }\lx@intercol\hfil\underline{\text{complex
part}}\hfil\lx@intercol\\\ \vspace{.25cm}\hfil\\\ \vdots\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&\hskip 8.5359pt\vdots&\hskip
8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip 2.84544pt\vdots\\\ 7\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 6\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
5\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 4\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
3\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 2\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
1\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 0\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
\hline\cr~{}\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&1&2&3\end{array}$

Then the structure of this spectral sequence implies that there are no non-
trivial differentials. Therefore $\displaystyle E_{p,q}^{2}=E_{p,q}^{\infty}$
in both the real and complex part. Furthermore, in the real case, there is
never more than one non-trivial group along a single diagonal $\displaystyle
p+q=i$ (for $\displaystyle i$ fixed), so there are no non-trivial extension
problems for $\displaystyle
KO_{i}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$. Thus the real
$\displaystyle K$-theory is as shown in the table. For the complex part, we
get (repeating what was obtained in [12]) that $\displaystyle
KU_{0}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ is an extension of
$\displaystyle{\mathbb{Z}}_{g}$ by $\displaystyle{\mathbb{Z}}_{g}$ and
$\displaystyle
KU_{1}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\cong{\mathbb{Z}}_{g}^{2}$.

It remains to show that $\displaystyle
KU_{0}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\cong{\mathbb{Z}}_{g}^{2}$.
We make use of the natural transformation $\displaystyle c\colon
KO_{i}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\rightarrow
KU_{i}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ which can be traced
back from the spectral sequence as follows.

The map $\displaystyle c$ on $\displaystyle\mathcal{A}$ commutes with the
chain maps $\displaystyle\partial_{i}$ and induces the map $\displaystyle
c\colon(E^{2}_{p,q})^{\scriptscriptstyle
O}\rightarrow(E^{2}_{p,q})^{\scriptscriptstyle U}$. The maps $\displaystyle c$
on each page of the spectral sequence induces maps on the following pages, and
ultimately on the $\displaystyle E^{\infty}$ page. Finally the map
$\displaystyle c\colon
KO_{i}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\rightarrow
KU_{i}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ commutes with the
filtrations and on each subfactor is equal to the map $\displaystyle c$
obtained on the $\displaystyle E^{\infty}$ page.

The real and complex parts of $\displaystyle\mathcal{A}$ are isomorphic in
degree 0 and the complexification map $\displaystyle c_{0}$ on
$\displaystyle\mathcal{A}$ actually implements this isomorphism, as seen in
Figure 6. Furthermore, the maps $\displaystyle\partial_{i}$ are the same and
commute with $\displaystyle c_{0}$. Thus $\displaystyle c$ is an isomorphism
from the first row of the spectral sequence for $\displaystyle
KO_{0}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ to the first row of
the spectral sequence for $\displaystyle
KU_{0}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$.

We now focus on the filtration of the $\displaystyle E^{\infty}$ page giving
$\displaystyle KO_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ and
$\displaystyle KU_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$. Since
$\displaystyle c$ commutes with the filtration we obtain the following
diagram.

$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{(E^{\infty}_{0,2})^{\scriptscriptstyle
O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{KO_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{(E^{\infty}_{2,0})^{\scriptscriptstyle
O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{0}$$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{(E^{\infty}_{0,2})^{\scriptscriptstyle
U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{KU_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{(E^{\infty}_{0,2})^{\scriptscriptstyle
U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{0\;}$

which can be rewritten as

$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{KO_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{0}$$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{KU_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{0\;}$

Now since the vertical map $\displaystyle c$ on the right is an isomorphism,
and the horizontal map from $\displaystyle
KO_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ is an isomorphism,
the exact sequence on the bottom has a splitting. This proves that
$\displaystyle
KU_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\cong{\mathbb{Z}}_{g}^{2}$,
and by periodicity $\displaystyle
KU_{i}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\cong{\mathbb{Z}}_{g}^{2}$
for all even $\displaystyle i$.

∎

######  Remark 3.2.

Comparing this result with the calculations of $\displaystyle
K^{\scriptscriptstyle{\it CR}}(\mathcal{O}_{n}^{\scriptstyle{\mathbb{R}}})$
from [1], we see that the $\displaystyle{\mathcal{C}R}$-module $\displaystyle
K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ decomposes as a direct
sum with four summands, each of which is isomorphic to $\displaystyle
K^{\scriptscriptstyle{\it CR}}(\mathcal{O}_{g+1}^{\scriptstyle{\mathbb{R}}})$
or a certain suspension thereof. Specifically,

$\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\cong
K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{g+1}^{\scriptstyle{\mathbb{R}}})\oplus(\Sigma^{-1}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{g+1}^{\scriptstyle{\mathbb{R}}}))^{2}\oplus\Sigma^{-2}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{g+1}^{\scriptstyle{\mathbb{R}}})\;.$

######  Remark 3.3.

In the special case that $\displaystyle M_{1}=T_{1}$, $\displaystyle
M_{2}=D_{2}$, $\displaystyle M_{3}=D_{3}$, the graph $\displaystyle\Lambda$
decomposes as a product graph (in the sense of Kumjian-Pask) of rank-1 graphs.
Specifically, we have

$\displaystyle\Lambda=\Lambda_{1}\times\Lambda_{2}\times\Lambda_{3}$

where $\displaystyle\Lambda_{1}$ is a graph with two vertices and
$\displaystyle 2n_{1}$ edges from each vertex to the other; and
$\displaystyle\Lambda_{2}$, $\displaystyle\Lambda_{3}$ are graphs with 1
vertex and $\displaystyle 2m_{i}$ loops. Therefore,

$\displaystyle C^{*}(\Lambda)=C^{*}(\Lambda_{1})\otimes
C^{*}(\Lambda_{2})\otimes C^{*}(\Lambda_{3})\;.$

It can further be shown that all factors in this product are isomorphic to
Cuntz algebras. Namely $\displaystyle
C^{*}(\Lambda_{1})\cong\mathcal{O}_{4n_{1}^{2}-1}$ and $\displaystyle
C^{*}(\Lambda_{i})\cong\mathcal{O}_{2m_{i}-1}$ (for $\displaystyle i=2,3$).
Similarly, at the level of real $\displaystyle C^{*}$-algebras we have

$\displaystyle
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda)=C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda_{1})\otimes
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda_{2})\otimes
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda_{3})\;$

where $\displaystyle
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda_{1})\cong\mathcal{O}^{\scriptstyle{\mathbb{R}}}_{4n_{1}^{2}-1}$
and $\displaystyle
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda_{i})\cong\mathcal{O}^{\scriptstyle{\mathbb{R}}}_{2m_{i}-1}$
(for $\displaystyle i=2,3$).

Therefore, this spectral sequence calculation above has given us an approach
to calculating the $\displaystyle K$-theory of these products which is
alternative to using the Künneth formula. The Künneth formula can be difficult
when there are more than 2 factors and when there is torsion involved. This is
especially true for the real case.

Also, in the more general case (without restriction on the forms of
$\displaystyle M_{i}$), we find a posteriori (from Proposition 3.1) that the
$\displaystyle K$-theory depends only on the value of $\displaystyle g$. Using
the classification theorems for purely infinite simple $\displaystyle
C^{*}$-algebras (see the manuscripts of Kirchberg [8] and Phillips [13] in the
complex case in and the work of the first author and others in the real case
[4]) it follows that the isomorphism classes of $\displaystyle\Lambda$ depend
only on the value of $\displaystyle g$. Therefore, in all cases the
$\displaystyle C^{*}$-algebra $\displaystyle C^{*}(\Lambda)$ is isomorphic to
an appropriate product of three Cuntz algebras (one with the same value of
$\displaystyle g$) and similarly the real $\displaystyle C^{*}$-algebra
$\displaystyle C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda)$ is isomorphic
to a product of three real Cuntz algebras.

