Introduction
Stable rank is a non-commutative analogue of topological dimension for $C^{*}$
-algebras introduced by Rieffel in the early 1980s [Reference Rieffel34], and widely used and studied ever since (see, for example [Reference Brown3, Reference Dădărlat, Nagy, Némethi and Pasnicu8, Reference Eilers and Elliott11, Reference Elliott, Ho and Toms12, Reference Farah and Rørdam15, Reference Jeong, Park and Shin18, Reference Osaka and Phillips24, Reference Putnam30, Reference Riedel33, Reference Rørdam36, Reference Suzuki44, Reference Villadsen46]). The condition of having stable rank 1, meaning that the invertible elements are dense in the $C^{*}$
-algebra, has attracted significant attention, in part due to its relevance to the classification of $C^{*}$
-algebras. Specifically, all separable, simple, unital, nuclear, $\mathcal {Z}$
-stable $C^{*}$
-algebras in the UCT class are classified by their Elliott invariant [Reference Castillejos, Evington, Tikuisis, White and Winter6, Reference Elliott and Niu13, Reference Kirchberg19, Reference Phillips29, Reference Sato, White and Winter40], and stable rank distinguishes two key cases: the stably finite $C^{*}$
-algebras in this class have stable rank 1 [Reference Rørdam36, Theorem 6.7], while the remainder are Kirchberg algebras with stable rank infinity [Reference Rieffel34, Proposition 6.5]. It follows that a simple $C^{*}$
-algebra whose stable rank is finite but not equal to 1 does not belong to the class of $C^{*}$
-algebras classified by their Elliott invariants.
Higher rank graphs (or $k$
-graphs) $\Lambda$
are generalizations of directed graphs. They give rise to an important class of $C^{*}$
-algebras $C^{*}(\Lambda )$
due to their simultaneous concreteness of presentation and diversity of structure [Reference Davidson and Yang9, Reference Kumjian, Pask and Sims22, Reference Pask, Raeburn, Rørdam and Sims26]. They provide good test cases for general theory [Reference Carlsen, Kang, Shotwell and Sims5, Reference Sims and Williams43] and have found unexpected applications in general $C^{*}$
-algebra theory. For example, the first proof that Kirchberg algebras in the UCT class have nuclear dimension 1 proceeded by realizing them as direct limits of 2-graph $C^{*}$
-algebras [Reference Ruiz, Sims and Sørensen39]. Nevertheless, and despite their deceptively elementary presentation in terms of generators and relations, $k$
-graph $C^{*}$
-algebras, in general, remain somewhat mysterious – for example, it remains an unanswered question whether all simple $k$
-graph $C^{*}$
-algebras are $\mathcal {Z}$
-stable and hence classifiable. This led us to investigate their stable rank. In this paper, we shed some light on how to compute the stable rank of $k$
-graph $C^{*}$
-algebras; though unfortunately, the simple $C^{*}$
-algebras to which our results apply all have stable rank either 1 or $\infty$
, so we obtain no new information about $\mathcal {Z}$
-stability or classifiability.
This paper focuses on unital $k$
-graph $C^{*}$
-algebras. For $k=1$
, i.e., for directed graph $C^{*}$
-algebras (unital or not), a complete characterization of stable rank has been obtained [Reference Deicke, Hong and Szymański10, Reference Jeong17, Reference Jeong, Park and Shin18]. In this paper, our main contribution is for $k\geq 2$
, a characterization of stable rank for $C^{*}$
-algebras associated with
(1) finite $k$
-graphs that have no cycle with an entrance, and(2) finite $k$
-graphs that are cofinal.
In the first case (1), we prove that such $k$
-graphs are precisely the ones for which the associated $C^{*}$
-algebra is stably finite. Partial results on how to characterize stably finite $k$
-graph $C^{*}$
-algebras have appeared in the past, see [Reference Clark, an Huef and Sims7, Reference Pask, Sierakowski and Sims28, Reference Schafhauser41]. It turns out that in the unital situation, all such $C^{*}$
-algebras are direct sums of matrix algebras over commutative tori of dimension at most $k$
; the precise dimensions of the tori are determined by the degrees of certain cycles (called initial cycles) in the $k$
-graph. Their $C^{*}$
-algebraic structure is therefore independent of the factorization property that determines how the one-dimensional subgraphs of a $k$
-graph fit together to give it its $k$
-dimensional nature.
We also settle the second case (2) where the $k$
-graphs are cofinal using our characterization of stable finiteness in combination with a technical argument on the von Neumann equivalence of (direct sums of) vertex projections. We initially obtained this result for selected $2$
-graphs using Python.
We now give a brief outline of the paper; Figure 1 may also help the reader to navigate. In § 1, we introduce terminology, including the notion (and examples) of an initial cycle. In § 2, we consider the stably finite case. In Proposition 2.7, we prove that the stable finiteness of $C^{*}(\Lambda )$
is equivalent to the condition that no cycle in the $k$
-graph $\Lambda$
has an entrance. In Theorem 2.5, we characterize the structure of $C^{*}(\Lambda )$
in the stably finite case and compute the stable rank of such algebras in Corollary 2.9. In § 3, we characterize which $k$
-graphs yield $C^{*}$
-algebras with stable rank 1 (Theorem 3.1 and Corollary 3.4) and show how the dimension of the tori that form the components of their spectra can be read off from (the skeleton of) the $k$
-graph, see Proposition 3.3.
Figure 1. Overview of some of our results. The ‘?’ indicates unknown stable rank.
In § 4, we look at $k$
-graphs which are cofinal. Firstly, in Proposition 4.1, we study the special case when $C^{*}(\Lambda )$
is simple. Then, in Theorem 4.4, we compute stable rank when $\Lambda$
is cofinal and contains a cycle with an entrance (so $C^{*}(\Lambda )$
is not stably finite). In § 5, we illustrate the difficulty in the remaining case where $\Lambda$
is not cofinal and contains a cycle with an entrance by considering 2-vertex 2-graphs with this property. We are able to compute the stable rank exactly for all but three classes of examples, for which the best we can say is that the stable rank is either 2 or 3.
1. Background
In this section, we recall the definition of stable rank, and the notions of stably finite and purely infinite $C^{*}$
-algebras. We also recall the definitions of $k$
-graphs and their associated $C^{*}$
-algebras. We discuss the path space of a locally convex $k$
-graph and describe initial cycles and their periodicity group. The reader familiar with these terms can skim through or skip this section.
1.1. Stable rank of $C^{*}$-algebras
Let $A$
be a unital $C^{*}$
-algebra. Following [Reference Blackadar2], let
The stable rank of $A$
, denoted $sr(A)$
, is the smallest $n$
such that $Lg_n(A)$
is dense in $A^{n}$
, or $\infty$
if there is no such $n$
. For a $C^{*}$
-algebra $A$
without a unit, we define its stable rank to be that of its minimal unitization $\widetilde {A}$
.