######  Lemma 3.4.

The Smith normal form of the matrices $\displaystyle\partial_{1}$,
$\displaystyle\partial_{2}$, $\displaystyle\partial_{3}$ (in the complex part
in degree 0) are equal to

$\displaystyle\mathcal{S}(\partial_{1})=\mathcal{S}(\partial_{3})^{T}=\begin{bmatrix}1&0&0&0&0&0\\\
0&g&0&0&0&0\end{bmatrix}\\\
\quad\text{and}\quad\mathcal{S}(\partial_{2})=\mathrm{diag}(1,1,g,g,0,0)\;.$

###### Proof.

We note that the proof in Case (1) is correct in [12].

In Case (2), proceed as in the proof of Lemma 6.1 of [12] where it is
necessary to compute the smith normal form of

$\displaystyle\partial_{1}=\begin{bmatrix}1&-2n_{1}&1&-2n_{2}&1-2m_{3}&0\\\
-2n_{1}&1&-2n_{2}&1&0&1-2m_{3}\end{bmatrix}\;.$

The list of the $\displaystyle 2\times 2$ minors (up to sign) of
$\displaystyle\partial_{1}$ is

(1)
$\begin{gathered}1-4n_{1}^{2},~{}1-4n_{2}^{2},~{}1-4n_{1}n_{2},~{}2(n_{1}-n_{2})\\\
1-2m_{3},~{}(1-2m_{3})^{2},~{}2n_{1}(1-2m_{3}),~{}2n_{2}(1-2m_{3})\end{gathered}$

As we are interested in the gcd of this list, we can clearly reduce everything
on the second line to just $\displaystyle 1-2m_{3}$. Furthermore, by Lemma 6.1
we can eliminate the last entry of the first row. Hence the
$\displaystyle\gcd$ of the $\displaystyle 2\times 2$ minors is $\displaystyle
g$ and $\displaystyle\mathcal{S}(\partial_{1})=\mathrm{diag}(1,g)$. The result
for $\displaystyle\partial_{3}$ is the same (up to transpose).

Now we consider $\displaystyle\partial_{2}$, where

$\displaystyle\partial_{2}=\begin{bmatrix}-1&2n_{2}&-(1-2m_{3})&0&0&0\\\
2n_{2}&-1&0&-(1-2m_{3})&0&0\\\ 1&-2n_{1}&0&0&-(1-2m_{3})&0\\\
-2n_{1}&1&0&0&0&-(1-2m_{3})\\\ 0&0&1&-2n_{1}&1&-2n_{2}\\\
0&0&-2n_{1}&1&-2n_{2}&1\end{bmatrix}\;$

Here the $\displaystyle\gcd$ of the list of $\displaystyle 2\times 2$ minors
is seen to be 1. Furthermore, each $\displaystyle 4\times 4$ minors is a
product of at least 2 factors from the list of $\displaystyle 2\times 2$
minors. It follows that the $\displaystyle\gcd$ of the list of $\displaystyle
4\times 4$ minors is $\displaystyle g^{2}$ and that
$\displaystyle\mathcal{S}(\partial_{2})=\mathrm{diag}(1,1,g,g,0,0)$.

For Case (3), we consider

$\displaystyle\partial_{1}=\begin{bmatrix}1&-2n_{1}&1&-2n_{2}&1&-2n_{3}\\\
-2n_{1}&1&-2n_{2}&1&-2n_{3}&1\end{bmatrix}\;.$

The list of the $\displaystyle 2\times 2$ minors is

(2) $\begin{gathered}1-4n_{1}^{2},~{}1-4n_{2}^{2},~{}1-4n_{3}^{3},\\\
~{}1-4n_{1}n_{2},~{}1-4n_{1}n_{3},~{}1-4n_{2}n_{3}\\\
~{}2(n_{1}-n_{2}),~{}2(n_{1}-n_{3}),~{}2(n_{2}-n_{3})\;.\end{gathered}$

Using Lemma 6.3, we can eliminate the third row of this set of formulae,
giving the desired result. This proves the result for
$\displaystyle\partial_{1}$ and $\displaystyle\partial_{3}$. The calculation
for $\displaystyle\partial_{2}$ is now similar to that in Case (2).

∎

##  4\. The Rank-3 case, with a non-trivial involution

Now we consider the same rank-3 graph $\displaystyle\Lambda$ with a non-
trivial involution $\displaystyle\gamma$. The involution $\displaystyle\gamma$
swaps the two vertices and this extends consistently to an involution on all
higher-degree edges. The next theorem shows the $\displaystyle K$-theory of
the real $\displaystyle C^{*}$-algebra $\displaystyle
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma)$.

Again, we have three cases to consider, depending on the structure of the
adjacency matrices.

  1. (1)

If $\displaystyle M_{1}=T_{1},M_{2}=D_{2},M_{3}=D_{3}$ then

$\displaystyle\displaystyle g$
$\displaystyle\displaystyle=\gcd(1-4n_{1}^{2},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle h$
$\displaystyle\displaystyle=\gcd(1-2n_{1},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle k$
$\displaystyle\displaystyle=\gcd(1+2n_{1},1-2m_{2},1-2m_{3})\;.$

  2. (2)

If $\displaystyle M_{1}=T_{1},M_{2}=T_{2},M_{3}=D_{3}$ then

$\displaystyle\displaystyle g$
$\displaystyle\displaystyle=\gcd(1-4n_{1}^{2},1-4n_{2}^{2},1-4n_{1}n_{2},1-2m_{3})$
$\displaystyle\displaystyle h$
$\displaystyle\displaystyle=\gcd(1-2n_{1},1-2n_{2},1-2m_{3})$
$\displaystyle\displaystyle k$
$\displaystyle\displaystyle=\gcd(1+2n_{1},1+2n_{2},1-2m_{3})\;.$

  3. (3)

If $\displaystyle M_{1}=T_{1},M_{2}=T_{2},M_{3}=T_{3}$ then

$\displaystyle\displaystyle g$
$\displaystyle\displaystyle=\gcd(1-4n_{1}^{2},1-4n_{2}^{2},1-4n_{3}^{2},1-4n_{1}n_{2},1-4n_{1}n_{3},1-4n_{2}n_{3})$
$\displaystyle\displaystyle h$
$\displaystyle\displaystyle=\gcd(1-2n_{1},1-2n_{2},1-2n_{3})$
$\displaystyle\displaystyle k$
$\displaystyle\displaystyle=\gcd(1+2n_{1},1+2n_{2},1+2n_{3})\;.$

Note that $\displaystyle g=hk$ in each case, because $\displaystyle 1-2n_{i}$
and $\displaystyle 1+2n_{i}$ are relatively prime, and by Lemmas 6.2 and 6.4.

######  Proposition 4.1.

Let $\displaystyle\Lambda$ be the rank-3 graph described above, with non-
trivial involution $\displaystyle\gamma$. Then $\displaystyle
K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma)$ is given by the
table below.

$\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$
$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\hline\cr
n&\makebox[28.45274pt][c]{0}&\makebox[28.45274pt][c]{1}&\makebox[28.45274pt][c]{2}&\makebox[28.45274pt][c]{3}&\makebox[28.45274pt][c]{4}&\makebox[28.45274pt][c]{5}&\makebox[28.45274pt][c]{6}&\makebox[28.45274pt][c]{7}\\\
\hline\cr\hline\cr
KO_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))&{\mathbb{Z}}_{h}\oplus{\mathbb{Z}}_{k}&{\mathbb{Z}}_{h}^{2}&{\mathbb{Z}}_{h}\oplus{\mathbb{Z}}_{k}&{\mathbb{Z}}_{k}^{2}&{\mathbb{Z}}_{h}\oplus{\mathbb{Z}}_{k}&{\mathbb{Z}}_{h}^{2}&{\mathbb{Z}}_{h}\oplus{\mathbb{Z}}_{k}&{\mathbb{Z}}_{k}^{2}\\\
\hline\cr
KU_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}^{2}\\\
\hline\cr\hline\cr\end{array}$

######  Remark 4.2.

Note that
$\displaystyle{\mathbb{Z}}_{g}\cong{\mathbb{Z}}_{h}\oplus{\mathbb{Z}}_{k}$.
So, similar to the previous examples, there is a direct sum decomposition.
Here it can be written as

$\displaystyle\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))\cong$
$\displaystyle\displaystyle\left(K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{h+1}^{\scriptstyle{\mathbb{R}}})\oplus\left(\Sigma^{-1}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{h+1}^{\scriptstyle{\mathbb{R}}})\right)^{2}\oplus\Sigma^{-2}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{h+1}^{\scriptstyle{\mathbb{R}}})\right)$
$\displaystyle\displaystyle\oplus\left(\Sigma^{-4}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{k+1}^{\scriptstyle{\mathbb{R}}})\oplus\left(\Sigma^{-5}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{k+1}^{\scriptstyle{\mathbb{R}}})\right)^{2}\oplus\Sigma^{-6}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{k+1}^{\scriptstyle{\mathbb{R}}})\right)\;.$

######  Remark 4.3.

Let $\displaystyle
A(n_{1},m_{2},m_{3})=C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma)$
be the real $\displaystyle C^{*}$-algebra obtained from a particular choice of
integers $\displaystyle n_{1},m_{2},m_{3}$ in Case (1). Let
$\displaystyle\widetilde{g}=\gcd(m_{2},m_{3})$. If $\displaystyle n_{1}$ and
$\displaystyle n^{\prime}_{1}$ are two positive integers satifying
$\displaystyle n_{1}\equiv n_{1}^{\prime}\pmod{\widetilde{g}}$ then we have

$\displaystyle\displaystyle\gcd(1-4n_{1}^{2},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle=\gcd(1-4(n_{1}^{\prime})^{2},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle\gcd(1-2n_{1},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle=\gcd(1-2n_{1}^{\prime},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle\gcd(1+n_{1},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle=\gcd(1+2n_{1}^{\prime},1-2m_{2},1-2m_{3})\;.$

Then it follows by Proposition 4.1 that $\displaystyle
K^{\scriptscriptstyle{\it CR}}(A(n_{1},m_{2},m_{3}))\cong
K^{\scriptscriptstyle{\it CR}}(A(n_{1}^{\prime},m_{2},m_{3}))$ and therefore
using [4] that $\displaystyle A(n_{1},m_{2},m_{3}))\cong
A(n_{1}^{\prime},m_{2},m_{3}))$.

Suppose on the other hand, we replace $\displaystyle n_{1}$ by $\displaystyle
n^{\prime}_{1}$ where $\displaystyle n_{1}\equiv-
n_{1}^{\prime}\pmod{\widetilde{g}}$. Then we have

$\displaystyle\displaystyle\gcd(1-4n_{1}^{2},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle=\gcd(1-4(n_{1}^{\prime})^{2},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle\gcd(1-2n_{1},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle=\gcd(1+2n_{1}^{\prime},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle\gcd(1+n_{1},1-2m_{2},1-2m_{3})$
$\displaystyle\displaystyle=\gcd(1-2n_{1}^{\prime},1-2m_{2},1-2m_{3})\;.$

Thus the roles of the $\displaystyle h$ and $\displaystyle k$ change places
and it follows by Proposition 4.1 that $\displaystyle
K^{\scriptscriptstyle{\it
CR}}(A(n_{1},m_{2},m_{3}))\cong\Sigma^{2}K^{\scriptscriptstyle{\it
CR}}(A(n_{1}^{\prime},m_{2},m_{3}))$. The real $\displaystyle C^{*}$-algebras
$\displaystyle A(n_{1},m_{2},m_{3})$ and $\displaystyle
A(n_{1}^{\prime},m_{2},m_{3})$ are not isomorphic in this case, though their
respective complexifications are, since $\displaystyle
KU_{*}(A(n_{1},m_{2},m_{3}))\cong KU_{*}(A(n_{1}^{\prime},m_{2},m_{3}))$.

###### Proof of Proposition 4.1.

The complex part $\displaystyle
KU_{*}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$ is the same as
what we obtained for $\displaystyle
KU(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$, since both are
isomorphic to the $\displaystyle K$-theory of the complex graph algebra
associated to $\displaystyle\Lambda$, that is to $\displaystyle
K_{*}(C^{*}(\Lambda))$. But to find $\displaystyle
KO_{*}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$ we go back to
the spectral sequence again.

By [2] there is again a chain complex

$\displaystyle
0\rightarrow\mathcal{A}\xrightarrow{\partial_{3}}\mathcal{A}^{3}\xrightarrow{\partial_{2}}\mathcal{A}^{3}\xrightarrow{\partial_{1}}\mathcal{A}\rightarrow
0$

the homology of which gives the $\displaystyle E^{2}$ page of a spectral
sequence which converges to $\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}(\Lambda))$, but this time we have
$\displaystyle\mathcal{A}=K^{\scriptscriptstyle{\it CR}}({\mathbb{C}})$.

The real part of this chain complex in even degrees is

$\displaystyle
0\rightarrow{\mathbb{Z}}\xrightarrow{\partial_{3}}{\mathbb{Z}}^{3}\xrightarrow{\partial_{2}}{\mathbb{Z}}^{3}\xrightarrow{\partial_{1}}{\mathbb{Z}}\rightarrow
0$

and the real part vanishes in the odd degrees. In degree 0 we have where

$\displaystyle\displaystyle\partial_{1}$
$\displaystyle\displaystyle=\begin{bmatrix}I-MO_{1}^{T}&I-MO_{2}^{T}&I-MO_{3}^{T}\end{bmatrix}$
$\displaystyle\displaystyle\partial_{2}$
$\displaystyle\displaystyle=\begin{bmatrix}-(I-MO_{2}^{T})&-(I-MO_{3}^{T})&0\\\
I-MO_{1}^{T}&0&-(I-MO_{3}^{T})\\\ 0&I-MO_{1}^{T}&I-MO_{2}^{T}&\end{bmatrix}$
$\displaystyle\displaystyle\partial_{3}$
$\displaystyle\displaystyle=\begin{bmatrix}I-MO_{1}^{T}\\\ -(I-MO_{2}^{T})\\\
I-MO_{3}^{T}\end{bmatrix}$

and the $\displaystyle 1\times 1$ matrices for $\displaystyle I-MO_{i}^{T}$
are found from $\displaystyle I-M_{i}^{T}$ using the instructions from Table 3
in Section 3D of [2].