A $C^{*}$
-algebra $A$
has stable rank one if and only if the set $A^{-1}$
of invertible elements in $A$
is dense in $A$
. We will make frequent use of the following key results concerning the stable rank of $C^{*}$
-algebras of functions on tori, matrix algebras, stable $C^{*}$
-algebras and direct sums later in the paper:
(1) $sr(C(\mathbb {T}^{\ell }))=\left \lfloor {\ell /2}\right \rfloor +1$
;(2) $sr(M_n(A))=\left \lceil {(sr(A)-1)/n}\right \rceil +1$
;(3) $sr(A\otimes \mathcal {K}) = 1$
if $sr(A) = 1$, and $sr(A \otimes \mathcal {K}) = 2$ if $sr(A) \not = 1$; and(4) $sr(A\oplus B)=\max (sr(A),\,sr(B))$
.
Stable rank is in general not preserved under Morita equivalence (unless the stable rank is one). For more details, see [Reference Rieffel34].
1.2. Stably finite and purely infinite C*-algebras
A projection in a $C^{*}$
-algebra is said to be infinite if it is (von Neumann) equivalent to a proper subprojection of itself. If a projection is not infinite, it is said to be finite. A unital $C^{*}$
-algebra $A$
is said to be finite if its unit is a finite projection, and stably finite if $M_n(A)$
is finite for each positive integer $n$
[Reference Rørdam, Larsen and Laustsen37, Definition 5.1.1]. We refer to [Reference Clark, an Huef and Sims7, Reference Schafhauser41] for results about stably finite graph $C^{*}$
-algebras.
A simple $C^{*}$
-algebra $A$
is purely infinite if every non-zero hereditary sub-$C^{*}$
-algebra of $A$
contains an infinite projection. For the definition when $A$
is non-simple, we refer the reader to [Reference Kirchberg and Rørdam20].
1.3. Higher rank graphs
Following [Reference Kumjian and Pask21, Reference Pask, Quigg and Raeburn25, Reference Raeburn, Sims and Yeend32], we recall some terminology for $k$
-graphs. For $k \ge 1$
, a $k$
-graph is a non-empty, countable, small category equipped with a functor $d : \Lambda \to \mathbb {N}^{k}$
satisfying the factorisation property: For all $\lambda \in \Lambda$
and $m,\,n\in \mathbb {N}^{k}$
such that $d(\lambda ) = m + n$
there exist unique $\mu ,\, \nu \in \Lambda$
such that $d(\mu ) = m$
, $d(\nu ) = n$
, and $\lambda = \mu \nu$
. When $d(\lambda ) = n$
we say $\lambda$
has degree $n$
, and we define $\Lambda ^{n} := d^{-1}(n)$
. If $k=1$
, then $\Lambda$
is isomorphic to the free category generated by the directed graph with edges $\Lambda ^{1}$
and vertices $\Lambda ^{0}$
. The generators of $\mathbb {N}^{k}$
are denoted $e_1,\, \ldots ,\,e_k$
, and $n_i$
denotes the $i^{\textrm {th}}$
coordinate of $n \in \mathbb {N}^{k}$
. For $m,\, n \in \mathbb {N}^{k}$
, we write $m \le n$
if $m_i \le n_i$
for all $i$
, and we write $m\vee n$
for the coordinatewise maximum of $m$
and $n$
, and $m \wedge n$
for the coordinatewise minimum of $m$
and $n$
.
If $\Lambda$
is a $k$
-graph, its vertices are the elements of $\Lambda ^{0}$
. The factorization property implies that the vertices are precisely the identity morphisms, and so can be identified with the objects. For each $\lambda \in \Lambda$
the source $s(\lambda )$
is the domain of $\lambda$
, and the range $r(\lambda )$
is the codomain of $\lambda$
(strictly speaking, $s(\lambda )$
and $r(\lambda )$
are the identity morphisms associated with the domain and codomain of $\lambda$
). Given $\lambda ,\,\mu \in \Lambda$
, $n \in \mathbb {N}^{k}$
and $E\subseteq \Lambda$
, we define
We say that a $k$
-graph $\Lambda$
is row-finite if $|v{\Lambda }^{n}|<\infty$
is finite for each $n\in \mathbb {N}^{k}$
and $v\in {\Lambda }^{0}$
, finite if $|\Lambda ^{n}|<\infty$
for all $n\in \mathbb {N}^{k}$
, and locally convex if for all distinct $i,\ j \in \{1,\ldots, k\}$
, and all paths $\lambda \in \Lambda ^{e_i}$
and $\mu \in \Lambda ^{e_j}$
such that $r(\lambda )=r(\mu )$
, the sets $s(\lambda )\Lambda ^{e_j}$
and $s(\mu )\Lambda ^{e_i}$
are non-empty.
Standing Assumption We have two standing assumptions. The first is that all of our $k$
-graphs $\Lambda$
are finite. This implies, in particular, that they are row-finite. The second is that all of our $k$
-graphs $\Lambda$
are locally convex.
A vertex $v$
is called a source if there exist $i\leq k$
such that $v\Lambda ^{e_i}=\emptyset$
. The term cycle, distinct from ‘generalised cycle’ [Reference Evans and Sims14], will refer to a path $\lambda \in \Lambda \setminus \Lambda ^{0}$
such that $r(\lambda ) = s(\lambda )$
.
We will occasionally illustrate $k$
-graphs as $k$
-coloured graphs. We refer to [Reference Hazlewood, Raeburn, Sims and Webster16] for the details, but in short there is a one-to-one correspondence between $k$
-graphs and $k$
-coloured graphs together with factorization rules for bi-coloured paths of length 2 satisfying an associativity condition [Reference Hazlewood, Raeburn, Sims and Webster16, Equation (3.2)].
1.4. Higher rank graph $C^{*}$-algebras
Let $\Lambda$
be a row-finite, locally convex $k$
-graph. Following [Reference Raeburn, Sims and Yeend32], a Cuntz–Krieger $\Lambda$
-family in a $C^{*}$
-algebra $B$
is a function $s : \lambda \mapsto s_\lambda$
from $\Lambda$
to $B$
such that
(1) $\{s_v : v\in \Lambda ^{0}\}$
is a collection of mutually orthogonal projections;(2) $s_\mu s_\nu = s_{\mu \nu }$
whenever $s(\mu ) = r(\nu )$;(3) $s^{*}_\lambda s_\lambda = s_{s(\lambda )}$
for all $\lambda \in \Lambda$; and(4) $s_v = \sum \nolimits _{\lambda \in v\Lambda ^{\leq n}} s_\lambda s_\lambda ^{*}$
for all $v\in \Lambda ^{0}$ and $n\in \mathbb {N}^{k}$.
The $C^{*}$
-algebra $C^{*}(\Lambda )$
is the universal $C^{*}$
-algebra generated by a Cuntz–Krieger $\Lambda$
-family. It is unital if and only if $|\Lambda ^{0}|<\infty$
, in which case $1=\sum \nolimits _{v\in \Lambda ^{0}}s_v$
. The universal family in $C^{*}(\Lambda )$
is denoted $\{s_\lambda :\lambda \in \Lambda \}$
.