Now, we consider Case (1) specifically, so that we have

$\displaystyle I-M_{i}=\begin{cases}\begin{bmatrix}1&-2n_{i}\\\
-2n_{i}&1\end{bmatrix}&i=1\\\ ~{}\\\ \begin{bmatrix}1-2m_{i}&0\\\
0&1-2m_{i}\end{bmatrix}&i=2,3.\end{cases}$

We then find that

$\displaystyle I-MO_{i}=\begin{cases}1-2n_{i}&i=1\\\
1-2m_{i}&i=2,3\end{cases}$

(the rule here is that we add the entries in the first row of each
$\displaystyle 2\times 2$ matrix to get a new $\displaystyle 1\times 1$
matrix). So we have

$\displaystyle\partial_{1}=\partial_{3}^{T}=\begin{bmatrix}1-2n_{1}&1-2m_{2}&1-2m_{3}\end{bmatrix}\quad\text{and}\quad\partial_{2}=\begin{bmatrix}-(1-2m_{2})&-(1-2m_{3})&0\\\
1-2n_{1}&0&-(1-2m_{3})\\\ 0&1-2n_{1}&1-2m_{2}\end{bmatrix}$

We claim that

$\displaystyle
S(\partial_{1})=S(\partial_{3}^{T})=\begin{bmatrix}h&0&0\end{bmatrix}\quad\text{and}\quad
S(\partial_{2})=\begin{bmatrix}h&0&0\\\ 0&h&0\\\ 0&0&0\end{bmatrix}$

where $\displaystyle h=\gcd(1-2n_{1},1-2m_{2},1-2m_{3})$. The statements about
$\displaystyle S(\partial_{1})$ and $\displaystyle S(\partial_{3})$ are clear,
but for $\displaystyle S(\partial_{2})$ first note that
$\displaystyle\partial_{2}$ has rank 2. The $\displaystyle\gcd$ of all the
entries of $\displaystyle\partial_{2}$ is $\displaystyle h$; while the
$\displaystyle\gcd$ of all the $\displaystyle 2\times 2$ minors

$\displaystyle(1-2n_{1})(1-2m_{2}),(1-2n_{1})(1-2m_{3}),(1-2m_{2})(1-m_{3}),(1-2n_{1})^{2},(1-2m_{2})^{2},(1-2m_{3})^{2}\;$

which is $\displaystyle h^{2}$. From this the statement about
$\displaystyle\mathcal{S}(\partial_{2})$ follows.

The result is that the homology of the chain complex in degree 0 is
$\displaystyle
H_{*}(\mathcal{C})=({\mathbb{Z}}_{h},{\mathbb{Z}}_{h}^{2},{\mathbb{Z}}_{h},0)$
in degrees $\displaystyle i=0,1,2,3$ and this gives us the 0th row of the
$\displaystyle E^{2}$ page of the real part of the spectral sequence.

For row 2 of the spectral sequence, Table 3 of [2] dictates that we subtract
instead of add the adjacent entries of $\displaystyle I-M_{i}$ so we have

$\displaystyle\partial_{1}^{T}=\begin{bmatrix}1+2n_{1}&1-2m_{2}&1-2m_{3}\end{bmatrix}\quad\text{and}\quad\partial_{2}=\begin{bmatrix}-(1-2m_{2})&1-2m_{3}&0\\\
1+2n_{1}&0&-(1-2m_{3})\\\ 0&-(1+2n_{1})&1-2m_{2}\end{bmatrix}$

Thus

$\displaystyle
S(\partial_{1})=S(\partial_{3}^{T})=\begin{bmatrix}k&0&0\end{bmatrix}\quad\text{and}\quad
S(\partial_{2})=\begin{bmatrix}k&0&0\\\ 0&k&0\\\ 0&0&0\end{bmatrix}$

where $\displaystyle k=\gcd(1+2n_{1},1-2m_{2},1-2m_{3})$. The homology of the
chain complex is $\displaystyle
H_{*}(\mathcal{C})=({\mathbb{Z}}_{k},{\mathbb{Z}}_{k}^{2},{\mathbb{Z}}_{k},0)$
in degrees $\displaystyle i=0,1,2,3$ and this gives us row 2 of the
$\displaystyle E^{2}$ page of the real part of the spectral sequence.

Rows 4 and 6 are the same as rows 0 and 2, respectively. So the $\displaystyle
E^{2}$ page of the spectral sequence is the following. For both the real and
complex parts, we have $\displaystyle E^{2}=E^{\infty}$, because no non-zero
differentials are possible.

$\displaystyle E^{2}_{p,q}$ (for $\displaystyle g$ odd)
$\displaystyle\displaystyle\begin{array}[]{ ccccc
}\lx@intercol\hfil\underline{\text{real part}}\hfil\lx@intercol\\\
\vspace{.25cm}\hfil\\\ \vdots\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&\hskip
8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip
2.84544pt\vdots\\\ 7\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\
6\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{k}&{\mathbb{Z}}_{k}^{2}&{\mathbb{Z}}_{k}&0\\\
5\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 4\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{h}&{\mathbb{Z}}_{h}^{2}&{\mathbb{Z}}_{h}&0\\\
3\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 2\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{k}&{\mathbb{Z}}_{k}^{2}&{\mathbb{Z}}_{k}&0\\\
1\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 0\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{h}&{\mathbb{Z}}_{h}^{2}&{\mathbb{Z}}_{h}&0\\\
\hline\cr~{}\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&1&2&3\end{array}\hskip
85.35826pt\begin{array}[]{ ccccc }\lx@intercol\hfil\underline{\text{complex
part}}\hfil\lx@intercol\\\ \vspace{.25cm}\hfil\\\ \vdots\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&\hskip 8.5359pt\vdots&\hskip
8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip 2.84544pt\vdots\\\ 7\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 6\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
5\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 4\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
3\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 2\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
1\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0\\\ 0\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{2}&{\mathbb{Z}}_{g}&0\\\
\hline\cr~{}\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&1&2&3\end{array}$

From the spectral sequence we immediately find the isomorphism class of
$\displaystyle KO_{j}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$
when $\displaystyle j$ is odd. Now, for $\displaystyle j$ even there is a
short exact sequence

$\displaystyle 0\rightarrow{\mathbb{Z}}_{h}\rightarrow
KO_{0}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))\rightarrow{\mathbb{Z}}_{k}\rightarrow
0\;,$

or

$\displaystyle 0\rightarrow{\mathbb{Z}}_{k}\rightarrow
KO_{0}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))\rightarrow{\mathbb{Z}}_{h}\rightarrow
0\;,$

depending on the parity of $\displaystyle j/2$. But since $\displaystyle h,k$
are relatively prime we must have $\displaystyle
KO_{0}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))\cong{\mathbb{Z}}_{h}\oplus{\mathbb{Z}}_{k}\cong{\mathbb{Z}}_{g}$
in both cases.

The proofs in Cases (2) and (3) proceed similarly. ∎

##  5\. Rank-4 graph with 2 vertices

Now let $\displaystyle\Lambda$ be the rank-4 graph with 2 vertices discussed
in Section 6 of [12]. In Proposition 6.4 of [12] some partial results are
described for $\displaystyle K_{*}(C^{*}(\Lambda))$. We again find that using
the real and complex $\displaystyle K$-theory together, we can complete these
computations. In this section we present the $\displaystyle K$-theory of both
real $\displaystyle C^{*}$-algebras $\displaystyle
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda)$ and $\displaystyle
C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma)$ where
$\displaystyle\gamma$ is the non-trivial involution. We also show the
additional complications for $\displaystyle k>4$ which present us from making
further progress.

The adjacency matrices $\displaystyle M_{i}$ for $\displaystyle\Lambda$ are
all of the form $\displaystyle T_{i}$ or $\displaystyle D_{i}$, as before. We
define $\displaystyle g$ as follows, according to the four possible cases. We
also define $\displaystyle h$ and $\displaystyle k$ for reference when
describing $\displaystyle
KO_{*}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$.

  1. (1)

If $\displaystyle M_{1}=T_{1},M_{2}=D_{2},M_{3}=D_{3},M_{4}=D_{4}$ then

$\displaystyle\displaystyle g$
$\displaystyle\displaystyle=\gcd\\{1-4n_{1}^{2},1-2m_{k}\mid
k\in\\{2,3,4\\}\\}$ $\displaystyle\displaystyle h$
$\displaystyle\displaystyle=\gcd\\{1-2n_{1},1-2m_{k}\mid k\in\\{2,3,4\\}\\}$
$\displaystyle\displaystyle k$
$\displaystyle\displaystyle=\gcd\\{1+2n_{1},1-2m_{k}\mid k\in\\{2,3,4\\}\\}$

  2. (2)

If $\displaystyle M_{1}=T_{1},M_{2}=T_{2},M_{3}=D_{3},M_{4}=D_{4}$ then

$\displaystyle\displaystyle g$
$\displaystyle\displaystyle=\gcd\\{1-4n_{i}^{2},1-4n_{i}n_{j},1-2m_{k}\mid
i,j\in\\{1,2\\},k\in\\{3,4\\}\\}$ $\displaystyle\displaystyle h$
$\displaystyle\displaystyle=\gcd\\{1-2n_{i},1-2m_{k}\mid
i\in\\{1,2\\},k\in\\{3,4\\}\\}$ $\displaystyle\displaystyle k$
$\displaystyle\displaystyle=\gcd\\{1+2n_{i},1-2m_{k}\mid
i\in\\{1,2\\}.,k\in\\{3,4\\}\\}\;.$

  3. (3)

If $\displaystyle M_{1}=T_{1},M_{2}=T_{2},M_{3}=T_{3},M_{4}=D_{4}$ then

$\displaystyle\displaystyle g$
$\displaystyle\displaystyle=\gcd\\{1-4n_{i}^{2},1-4n_{i}n_{j},1-2m_{4}\mid
i,j\in\\{1,2,3\\}\\}$ $\displaystyle\displaystyle h$
$\displaystyle\displaystyle=\gcd\\{1-2n_{i},1-2m_{4}\mid i\in\\{1,2,3\\}\\}$
$\displaystyle\displaystyle k$
$\displaystyle\displaystyle=\gcd\\{1+2n_{i},1-2m_{4}\mid
i\in\\{1,2,3\\}\\}\;.$

  4. (4)

If $\displaystyle M_{1}=T_{1},M_{2}=T_{2},M_{3}=T_{3},M_{4}=T_{4}$ then

$\displaystyle\displaystyle g$
$\displaystyle\displaystyle=\gcd\\{1-4n_{i}^{2},1-4n_{i}n_{j}\mid
i,j\in\\{1,2,3,4\\}\\}$ $\displaystyle\displaystyle h$
$\displaystyle\displaystyle=\gcd\\{1-2n_{i}\mid i\in\\{1,2,3,4\\}\\}$
$\displaystyle\displaystyle k$ $\displaystyle\displaystyle=\gcd\\{1+2n_{i}\mid
i\in\\{1,2,3,4\\}\\}\;.$

Then using the methods of the two previous sections, we obtain the following
propositions.

######  Proposition 5.1.

For the rank-4 graph described above, with non-trivial involution
$\displaystyle\gamma$ we have $\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}(\Lambda))$ and $\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma)$ given by the
table below, for $\displaystyle g\geq 3$. If $\displaystyle g=1$, then
$\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))=K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma)=0$.

$\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$
$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\hline\cr
n&\makebox[28.45274pt][c]{0}&\makebox[28.45274pt][c]{1}&\makebox[28.45274pt][c]{2}&\makebox[28.45274pt][c]{3}&\makebox[28.45274pt][c]{4}&\makebox[28.45274pt][c]{5}&\makebox[28.45274pt][c]{6}&\makebox[28.45274pt][c]{7}\\\
\hline\cr\hline\cr
KO_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}\\\
\hline\cr
KU_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}\\\
\hline\cr\hline\cr\end{array}$

$\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$
$\displaystyle\begin{array}[]{|c|c|c|c|c|c|c|c|c|}\hline\cr\hline\cr
n&\makebox[28.45274pt][c]{0}&\makebox[28.45274pt][c]{1}&\makebox[28.45274pt][c]{2}&\makebox[28.45274pt][c]{3}&\makebox[28.45274pt][c]{4}&\makebox[28.45274pt][c]{5}&\makebox[28.45274pt][c]{6}&\makebox[28.45274pt][c]{7}\\\
\hline\cr\hline\cr
KO_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))&{\mathbb{Z}}_{h}\oplus{\mathbb{Z}}_{k}^{3}&{\mathbb{Z}}_{h}^{3}\oplus{\mathbb{Z}}_{k}&{\mathbb{Z}}_{h}^{3}\oplus{\mathbb{Z}}_{k}&{\mathbb{Z}}_{h}\oplus{\mathbb{Z}}_{k}^{3}&{\mathbb{Z}}_{h}\oplus{\mathbb{Z}}_{k}^{3}&{\mathbb{Z}}_{h}^{3}\oplus{\mathbb{Z}}_{k}&{\mathbb{Z}}_{h}^{3}\oplus{\mathbb{Z}}_{k}&{\mathbb{Z}}_{h}\oplus{\mathbb{Z}}_{k}^{3}\\\
\hline\cr
KU_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}&{\mathbb{Z}}_{g}^{4}\\\
\hline\cr\hline\cr\end{array}$

###### Proof.

We first consider the spectral sequence for $\displaystyle
K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$. The incidence matrices
$\displaystyle M_{i}$ for $\displaystyle\Lambda$ are of the form
$\displaystyle D_{i}$ and $\displaystyle T_{i}$ as above, for $\displaystyle
i=1,2,3,4$. The chain complex we consider is

$\displaystyle
0\rightarrow\mathcal{A}\xrightarrow{\partial_{4}}\mathcal{A}^{4}\xrightarrow{\partial_{3}}\mathcal{A}^{6}\xrightarrow{\partial_{2}}\mathcal{A}^{4}\xrightarrow{\partial_{1}}\mathcal{A}\rightarrow
0$

where $\displaystyle\mathcal{A}=K^{\scriptscriptstyle{\it
CR}}({\mathbb{R}})\oplus K^{\scriptscriptstyle{\it CR}}({\mathbb{R}})$. The
analysis of Propositions 6.3 and 6.4 of [12] obtains the following chain
complex in the degree 0 complex part,

$\displaystyle
0\rightarrow{\mathbb{Z}}^{2}\xrightarrow{\partial_{4}}{\mathbb{Z}}^{8}\xrightarrow{\partial_{3}}{\mathbb{Z}}^{12}\xrightarrow{\partial_{2}}{\mathbb{Z}}^{8}\xrightarrow{\partial_{1}}{\mathbb{Z}}^{2}\rightarrow
0;.$