1.5. The path space of a $k$-graph
Let $\Lambda$
be a $k$
-graph. For each path $\lambda \in \Lambda$
, and $m \le n \le d(\lambda )$
, we denote by $\lambda (m,\,n)$
the unique element of $\Lambda ^{n-m}$
such that $\lambda = \lambda '\lambda (m,\,n)\lambda ''$
for some $\lambda ',\, \lambda ''\in \Lambda$
with $d(\lambda ') = m$
and $d(\lambda ') = d(\lambda ) - n$
. We abbreviate $\lambda (m,\,m)$
by $\lambda (m)$
. A $k$
-graph morphism between two $k$
-graphs is a degree preserving functor.
Following [Reference Evans and Sims14], for each $m \in (\mathbb {N} \cup \{\infty \})^{k}$
, we define a $k$
-graph $\Omega _{k,m}$
by $\Omega _{k,m} = \{(p,\,q) \in \mathbb {N}^{k} \times \mathbb {N}^{k} : p \le q \le m\}$
with range map $r(p,\,q) = (p,\,p)$
, source map $s(p,\,q) = (q,\,q)$
, and degree map $d(p,\,q) = q-p$
. We identify $\Omega _{k,m}^{0}$
with $\{p \in \mathbb {N}^{k} : p \le m\}$
via the map $(p,\,p) \mapsto p$
. Given a $k$
-graph and $m \in \mathbb {N}^{k}$
there is a bijection from $\Lambda ^{m}$
to the set of morphisms $x : \Omega _{k,m} \to \Lambda$
, given by $\lambda \mapsto ((p,\,q)\mapsto \lambda (p,\,q))$
; the inverse is the map $x \mapsto x(0,\,m)$
. Thus, for each $m\in \mathbb {N}^{k}$
, we may identify the collection of $k$
-graph morphisms from $\Omega _{k,m}$
to $\Lambda$
with $\Lambda ^{m}$
. We extend this idea beyond $m\in \mathbb {N}^{k}$
as follows: Given $m \in (\mathbb {N} \cup \{\infty \})^{k} \setminus \mathbb {N}^{k}$
, we regard each $k$
-graph morphism $x : \Omega _{k,m} \to \Lambda$
as a path of degree $m$
in $\Lambda$
and write $d(x){:=}m$
and $r(x){:=}x(0)$
; we denote the set of all such paths by $\Lambda ^{m}$
. We denote by $W_\Lambda$
the collection $\bigcup _{m \in (\mathbb {N} \cup \{\infty \})^{k}} \Lambda ^{m}$
of all paths in $\Lambda$
; our conventions allow us to regard $\Lambda$
as a subset of $W_\Lambda$
. We call $W_\Lambda$
the path space of $\Lambda$
. We set
and for $v\in \Lambda ^{0}$
, we define $v\Lambda ^{\leq \infty }:=\{x\in \Lambda ^{\leq \infty }: r(x)=v\}$
. Given a cycle $\tau$
, we define $\tau ^{\infty }$
(informally written as $\tau ^{\infty }{:=}\tau \tau \tau \ldots$
) to be the unique element of $W_\Lambda$
such that $d(\tau ^{\infty })_i$
is equal to $\infty$
when $d(\tau )_i > 0$
and equal to $0$
when $d(\tau )_i = 0$
, and such that $(\tau ^{\infty })(n\cdot d(\tau ),\, (n+1)\cdot d(\tau )) = \tau$
for all $n \in \mathbb {N}$
.
1.6. Initial cycles and their periodicity group
In this section, we introduce initial cycles and their associated periodicity group. As we will see in Corollary 2.9 and Theorem 3.1, the periodicity group plays an important role in the characterization of stable rank.
Let $\lambda$
be a cycle in a row-finite, locally convex $k$
-graph $\Lambda$
. We say $\lambda$
is a cycle with an entrance if there exists $\tau \in r(\lambda )\Lambda$
such that $d(\tau )\leq d(\lambda ^{\infty })$
and $\tau \neq \lambda ^{\infty }(0,\,d(\tau ))$
.
Definition 1.1 Evans and Sims [Reference Evans and Sims14]
Let $\Lambda$
be a finite, locally convex $k$
-graph that has no cycle with an entrance. We call $\mu \in \Lambda$
an initial cycle if $r(\mu ) = s(\mu )$
and if $r(\mu ) \Lambda ^{e_i} = \emptyset$
whenever $d(\mu )_i = 0$
.
Remark 1.2 While the wording of Definition 1.1 differs from that of [Reference Evans and Sims14], we will show (in Proposition 2.7) that for any $k$
-graph $\Lambda$
, a path $\mu \in \Lambda$
is an initial cycle in the sense of Definition 1.1 if and only if it is an initial cycle in the sense of [Reference Evans and Sims14].
Remark 1.3 An initial cycle may be trivial, in the sense that it has degree 0, so it is in fact a vertex. This vertex must then be a source, as for example, $w_4$
in Figure 5. It is not true that every source is an initial cycle; for example, $w_3$
in Figure 5 is a source but not an initial cycle.
As in [Reference Evans and Sims14], we let $\operatorname {IC}(\Lambda )$
denote the collection of all initial cycles in $\Lambda$
; if $\Lambda ^{0}$
is finite and $\Lambda$
has no cycle with an entrance, then $\operatorname {IC}(\Lambda )$
is non-empty – see Lemma 2.2. A vertex $v\in \Lambda ^{0}$
is said to be on the initial cycle $\mu$
if $v=\mu (p)$
for some $p\leq d(\mu )$
Footnote 1. We let $(\mu ^{\infty })^{0}$
denote the collection of all vertices on an initial cycle $\mu$
and let $\sim$
be the equivalence relation on $\operatorname {IC}(\Lambda )$
given by $\mu \sim \nu \Leftrightarrow (\mu ^{\infty })^{0}=(\nu ^{\infty })^{0}$
.
Remark 1.4 For a finite, locally convex $k$
-graph that has no cycle with an entrance, each initial cycle is an ‘initial segment’ in the following sense:
(1) Every path with range on the initial cycle is in the initial cycle, so paths can not ‘enter’ an initial cycle (see Lemma 2.1).
Without the assumption that $\Lambda ^{0}$
is finite and $\Lambda$
has no cycle with an entrance property (1) might fail. This is, for example, the case for the $1$
-graph with one vertex and two edges representing the Cuntz algebra $\mathcal {O}_2$
. This suggests that, in general, a different terminology should perhaps be used.
As in [Reference Evans and Sims14], we associate a group $G_\mu$
to each initial cycle $\mu$
. Let $\Lambda$
be a finite, locally convex $k$
-graph that has no cycle with an entrance. Let $\mu$
be an initial cycle in $\Lambda$
. If $\mu$
is not a vertex, we define
otherwise, we let $G_\mu {:=}\{0\}$
.
Definition 1.5 By [Reference Evans and Sims14, Lemma 5.8], $G_\mu$
is a subgroup of $\mathbb {Z}^{k}$
, and hence isomorphic to $\mathbb {Z}^{\ell _\mu }$
for some $\ell _\mu \in \{0,\,\ldots ,\, k\}$
: we often refer to $\ell _\mu$
as the rank of $G_\mu$
.