Now, following the method of calculation of [12] but using the corrections as
in Section 3 we find the following

$\displaystyle\displaystyle\mathcal{S}(\partial_{1})=\mathcal{S}(\partial_{3})^{T}$
$\displaystyle\displaystyle=\begin{bmatrix}1&0&\textbf{0}\\\
0&g&\textbf{0}\end{bmatrix}\text{~{}in~{}}M_{2,8}({\mathbb{R}})$
$\displaystyle\displaystyle\text{and}\quad\mathcal{S}(\partial_{2})=\mathcal{S}(\partial_{4})^{T}$
$\displaystyle\displaystyle=\begin{bmatrix}I_{3}&0_{3}&\textbf{0}\\\
0_{3}&gI_{3}&\textbf{0}\\\
\textbf{0}&\textbf{0}&\textbf{0}\end{bmatrix}\text{~{}in~{}}M_{8,12}({\mathbb{R}})$

Thus in the even degree complex part we have $\displaystyle
H_{*}(\mathcal{C})=({\mathbb{Z}}_{g},{\mathbb{Z}}_{g}^{3},{\mathbb{Z}}_{g}^{3},{\mathbb{Z}}_{g},0)$.
The real part of the chain complex in degrees 0 and 4 are the same. The real
part in degrees 1 and 2 are the same modulo 2, so $\displaystyle
H_{*}(\mathcal{C})=0$ in those degrees. The $\displaystyle E^{2}$ page of the
spectral sequence in the real and complex parts are then given by

$\displaystyle E^{2}_{p,q}$ (for $\displaystyle g$ odd)
$\displaystyle\displaystyle\begin{array}[]{ cccccc
}\lx@intercol\hfil\underline{\text{real part}}\hfil\lx@intercol\\\
\vspace{.25cm}\hfil\\\ \vdots\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&\hskip
8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip
8.5359pt\vdots&\hskip 2.84544pt\vdots\\\ 7\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 6\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 5\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 4\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&0\\\
3\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 2\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 1\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 0\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&0\\\
\hline\cr~{}\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&1&2&3&4\end{array}\hskip
85.35826pt\begin{array}[]{ cccccc }\lx@intercol\hfil\underline{\text{complex
part}}\hfil\lx@intercol\\\ \vspace{.25cm}\hfil\\\ \vdots\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&\hskip 8.5359pt\vdots&\hskip
8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip
2.84544pt\vdots\\\ 7\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\
6\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&0\\\
5\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 4\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&0\\\
3\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 2\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&0\\\
1\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 0\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&0\\\
\hline\cr~{}\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&1&2&3&4\end{array}$

We note that again, the complexification map $\displaystyle c$ is an
isomorphism on $\displaystyle E^{2}_{p,0}$, the bottom row of the spectral
sequence. In the real part of this spectral sequence all differentials must
vanish but in the complex part, while $\displaystyle d_{2}=0$ there appears to
be possible non-zero differentials $\displaystyle d_{3}$. We use the map c to
show that $\displaystyle d_{3}=0$. The complexification map $\displaystyle c$
is an isomorphism in degree 0 on $\displaystyle\mathcal{A}$ and passes to a
map $\displaystyle c$ which is an isomorphism in on the first row of the
$\displaystyle E^{2}=E^{3}$ pages of the spectral sequence. Furthermore,
$\displaystyle c$ commutes with $\displaystyle d_{3}$. In particular there is
a commutative diagram

$\displaystyle\textstyle{(E_{3,0}^{3})^{\scriptscriptstyle
O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{d_{3}}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{(E_{0,2}^{3})^{\scriptscriptstyle
O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$or$\displaystyle\textstyle{{\mathbb{Z}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{d_{3}}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{(E_{3,0}^{3})^{\scriptscriptstyle
U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{d_{3}}$$\displaystyle\textstyle{(E_{0,2}^{3})^{\scriptscriptstyle
U}}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{d_{3}}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}}$

Since $\displaystyle(E_{0,2}^{3})^{\scriptscriptstyle O}=0$ and $\displaystyle
c_{3,0}^{3}$ is an isomorphism, the commutative diagram forces $\displaystyle
d_{3}\colon(E_{3,0}^{3})^{\scriptscriptstyle
U}\rightarrow(E_{0,2}^{3})^{\scriptscriptstyle U}$ to vanish. By periodicity
$\displaystyle d_{3}$ vanishes on all
$\displaystyle(E_{3,i}^{3})^{\scriptscriptstyle U}$. Thus $\displaystyle
E^{2}=E^{3}=E^{\infty}$ on both the real and complex parts.

Now, in the real part there are no extension problems so the calculation of
$\displaystyle KO_{*}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ is
complete. But in the complex part, there are questions of extensions for both
$\displaystyle KU_{0}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ and
$\displaystyle KU_{1}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$. In
both cases, we use the complexification map $\displaystyle c$ as in the rank 3
case to show that the extension is split. First we use the $\displaystyle
p+q=2$ diagonal of the spectral sequence to find a diagram involving
$\displaystyle c\colon
KO_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\rightarrow
KU_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$.

$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{KO_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{0}$$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{KU_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{0}$

The vertical map $\displaystyle c$ on the right is an isomorphism, coming from
the first row of the spectral sequence. This shows that the extension on the
bottom of the diagram has a splitting and thus that $\displaystyle
KU_{2}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\cong{\mathbb{Z}}_{g}^{2}$.

Using the $\displaystyle p+q=3$ diagonal, we obtain the diagram

$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{KO_{3}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\scriptstyle{c}$$\displaystyle\textstyle{0}$$\displaystyle\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{KU_{3}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{{\mathbb{Z}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\displaystyle\textstyle{0}$

where again the vertical map $\displaystyle c$ is an isomorphism and again we
find that $\displaystyle
KU_{3}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))={\mathbb{Z}}_{g}^{2}$.
Therefore $\displaystyle
KU_{i}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))={\mathbb{Z}}_{g}^{2}$
for all $\displaystyle i$. This completes the calculation of $\displaystyle
K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$.

For $\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$, the
$\displaystyle E^{2}$ page of the spectral sequence in the real and complex
parts are as follows.

$\displaystyle E^{2}_{p,q}$ (for $\displaystyle g$ odd)
$\displaystyle\displaystyle\begin{array}[]{ cccccc
}\lx@intercol\hfil\underline{\text{real part}}\hfil\lx@intercol\\\
\vspace{.25cm}\hfil\\\ \vdots\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&\hskip
8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip
8.5359pt\vdots&\hskip 2.84544pt\vdots\\\ 7\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 6\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{k}&{\mathbb{Z}}_{k}^{3}&{\mathbb{Z}}_{k}^{3}&{\mathbb{Z}}_{k}&0\\\
5\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 4\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{h}&{\mathbb{Z}}_{h}^{3}&{\mathbb{Z}}_{h}^{3}&{\mathbb{Z}}_{h}&0\\\
3\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 2\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{k}&{\mathbb{Z}}_{k}^{3}&{\mathbb{Z}}_{k}^{3}&{\mathbb{Z}}_{k}&0\\\
1\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 0\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{h}&{\mathbb{Z}}_{h}^{3}&{\mathbb{Z}}_{h}^{3}&{\mathbb{Z}}_{h}&0\\\
\hline\cr~{}\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&0&1&2&3&4\end{array}\hskip
85.35826pt\begin{array}[]{ cccccc }\lx@intercol\hfil\underline{\text{complex
part}}\hfil\lx@intercol\\\ \vspace{.25cm}\hfil\\\ \vdots\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&\hskip 8.5359pt\vdots&\hskip
8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip 8.5359pt\vdots&\hskip
2.84544pt\vdots\\\ 7\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\
6\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&0\\\
5\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 4\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&0\\\
3\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 2\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&0\\\
1\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&0&0&0&0\\\ 0\hfil\kern
5.0pt\vline\kern-5.0pt\hfilneg&{\mathbb{Z}}_{g}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}^{3}&{\mathbb{Z}}_{g}&0\\\
\hline\cr~{}\hfil\kern 5.0pt\vline\kern-5.0pt\hfilneg&0&1&2&3&4\end{array}$

The complex part of this is the same as what we analyzed in the first part of
this proof. For the real part, since $\displaystyle h$ and $\displaystyle k$
are relatively prime, we have $\displaystyle d_{r}=0$ for all $\displaystyle
r$, so $\displaystyle(E^{2}_{p,q})^{\scriptscriptstyle
O}=(E^{\infty}_{p,q})^{\scriptscriptstyle O}$. Furthermore, along each
diagonal $\displaystyle p+q=n$, the extensions determining $\displaystyle
KO_{n}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))$ must be direct
sums, again because $\displaystyle h$ and $\displaystyle k$ are relatively
prime. ∎

######  Remark 5.2.

The $\displaystyle{\mathcal{C}R}$-modules $\displaystyle
K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ and $\displaystyle
K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma)$ can be seen to
have decompositions as a direct sum with eight summands,

$\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))\cong
K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{g+1}^{\scriptstyle{\mathbb{R}}})\oplus(\Sigma^{-1}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{g+1}^{\scriptstyle{\mathbb{R}}}))^{3}\oplus(\Sigma^{-2}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{g+1}^{\scriptstyle{\mathbb{R}}}))^{3}\oplus\Sigma^{-3}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{g+1}^{\scriptstyle{\mathbb{R}}})\;$

and

$\displaystyle\displaystyle K^{\scriptscriptstyle{\it
CR}}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda,\gamma))\cong$
$\displaystyle\displaystyle\left(K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{h+1}^{\scriptstyle{\mathbb{R}}})\oplus(\Sigma^{-1}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{h+1}^{\scriptstyle{\mathbb{R}}}))^{3}\oplus(\Sigma^{-2}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{h+1}^{\scriptstyle{\mathbb{R}}}))^{3}\oplus\Sigma^{-3}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{h+1}^{\scriptstyle{\mathbb{R}}})\right)$
$\displaystyle\displaystyle\oplus\left(K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{k+1}^{\scriptstyle{\mathbb{R}}})\oplus(\Sigma^{-1}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{k+1}^{\scriptstyle{\mathbb{R}}}))^{3}\oplus(\Sigma^{-2}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{k+1}^{\scriptstyle{\mathbb{R}}}))^{3}\oplus\Sigma^{-3}K^{\scriptscriptstyle{\it
CR}}(\mathcal{O}_{k+1}^{\scriptstyle{\mathbb{R}}})\right)\;.$

######  Remark 5.3.

We note that if $\displaystyle k\geq 5$, there will be an extra non-zero
column to these spectral sequences. We can still analyze the spectral sequence
and find that $\displaystyle(E^{2}_{p,q})^{\scriptscriptstyle
O}=(E^{\infty}_{p,q})^{\scriptscriptstyle O}$ and
$\displaystyle(E^{2}_{p,q})^{\scriptscriptstyle
U}=(E^{\infty}_{p,q})^{\scriptscriptstyle U}$, using similar arguments as in
the previous cases. However, there will be extension problems that we are
unable to determine. For example, $\displaystyle
KO_{0}(C^{*}_{\scriptscriptstyle{\mathbb{R}}}(\Lambda))$ will be extension of
$\displaystyle{\mathbb{Z}}_{g}$ by $\displaystyle{\mathbb{Z}}_{g}$ with no
clear way to determine the isomorphism class of group.

Furthermore, if $\displaystyle k\geq 6$, there is a more fundamental problem.
There will be two extra non-zero columns of the spectral sequences used in
these computations. As a result, there will be the possibility of a non-zero
differential, say $\displaystyle d_{5}\colon E^{5}_{5,0}\rightarrow
E^{5}_{0,4}$, in both the real and complex parts. We have no clear way of
determining the value of this differential. In general, we have no
understanding of how the differential maps $\displaystyle d_{r}$ relate to the
structure of the higher rank graph.

##  6\. Appendix: Number Theory Lemmas

######  Lemma 6.1.

Let $\displaystyle n_{1}$ and $\displaystyle n_{2}$ be positive integers,
greater than 1. Then $\displaystyle g_{1}=g_{2}=g_{3}$ where

$\displaystyle\displaystyle g_{1}$
$\displaystyle\displaystyle=\gcd(1-4n_{1}^{2},1-4n_{2}^{2},1-4n_{1}n_{2})$
$\displaystyle\displaystyle g_{2}$
$\displaystyle\displaystyle=\gcd(1-4n_{1}^{2},1-4n_{2}^{2},2n_{1}-2n_{2})$
$\displaystyle\displaystyle g_{3}$
$\displaystyle\displaystyle=\gcd(1-4n_{1}^{2},1-4n_{2}^{2},1-4n_{1}n_{2},2n_{1}-2n_{2})$

###### Proof.

Let Suppose that $\displaystyle p^{k}$ is a odd prime power such that
$\displaystyle p^{k}|\gcd(1-4n_{1}^{2},1-4n_{2}^{2})$. Since $\displaystyle
1-4n_{1}^{2}=(1-2n_{1})(1+2n_{1})$, and since $\displaystyle 1-2n_{1}$ and
$\displaystyle 1+2n_{1}$ are relatively prime, we have either $\displaystyle
p^{k}|(1-2n_{1})$ or $\displaystyle p^{k}|(1+2n_{1})$. If $\displaystyle
p^{k}|(1-2n_{1})$ then $\displaystyle p^{k}$ divides

$\displaystyle(1-2n_{1})(1+2n_{2})=(1-4n_{1}n_{2})-2(n_{1}-n_{2})\;.$

It follows that if $\displaystyle p^{k}$ divides one of $\displaystyle
1-4n_{1}n_{2}$ and $\displaystyle 2(n_{1}-n_{2})$, then it divides both.
Similarly, if $\displaystyle p^{k}|(1+2n_{1})$ we find that $\displaystyle
p^{k}$ divides

$\displaystyle(1+2n_{1})(1-2n_{2})=(1-4n_{1}n_{2})+2(n_{1}-n_{2})$

and the same conclusion is made. This proves the lemma. ∎

######  Lemma 6.2.

Let $\displaystyle n_{1}$ and $\displaystyle n_{2}$ be positive integers,
greater than 1. Then $\displaystyle\gcd(h,k)=1$ and $\displaystyle g=hk$ where

$\displaystyle\displaystyle g$
$\displaystyle\displaystyle=\gcd(1-4n_{1}^{2},1-4n_{2}^{2},1-4n_{1}n_{2})$
$\displaystyle\displaystyle h$
$\displaystyle\displaystyle=\gcd(1-2n_{1},1-2n_{2})$
$\displaystyle\displaystyle k$
$\displaystyle\displaystyle=\gcd(1+2n_{1},1+2n_{2})\;.$

###### Proof.

The first statement follows since $\displaystyle 1-2n_{1}$ and $\displaystyle
1+2n_{1}$ are relatively prime.