Remark 1.6 It turns out that $\ell _\mu = |\{i\leq k: d(\mu )_i> 0\}|$
– see Proposition 3.3.
1.7. Examples of initial cycles
Figure 2 illustratesFootnote 2 two examples of $2$
-graphs $\Lambda _1$
and $\Lambda _2$
containing lots of initial cycles. In fact, a cycle in either $\Lambda _1$
or $\Lambda _2$
is an initial cycle precisely if it contains edges of both colours. Each initial cycle in either $\Lambda _i$
visits every vertex, so any two initial cycles in $\Lambda _i$
are $\sim$
- equivalent. A computation shows that for each initial cycle $\mu$
in either $\Lambda _i$
, $G_{\mu }\cong \mathbb {Z}^{2}$
. Notice that each vertex on a cycle in either $\Lambda _i$
has exactly one red (dashed) and one blue (solid) incoming and outgoing edge and exactly one infinite path with range at that vertex.
Figure 2. Two $2$
-graphs $\Lambda _1,\, \Lambda _2$
, each containing lots of initial cycles, but only one such up to $\sim$
- equivalence.
Below, we illustrate how the stable rank of each $C^{*}(\Lambda _i)$
can be computed. For this, we need some definitions and a lemma.
(1) Fix an integer $n \ge 1$
, let $L_n$ denote the connected $1$-graph with $n$ vertices $v_0 ,\, \ldots ,\, v_{n-1}$ and $n$ morphisms $f_0 ,\, \ldots ,\, f_{n-1}$ of degree $1$ such that $s ( f_i ) = v_{i+1\!\!\pmod {n}}$ and $r ( f_i ) = v_{i}$ for $0 \le i \le n-1$.(2) Let $( \Lambda _1 ,\, d_1 )$
and $( \Lambda _2 ,\, d_2 )$ be $k_1$-, $k_2$-graphs respectively, then $( \Lambda _1 \times \Lambda _2 ,\, d_1 \times d_2 )$ is a $(k_1 + k_2)$-graph where $\Lambda _1 \times \Lambda _2$ is the product category and $d_1 \times d_2 : \Lambda _1 \times \Lambda _2 \rightarrow \mathbb {N}^{k_1 + k_2}$ is given by $d_1 \times d_2 ( \lambda _1 ,\, \lambda _2 ) = ( d_1 ( \lambda _1 ) ,\, d_2 ( \lambda _2 ) ) \in \mathbb {N}^{k_1} \times \mathbb {N}^{k_2}$ [Reference Kumjian and Pask21, Proposition 1.8].(3) Let $f : \mathbb {N}^{\ell } \rightarrow \mathbb {N}^{k}$
be a monoid morphism. If $( \Lambda ,\,d)$ is a $k$-graph we may form the $\ell$-graph $f^{*} ( \Lambda )$ as follows: $f^{*} ( \Lambda ) = \{ ( \lambda ,\, n ) : d ( \lambda ) = f(n) \}$ with $d ( \lambda ,\, n ) = n$, $s ( \lambda ,\, n ) = s ( \lambda )$ and $r ( \lambda ,\, n ) = r ( \lambda )$ [Reference Kumjian and Pask21, Example 1.10].(4) Let $\Lambda$
be a $k$–graph and define $g_i : \mathbb {N} \to \mathbb {N}^{k}$ by $g_i (n) = n e_i$ for $1 \leq i \leq k$ (so $\ell =1$). The $1$-graphs $\Lambda _i : = g_i^{*} ( \Lambda )$ are called the coordinate graphs of $\Lambda$.
Lemma 1.7 Kumjian and Pask [Reference Kumjian and Pask21, Proposition 1.11, Corollary 3.5(iii), Corollary 3.5(iv)]
(1) Let $( \Lambda _i ,\, d_i )$
be $k_i$-graphs for $i = 1,\, 2,$ then $C^{*} ( \Lambda _1 \times \Lambda _2 ) \cong C^{*} ( \Lambda _1 ) \otimes C^{*} ( \Lambda _2 )$ via the map $s_{(\lambda _1 , \lambda _2)} \mapsto s_{\lambda _1} \otimes s_{\lambda _2}$ for $( \lambda _1 ,\, \lambda _2 ) \in \Lambda _1 \times \Lambda _2$.(2) Let $\Lambda$
be a $k$-graph and $f : \mathbb {N}^{\ell } \rightarrow \mathbb {N}^{k}$ a surjective monoid morphism. Then $C^{*} (f^{*} ( \Lambda ) ) \cong C^{*} ( \Lambda ) \otimes C ( \mathbb {T}^{\ell - k})$.
Let $\Lambda$
be a $1$
-graph and define $f_1 : \mathbb {N}^{2} \rightarrow \mathbb {N}$
by $( m_1,\, m_2) \mapsto m_1 + m_2$
. Then $f_1^{*} ( L_6 )$
is isomorphic to the $2$
-graph $\Lambda$
shown on the left in Figure 2. The $2$
-graph $L_6 \times L_1$
is isomorphic to the $2$
-graph shown on the right in Figure 2. Using Lemma 1.7 and that $C^{*}(L_6)\cong M_6(C(\mathbb {T}))$
[Reference an Huef and Raeburn1, Lemma 2.4], we get $C^{*}(\Lambda _i)\cong C^{*} (L_6) \otimes C(\mathbb {T}) \cong M_6(C(\mathbb {T}^{2}))$
, $i=1,\,2$
, so both have stable rank 2 as discussed in § 1.1.
2. Structure and stable rank in the stably finite case
In this section, we study finite $k$
-graphs whose $C^{*}$
-algebras are stably finite, corresponding to boxes 1 and 2 in Figure 1. In Proposition 2.7, we show that stable finiteness is equivalent to the lack of infinite projections and provide a characterization in terms of properties of the $k$
-graph. We also provide a structure result and compute stable rank of such $C^{*}$
-algebras – see Theorems 2.5 and 2.6. We begin with four technical lemmas needed to prove Theorem 2.5.
Lemma 2.1 Let $\Lambda$
be a finite, locally convex $k$
-graph that has no cycle with an entrance. Let $v\in \Lambda ^{0}$
be a vertex on an initial cycle $\mu \in \Lambda$
. Then
(1) there exist paths $\iota _v,\,\tau _v\in \Lambda$
such that $\mu =\iota _v\tau _v$ and $s(\iota _v)=v=r(\tau _v);$(2) the path $\mu _v{:=} \tau _v\iota _v$
satisfies $r(\mu _v)= s(\mu _v),$ and $r(\mu _v) \Lambda ^{e_i} = \emptyset$ whenever $d(\mu _v)_i = 0;$(3) if $f\in \Lambda ^{e_i}$
is an edge with range $v$ on $\mu ,$ then $f=\mu _{v}(0,\, e_i)$ and $\mu =\nu 'f\nu ''$ for some $\nu ',\,\nu ''\in \Lambda ;$(4) if $n\leq d(\mu )$
and $\lambda \in v\Lambda ^{\leq n}$, then $\lambda =\mu _{v}(0,\, d(\lambda ))$; and(5) $s_{\tau _v}s_{\tau _v}^{*}=s_v$
and $s_{\tau _v}^{*}s_{\tau _v}=s_{s(\mu )}$.