Let $\displaystyle p^{\ell}$ be a prime power and if $\displaystyle
p^{\ell}|hk$, then $\displaystyle p^{\ell}|h$ or $\displaystyle p^{\ell}|k$.
As $\displaystyle 1-4n_{i}^{2}=(1+2n_{i})(1-2n_{i})$, it follows that
$\displaystyle p^{\ell}|(1-4n_{i}^{2})$ for both $\displaystyle i$.
Furthermore, working modulo $\displaystyle p^{\ell}$, if $\displaystyle
p^{\ell}|h$ we have $\displaystyle 2n_{1}\equiv 2n_{2}\equiv 1$ so
$\displaystyle 4n_{1}n_{2}\equiv 1$. If $\displaystyle p^{\ell}|k$ we have
$\displaystyle 2n_{1}\equiv 2n_{2}\equiv-1$ so also $\displaystyle
4n_{1}n_{2}\equiv 1$. Either way, $\displaystyle p^{\ell}|(1-4n_{1}n_{2})$,
which implies $\displaystyle p^{\ell}|g$.

Conversely, suppose that $\displaystyle p^{\ell}|g$. So $\displaystyle
p^{\ell}|(1-2n_{i})$ or $\displaystyle p^{\ell}|(1+2n_{i})$, for each
$\displaystyle i$. If $\displaystyle p^{\ell}$ divides both $\displaystyle
1-2n_{1}$ and $\displaystyle 1-2n_{2}$, then by subtracting we find that
$\displaystyle p^{\ell}|2(n_{1}-n_{2})$. Similarly, if $\displaystyle
p^{\ell}$ divides both $\displaystyle 1+2n_{1}$ and $\displaystyle 1+2n_{2}$,
we also find $\displaystyle p^{\ell}|2(n_{1}-n_{2})$. In either of these cases
we have $\displaystyle p^{\ell}|g$, using Lemma 6.1.

If on the other hand $\displaystyle p^{\ell}$ divides both $\displaystyle
1-2n_{1}$ and $\displaystyle 1+2n_{2}$, then modulo $\displaystyle p^{\ell}$
we have

$\displaystyle 1\equiv
4n_{1}n_{2}\equiv(2n_{1})(2n_{2})\equiv(1)(-1)\equiv-1\;.$

This is a contradiction, as $\displaystyle p$ is odd. ∎

Using the same methods, we obtain the following extensions to these results.

######  Lemma 6.3.

Let $\displaystyle n_{1},\dots,n_{\ell}$ be positive integers, greater than 1.
Then $\displaystyle g_{1}=g_{2}=g_{3}$ where

$\displaystyle\displaystyle g_{1}$
$\displaystyle\displaystyle=\gcd\\{1-4n_{i}^{2},1-4n_{i}n_{j}\mid
i,j\in\\{1,\dots\ell\\}\\}$ $\displaystyle\displaystyle g_{2}$
$\displaystyle\displaystyle=\gcd\\{1-4n_{i}^{2},2n_{i}-2n_{j}\mid
i,j\in\\{1,\dots\ell\\}\\}$ $\displaystyle\displaystyle g_{3}$
$\displaystyle\displaystyle=\gcd\\{1-4n_{i}^{2},1-4n_{i}n_{j},2n_{i}-2n_{j}\mid
i,j\in\\{1,\dots\ell\\}\\}$

######  Lemma 6.4.

Let $\displaystyle n_{1},\dots,n_{\ell}$ be positive integers, greater than 1.
Then $\displaystyle\gcd(h,k)=1$ and $\displaystyle g=hk$ where

$\displaystyle\displaystyle g$
$\displaystyle\displaystyle=\gcd\\{1-4n_{i}^{2},1-4n_{i}n_{j}\mid
i,j\in\\{1,\dots\ell\\}\\}$ $\displaystyle\displaystyle h$
$\displaystyle\displaystyle=\gcd\\{1-2n_{i}\mid i\in\\{1,\dots\ell\\}\\}$
$\displaystyle\displaystyle k$ $\displaystyle\displaystyle=\gcd\\{1+2n_{i}\mid
i\in\\{1,\dots\ell\\}\\}$

## References

  * [1] J. Boersema, Real $\displaystyle C^{*}$-algebras, united $\displaystyle K$-theory, and the Künneth formula, K-Theory 26 (2002), no. 4, 345–402.
  * [2] J. Boersema and E. Gillaspy, K-theory for real $\displaystyle k$-graph $\displaystyle C^{*}$-algebras, Ann. K-Theory 7 (2022), no. 2, 395–440.
  * [3] J. Boersema, S. Browne, and E. Gillaspy, The stable exotic Cuntz algebras are higher-rank graph algebras, Proceedings of the American Mathematical Society, Series B 11 (2024), 47–62.
  * [4] J. Boersema, E. Ruiz, and P. Stacey, The classification of real purely infinite simple $\displaystyle C^{*}$-algebras. Doc. Math. 16 (2011), 619–655.
  * [5] D.G. Evans, On the K-theory of higher rank graph $\displaystyle C^{*}$-algebras, New York J. Math. 14 (2008), 1–31.
  * [6] C. Farsi, A. Kumjian, D. Pask and A. Sims, Ample groupoids: equivalence, homology, and Matui’s HK-conjecture, Münster J. Math. 12 (2019), 411–451.
  * [7] R. Hazlewood, I. Raeburn, A. Sims and S.B.G. Webster _Remarks on some fundamental results about higher-rank graphs and their $\displaystyle C^{*}$-algebras_, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 2, 575–597.
  * [8] E. Kirchberg, The classification of purely infinite $\displaystyle C^{*}$-algebras using Kasparov’s theory, Preprint (1994).
  * [9] A. Kumjian and D. Pask, Higher rank graph $\displaystyle C^{*}$-algebras, New York J. Math. 14 (2000), 1–20.
  * [10] N. Larsen and A. Vdovina, Higher dimensional digraphs from cube complexes and their spectral theory, Groups Geom. Dyn. (2024), published Online First.
  * [11] M. V. Lawson, A. Sims, A. Vdovina, Higher dimensional generalizations of the Thompson groups via higher rank graphs, 228 J. Pure Appl. Algebra (2023), no. 1 Paper No. 107456, 40 pp.
  * [12] A. Mutter, A.-C. Radu, and A. Vdovina, $\displaystyle C^{*}$-algebras of higher-rank graphs from groups acting on buildings, and explicit computation of their K-theory, Publications Mathemátiques 68 (2024), no. 1, 187–217.
  * [13] N.C. Phillips., A classification theorem for nuclear purely infinite simple $\displaystyle C^{*}$-algebras, Doc. Math. 5 (2000), 49–114.
  * [14] S.C. Power, Classifying higher rank analytic Toeplitz algebras, New York J. Math. 13 (2007), 271–298.
  * [15] I. Raeburn, A. Sims and T. Yeend, Higher-rank graphs and their $\displaystyle C^{*}$-algebras, Proc. Edinburgh Math. Soc. 46 (2003), 99–115.
  * [16] N. Rungtanapirom, J. Stix, A. Vdovina, Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes, Internat. J. Algebra Comput. 29 (2019), no. 6, 951–1007.