Proof. (1). Since $v$
is a vertex on $\mu$
, we have $v=\mu (p)$
for some $p\leq d(\mu )$
. Set $\iota _v{:=} \mu (0,\,p)$
and $\tau _v{:=} \mu (p,\,d(\mu ))$
. Then $\mu =\iota _v\tau _v$
and $s(\iota _v)=v=r(\tau _v)$
.
(2). By property (1), $s(\iota _v)=v=r(\tau _v)$
, so the path $\mu _v{:=} \tau _v\iota _v\in \Lambda$
satisfies $r(\mu _v)=s(\mu _v)$
. Suppose $r(\mu _v) \Lambda ^{e_i}$
is non-empty, say $\alpha \in r(\mu _v) \Lambda ^{e_i}$
. Then $(\iota _v\alpha )(0,\,e_i)\in r(\mu )\Lambda ^{e_i}$
. Since $\mu$
is an initial cycle, it follows that $d(\mu )_i \neq 0$
.
(3). Suppose $f\in \Lambda ^{e_i}$
is an edge with range $v$
on $\mu$
. Since $r(f)=v=r(\mu _{v})$
, we have $f \in r(\mu _{v})\Lambda ^{e_i}$
. Now property (2) ensures that $d(\mu _v)_i \neq 0$
. Since $d(\mu _v)_i > 0$
, there exists a path $\lambda \in s(f)\Lambda ^{\leq d(\mu _v)-e_i}$
. Hence $f\lambda \in \Lambda ^{\leq e_i}\Lambda ^{\leq d(\mu _v)-e_i}=\Lambda ^{\leq d(\mu _v)}$
. Now using that $\mu _v\in v\Lambda$
is a cycle and that $\Lambda$
has no cycle with an entrance, it follows that $v\Lambda ^{\leq d(\mu _v)}= \{\mu _v\}$
. Hence $\mu _v=f\lambda$
. Since $f\lambda =\mu _v=\tau _v\iota _v$
, we get $f=\tau _v(0,\,e_i)$
if $d(\tau _v)_i>0$
and $f=\iota _v(0,\,e_i)$
if $d(\tau _v)_i=0$
.
(4). Fix $n\leq d(\mu )$
and $\lambda \in v\Lambda ^{\leq n}$
. If $d(\lambda ) = 0$
then $\lambda = v$
and the statement is trivial, so we may assume that $\lambda \not \in \Lambda ^{0}$
. Write $d(\lambda )=e_{i_1}+\ldots +e_{i_m}$
where $i_1\ldots ,\,i_m\in \{1,\,\ldots ,\,k\}$
. By the factorization property $\lambda =\lambda _1\ldots \lambda _m$
for some $\lambda _j \in \Lambda ^{e_{i_j}}$
. Repeated applications of part (3) give $\lambda _j = \mu _{r(\lambda _j)}(0,\, e_{i_j})$
for $j \le m$
. Since $d(\lambda )\leq d(\mu )$
, it follows that $\lambda =\mu _{v}(0,\, d(\lambda ))$
.
(5). Since $s(\tau _v)=s(\mu )$
we have $s_{\tau _v}^{*}s_{\tau _v}=s_{s(\mu )}$
. For $n{:=}d(\tau _v)$
notice that $n\leq d(\mu )$
. Fix $\lambda \in v\Lambda ^{\leq n}$
. Using part (4) we have $\lambda =\mu _{v}(0,\, d(\lambda ))$
. Since $\tau _v\in v\Lambda ^{\leq n}$
and since $d(\lambda )\leq n$
, we have $\tau _v=\mu _{v}(0,\, n)=\lambda \mu _{v}(d(\lambda ),\, n)$
. But both $\tau _v$
and $\lambda$
belong to $\Lambda ^{\leq n}$
, so $\tau _v=\lambda$
. Consequently $v\Lambda ^{\leq n}=\{\tau _v\}$
and $s_{\tau _v}s_{\tau _v}^{*}=\sum \nolimits _{\lambda \in v\Lambda ^{\leq n}}s_\lambda s_\lambda ^{*}=s_{v}$
.
Lemma 2.2 Let $\Lambda$
be a finite, locally convex $k$
-graph that has no cycle with an entrance. Let $N{:=}(|\Lambda ^{0}|,\,\ldots ,\, |\Lambda ^{0}|)\in \mathbb {N}^{k}$
. Then
(1) $\sum \nolimits _{\lambda \in \Lambda ^{\leq N}} s_\lambda s_\lambda ^{*} = 1_{C^{*}(\Lambda )};$
and(2) for every $\lambda \in \Lambda ^{\leq N},$
$s(\lambda )$ is a vertex on an initial cycle.
Proof. For part (1) use $1=\sum \nolimits _{v\in \Lambda ^{0}}s_v=\sum \nolimits _{v\in \Lambda ^{0}}\sum \nolimits _{\lambda \in v\Lambda ^{\leq N}}s_\lambda s_\lambda ^{*}=\sum \nolimits _{\lambda \in \Lambda ^{\leq N}}s_\lambda s_\lambda ^{*}.$
For part (2), we refer to the second paragraph of the proof of [Reference Evans and Sims14, Proposition 5.9].
Recall that $(\mu ^{\infty })^{0}$
denotes the collection of all vertices on an initial cycle $\mu$
. For the terminology $\tau _{v},\, v\in \Lambda ^{0}$
in the following lemma, see Lemma 2.1.
Lemma 2.3 Let $\Lambda$
be a finite, locally convex $k$
-graph that has no cycle with an entrance. Fix an initial cycle $\mu \in \Lambda$
. Let $N{:=}(|\Lambda ^{0}|,\,\ldots ,\, |\Lambda ^{0}|)\in \mathbb {N}^{k}$
and for each $\lambda ,\,\nu \in \Lambda ^{\leq N}(\mu ^{\infty })^{0}$
set (using Lemma 2.1)
Then the $\theta _{\lambda ,\nu }$
are matrix units, i.e., $\theta _{\lambda ,\nu }^{*}=\theta _{\nu ,\lambda }$
and $\theta _{\lambda ,\nu }\theta _{\gamma , \eta }=\delta _{\nu ,\gamma }\theta _{\lambda , \eta }$
in $C^{*}(\Lambda )$
.
Proof. Firstly, note that each $\theta _{\lambda ,\nu }$
makes sense because the source of $\lambda \in \Lambda ^{\leq N}(\mu ^{\infty })^{0}$
is a vertex on $\mu$
and $\tau _{s(\lambda )}$
is a path on the initial cycle $\mu$
with range $s(\lambda )$
.
Fix $\lambda ,\, \nu ,\,\gamma ,\, \eta \in \Lambda ^{\leq N}(\mu ^{\infty })^{0}$
. Clearly, $\theta _{\lambda ,\nu }^{*}=\theta _{\nu ,\lambda }$
. We claim that $\theta _{\lambda ,\nu }\theta _{\gamma , \eta }=\delta _{\nu ,\gamma }\theta _{\lambda , \eta }$
. To see this let $v{:=}r(\mu )$
. Then
is non-zero only if $r(\nu )=r(\gamma )$
. Since $s_{r(\nu )} = \sum \nolimits _{\alpha \in {r(\nu )}\Lambda ^{\leq N}} s_\alpha s_\alpha ^{*}$
, we have $(s_\nu s_\nu ^{*})(s_\gamma s_\gamma ^{*})=\delta _{\nu ,\gamma }s_\nu s_\nu ^{*}$
, so if (2.1) is non-zero, then $\nu =\gamma$
and then
Hence $\theta _{\lambda ,\nu }\theta _{\gamma , \eta }=s_{\lambda \tau _{s(\lambda )}}(\delta _{\nu ,\gamma }s_{v})s_{\eta \tau _{s(\eta )}}^{*}=\delta _{\nu ,\gamma }\theta _{\lambda , \eta }$
as claimed.
The next lemma is of general nature. See [Reference Kumjian, Pask and Sims22, Lemma 3.3] for a special case of this result.
Lemma 2.4 Suppose that $\{e_{ij}^{(k)} : 1\leq k\leq r,\, 1\leq i,\,j\leq n_k\}$
is a system of matrix units in a unital $C^{*}$
-algebra $A$
, in the sense that
(1) $e_{ij}^{(k)}e_{jl}^{(k)}=e_{il}^{(k)};$
(2) $e_{ij}^{(k)}e_{mn}^{(l)}=0$
if $k\neq l$ or if $j\neq m;$(3) $(e_{ij}^{(k)})^{*}=e_{ji}^{(k)};$
and(4) $\sum \nolimits _{k=1}^{r} \sum \nolimits _{i=1}^{n_k} e_{ii}^{(k)}=1$
.
For $k \leq r$
let $p^{(k)}{:=} \sum \nolimits _{i=1}^{n_k} e_{ii}^{(k)}$
. Suppose that for each $a\in A$
, $a=\sum \nolimits _{k=1}^{r} p^{(k)}ap^{(k)}$
. Then, for $1\leq k \leq r$
, each $e_{11}^{(k)}$
is a projection and $A\cong \bigoplus _{k=1}^{r} M_{n_k}(e_{11}^{(k)}Ae_{11}^{(k)})$
.
Proof. Clearly $A\cong \bigoplus _{k=1}^{r}p^{(k)}Ap^{(k)}$
via $a\mapsto (p^{(1)}ap^{(1)},\,\ldots ,\, p^{(r)}ap^{(r)})$
and inverse $(a^{(1)},\, \ldots ,\, a^{(r)})\mapsto \sum \nolimits _{k=1}^{r} a^{(k)}$
. Routine calculations show that for each $k\in \{1,\,\ldots ,\,r\}$
, the elements $v_i{:=}e_{i1}^{(k)}$
, $i=1,\, \ldots ,\, n_k$
, satisfy
By [Reference Kumjian, Pask and Sims22, Lemma 3.3], $p^{(k)}Ap^{(k)}\cong M_{n_k}(e_{11}^{(k)}Ae_{11}^{(k)})$
completing the proof.
We now characterize the structure of $k$
-graph $C^{*}$
-algebras $C^{*}(\Lambda )$
such that $\Lambda$
is finite and has no cycle with an entrance. For the notions of $\operatorname {IC}(\Lambda )$
, $\sim$
, $(\mu ^{\infty })^{0}$
, and $\ell _\mu$
see § 1.6. We note that in Theorem 2.5, $\operatorname {IC}(\Lambda )\neq \emptyset$
and $\ell _\mu = |\{i\leq k: d(\mu )_i> 0\}|$
(see Lemma 2.2(2) and Proposition 3.3).
Theorem 2.5 Structure theorem
Let $\Lambda$
be a finite, locally convex $k$
-graph that has no cycle with an entrance. For $N{:=}(|\Lambda ^{0}|,\,\ldots ,\, |\Lambda ^{0}|)\in \mathbb {N}^{k},$
and each $s_{r(\mu )}C^{*}(\Lambda )s_{r(\mu )}\cong C(\mathbb {T}^{\ell _\mu })$
.
Proof. Evidently, $\operatorname {IC}(\Lambda )/\negthickspace \sim$
is finite since $\Lambda ^{0}$
is finite. Let $I$
be a maximal collection of initial cycles satisfying $(\mu ^{\infty })^{0}\cap (\nu ^{\infty })^{0}=\emptyset$
for any $\mu \neq \nu \in I$
. For each initial cycle $\mu \in I$
let $\{\theta _{\lambda ,\nu }^{(\mu )}\}$
be the matrix units of Lemma 2.3. We first prove that $\{\theta _{\lambda ,\nu }^{(\mu )}: \mu \in I \}$
is a system of matrix units, i.e.,
(1) $\theta _{\lambda \lambda '}^{(\mu )}\theta _{\lambda '\lambda ''}^{(\mu )}=\theta _{\lambda \lambda ''}^{(\mu )}$
;(2) $\theta _{\lambda \lambda '}^{(\mu )}\theta _{\eta \eta '}^{(\nu )}=0$
if $\mu \neq \nu$ or if $\lambda '\neq \eta$;(3) $(\theta _{\lambda \lambda '}^{(\mu )})^{*}=\theta _{\lambda '\lambda }^{(\mu )}$
; and(4) $\sum \nolimits _{\mu \in I}\sum \nolimits _{\lambda \in \Lambda ^{\leq N}(\mu ^{\infty })^{0}} \theta _{\lambda \lambda }^{(\mu )}=1$
.
We start with property (4). Fix $\mu \in I$
and $\lambda \in \Lambda ^{\leq N}(\mu ^{\infty })^{0}$
. Using Lemma 2.1(5), we have $s_{\tau _{s(\lambda )}}s_{\tau _{s(\lambda )}}^{*}=s_{s(\lambda )}$
. Hence $\theta _{\lambda ,\lambda }^{(\mu )}= s_{\lambda \tau _{s(\lambda )}}s_{\lambda \tau _{s(\lambda )}}^{*}=s_\lambda s_\lambda ^{*}$
. Lemma 2.2(1) now gives
Properties (1)–(3) follow from Lemma 2.2 and that $\theta _{\lambda \lambda }^{(\mu )}\theta _{\lambda \lambda }^{(\nu )}=0$
whenever $\mu \neq \nu$
in $I$
(the latter is a consequence of property (4)).
For each $\mu \in I$
, define $p^{(\mu )}{:=} \sum \nolimits _{\lambda \in \Lambda ^{\leq N}(\mu ^{\infty })^{0}} \theta _{\lambda \lambda }^{(\mu )}$
. Then (2.2) gives $\sum \nolimits _{\mu \in I} p^{(\mu )}=1$
. We claim that
is all of $C^{*}(\Lambda )$
. Clearly $A$
is a closed linear subspace of $C^{*}(\Lambda )$
. Fix $\alpha ,\, \beta \in \Lambda ^{\leq N}$
such that $s(\alpha )=s(\beta )$
. Using Lemma 2.2(2), it follows that $s(\alpha )\in (\mu ^{\infty })^{0}$
for some $\mu \in I$
. Since $s_\alpha s_\alpha ^{*}\leq p^{(\mu )}$
, we get
so $s_\alpha s_\beta ^{*}\in A$
. Using that $\operatorname {span}\{s_\alpha s_\beta ^{*}: \alpha ,\, \beta \in \Lambda ^{\leq N},\, s(\alpha )=s(\beta )\}$
is dense in $C^{*}(\Lambda )$
, we get $A=C^{*}(\Lambda )$
as claimed.
For each $\mu \in I$
, Lemma 2.1(3)–(4) implies that $r(\mu )\Lambda ^{\leq N}$
contains exactly one path which we denote by $\lambda _\mu$
. As in the proof of (4), we have $\theta _{\lambda _\mu ,\lambda _\mu }^{(\mu )}=s_{\lambda _\mu } s_{\lambda _\mu } ^{*}$
, so $s_{r(\mu )}= \sum \nolimits _{\lambda \in r(\mu )\Lambda ^{\leq N}} s_\lambda s_\lambda ^{*}=\theta _{\lambda _\mu ,\lambda _\mu }^{(\mu )}$
. Identifying $I$
with $\operatorname {IC}(\Lambda )/\negthickspace \sim$
via the map $\mu \mapsto [\mu ]$
, Lemma 2.4 provides an isomorphism
To see that each $s_{r(\mu )}C^{*}(\Lambda )s_{r(\mu )}\cong C(\mathbb {T}^{\ell _\mu })$
, see the proof of [Reference Evans and Sims14, Proposition 5.9].
The main result of this section is the characterization of stable rank for $k$
-graph $C^{*}$
-algebras $C^{*}(\Lambda )$
such that $\Lambda$
is finite and has no cycle with an entrance. Recall the notion of the floor and ceiling functions: for $x \in \mathbb {R}$
, we write $\lfloor x\rfloor {:=} \max \{n \in \mathbb {Z} : n \le x\}$
and $\lceil x\rceil {:=} \min \{n \in \mathbb {Z} : n \ge x\}$
.
Theorem 2.6 Let $\Lambda$
be a finite, locally convex $k$
-graph that has no cycle with an entrance. For $N{:=}(|\Lambda ^{0}|,\,\ldots ,\, |\Lambda ^{0}|)\in \mathbb {N}^{k},$
Proof. By Theorem 2.5 and property (4) from § 1.1,
and each $s_{r(\mu )}C^{*}(\Lambda )s_{r(\mu )}\cong C(\mathbb {T}^{\ell _\mu })$
. Now using property (2) from § 1.1, we get
Finally, property (1) from § 1.1, gives $sr(s_{r(\mu )}C^{*}(\Lambda )s_{r(\mu )}) -1=\left \lfloor {\ell _\mu /2}\right \rfloor$
for each $[\mu ] \in \operatorname {IC}(\Lambda )/\negthickspace \sim$
, completing the proof.
Our next Proposition 2.7 characterizes stable finiteness of $C^{*}$
-algebras of finite, locally convex $k$
-graphs. For other results on stable finiteness of $C^{*}$
-algebras associated with row-finite $k$
-graphs with no sources, see [Reference Clark, an Huef and Sims7, Reference Evans and Sims14]. Note that the $C^{*}$
-algebras satisfying the hypotheses of Proposition 2.7 are exactly those shown in Figure 1 in boxes 3 and 4.
We briefly introduce relevant terminology. Following [Reference Raeburn and Sims31], we write $\operatorname {MCE}(\mu ,\, \nu ){:=} \mu \Lambda \cap \nu \Lambda \cap \Lambda ^{d(\mu )\vee d(\nu )}$
for the set of all minimal common extensions of $\mu ,\,\nu \in \Lambda$
. The cycle $\lambda$
is a cycle with an entrance in the sense of [Reference Evans and Sims14, Definition 3.5] if there exists a path $\tau \in r(\lambda )\Lambda$
such that $\operatorname {MCE}(\tau ,\, \lambda )=\emptyset$
Footnote 3.
Proposition 2.7 Let $\Lambda$
be a finite, locally convex $k$
-graph. With notation as above, the following are equivalent:
(1) $\Lambda$
has a cycle $\mu$ with an entrance;(2) $\Lambda$
has a cycle $\mu$ with an entrance in the sense of [Reference Evans and Sims14, Definition 3.5];(3) $C^{*}(\Lambda )$
contains an infinite projection; and(4) $C^{*}(\Lambda )$
is not stably finite.
Proof. To prove (1)$\Rightarrow$
(2) let $\mu$
be a cycle with an entrance $\tau$
, so $\tau \in r(\mu )\Lambda$
satisfies $d(\tau )\leq d(\mu ^{\infty })$
and $\tau \neq \mu ^{\infty }(0,\,d(\tau ))$
. Fix $n\geq 1$
such that $nd(\mu )\geq d(\tau )$
. Clearly $\tau \in r(\mu ^{n})\Lambda$
. Then $\tau \neq \mu ^{\infty }(0,\,d(\tau ))=\mu ^{n}(0,\,d(\tau ))$
, so $\operatorname {MCE}(\mu ^{n},\, \tau )= (\mu ^{n}\Lambda \cap \Lambda ^{d(\mu ^{n})\vee d(\tau )})\cap \tau \Lambda \subseteq \{\mu ^{n}\} \cap \tau \Lambda =\emptyset$
. For the proof of (2)$\Rightarrow$
(3), see [Reference Evans and Sims14, Corollary 3.8].
The implication (3)$\Rightarrow$
(4) follows from [Reference Rørdam, Larsen and Laustsen37, Lemma 5.1.2]. It remains to prove (4)$\Rightarrow$
(1). We establish the contrapositive. Suppose that condition (1) does not hold, that is $\Lambda$
has no cycle with an entrance. Theorem 2.5 gives that $C^{*}(\Lambda )$
is isomorphic to a direct sum of matrix algebras over commutative $C^{*}$
-algebras, hence stably finite, so condition (4) does not hold.
Remark 2.8 Our main results are for finite $k$
-graphs so Proposition 2.7 is stated in that context, but some of the implications hold more generally.
(1) Only the proof of (4) ${\implies }$
(1) uses that $\Lambda$ is finite. The proofs of the implications (1) ${\implies }$ (2) ${\implies }$ (3) ${\implies }$ (4) are valid for any locally convex row-finite $k$-graph.(2) Similarly, while (1) and (2) are not equivalent for arbitrary $k$
-graphs, they are equivalent for locally convex $k$-graphs, whether finite or not. To see this, suppose that $\lambda$ is a cycle in a locally convex $k$-graph and $\tau \in r(\lambda )\Lambda$ satisfies $\operatorname {MCE}(\lambda ,\,\tau ) = \emptyset$. Let $I = \{i \le k : d(\lambda )_i > 0\}$, let $m_I = \sum \nolimits _{i \in I} d(\tau )_i e_i$ and $m' := d(\tau ) - m_I$, and factorize $\tau = \tau _I \tau '$ with $d(\tau _I) = m_I$. If $\tau _I \not = (\lambda ^{\infty })(0,\, m_I)$, then $\lambda$ is a cycle with an entrance as required, so we may assume that $\tau _I = (\lambda ^{\infty })(0,\,m_I)$. So replacing $\lambda$ with $(\lambda ^{\infty })(m_I,\, m_I + d(\lambda ))$ and $\tau$ with $\tau '$ we may assume that $d(\tau ) \wedge d(\lambda ) = 0$. Since $\Lambda$ is locally convex, a quick inductive argument shows that there exists $\mu \in s(\tau )\Lambda ^{d(\lambda )} \not = \emptyset$. Factorise $\tau \mu = \alpha \beta$ with $d(\alpha ) = d(\mu ) = d(\lambda )$. Since $\operatorname {MCE}(\tau ,\,\lambda ) = \emptyset$, we must have $\alpha \not = \lambda$ and in particular $d(\alpha ) = d(\lambda ) < d(\lambda ^{\infty })$ and $\alpha \not = (\lambda ^{\infty })(0,\, d(\alpha ))$. So once again $\lambda$ is a cycle with an entrance.
Corollary 2.9 Let $\Lambda$
be a finite, locally convex $k$
-graph. Suppose that $\Lambda$
has no cycle with an entrance (i.e., $C^{*}(\Lambda )$
is stably finite). For $N{:=}(|\Lambda ^{0}|,\,\ldots ,\, |\Lambda ^{0}|)\in \mathbb {N}^{k},$
Remark 2.10 A cycle with an incoming edge may fail to be a cycle with an entrance. This is, for example, the case for any of the red (dashed) cycles in Figure 3.
Figure 3. An example of a $2$
-graph $\Lambda$
with $C^{*}(\Lambda )$
of stable rank 2.
Example 2.11 In this example, we consider the $2$
-graph $\Lambda$
in Figure 3. As before, we refer to [Reference Hazlewood, Raeburn, Sims and Webster16] for details on how to illustrate 2-graphs as a 2-coloured graph. Here we use blue (solid) and red (dashed) as the first and second colour. We use our results to compute the structure and stable rank of $C^{*}(\Lambda )$
. Firstly, notice that red the cycle $\nu \in v\Lambda ^{e_2}$
based at $v$
is not an initial cycle because $d(\nu )_1 = 0$
but $r(\nu ) \Lambda ^{e_1} \neq \emptyset$
. However, the cycle $\mu \in v_0\Lambda ^{e_1+e_2}$
is an initial cycle. There are many other initial cycles, but they are all $\sim$
-equivalent to $\mu$
. So $\operatorname {IC}(\Lambda )/\negthickspace \sim = \{[\mu ]\}$
. Since the vertices on a path $\lambda \in v_0\Lambda$
alternate between $v_0$
and $v_1$
as we move along the path, it follows that $\mu ^{\infty }(m)=v_{m_1+m_2\pmod {2}}$
for each $m\in \mathbb {N}^{2}$
. Hence
We deduce that $\ell _\mu =\text {rank}(G_\mu )=2$
. Now set $N=(|\Lambda ^{0}|,\,|\Lambda ^{0}|) =(3,\,3)$
. As mentioned, modulo $\sim$
, there is only one initial cycle $\mu$
, so any path in $\Lambda ^{\leq N}$
has its source on $\mu$
. Hence $\Lambda ^{\leq N}(\mu ^{\infty })^{0}=\Lambda ^{\leq N}=v\Lambda ^{\leq N}\sqcup v_0\Lambda ^{\leq N}\sqcup v_1\Lambda ^{\leq N}$
. By Lemma 2.1, $|v_0\Lambda ^{\leq N}|=|v_1\Lambda ^{\leq N}|=1$
. Using the factorization property to push the red edges to the start of a path and uniqueness of such paths on $\mu$
, we have $|v\Lambda ^{\leq N}|=|v\Lambda ^{(3,3)}|=|v\Lambda ^{(3,0)}|=|v\Lambda ^{(1,0)}|=2$
. Hence $C^{*}(\Lambda )\cong M_{4}(C(\mathbb {T}^{2}))$
and $sr(C^{*}(\Lambda ))=\left \lceil \left \lfloor \frac {2}{2}\right \rfloor / 4\right \rceil +1=2$
by Theorem 2.6.
Example 2.12 In the following let $\Lambda _1,\, \Lambda _2$
and $\Lambda _3,\, \Lambda _4,\, \Lambda _5$
be the $2$
-graphs in Figure 2 and Figure 4 respectively. Up to a swap of the colours these five examples make up all the examples of $2$
-graphs on 6 vertices with only one initial cycle up to $\sim$
- equivalence and with all vertices on that initial cycle. Let $\mu _i$
denote such an initial cycle in $\Lambda _i$
. For $0 \le j \le n-1$
define $f_j : \mathbb {N}^{2} \to \mathbb {N}$
by $f_j ( m_1 ,\, m_2 ) = m_1 + j m_2$
. With the terminology of § 1.7 one can show that $\Lambda _3=f_5^{*} ( L_6 )$
, that $\Lambda _4=L_2 \times L_3$
, and that $\Lambda _5=f_2^{*} ( L_6 )$
. Hence
In particular $C^{*}(\Lambda _i)\cong C^{*}(\Lambda _j)$
for all $i,\,j$
. These five examples indicate how the number of colours and vertices impacts the structure of the corresponding $C^{*}$
-algebras. The next Proposition 2.13 verifies this.
Figure 4. Another three examples of $2$
-graphs with lots of initial cycles, but only one such up to $\sim$
- equivalence.
In the following, ‘the number of vertices’ on an initial cycle $\mu$
means $|(\mu ^{\infty })^{0}|$
, and ‘the number of colours’ means $|\{i\leq k: d(\mu )_i> 0\}|$
. The proof borrows material from an independent result (Proposition 3.3).
Proposition 2.13 Let $\Lambda$
be a finite, locally convex $k$
-graph on $n=|\Lambda ^{0}|$
vertices. Suppose that $\Lambda$
has no cycle with an entrance and $\Lambda$
has exactly one initial cycle, up to $\sim$
- equivalence, with $n$
vertices and $\ell$
colours. Then
Proof. Let $\mu$
be an initial cycle in $\Lambda$
. By Theorem 2.5 and Theorem 2.6, we have
where $\ell _\mu$
is the rank of the periodicity group associated with $\mu$
(Definition 1.5). By the factorization property $\Lambda$
has no sources, so $\Lambda ^{\leq (n,\ldots , n)}=\Lambda ^{(n,\ldots , n)}$
. Lemma 2.1(3) implies that $|\Lambda ^{(n,\ldots , n)}|=n$
. By Proposition 3.3, $\ell _\mu =\ell$
. Combining these results gives $C^{*}(\Lambda )\cong M_n(C(\mathbb {T}^{\ell }))$
, and $sr(C^{*}(\Lambda ))=\lceil \lfloor \ell /2\rfloor /n\rceil +1$
.
3. Stable rank one
In this section, we characterize which finite $k$
-graphs have $C^{*}$
-algebras of stable rank 1 – see Theorem 3.1 and Corollary 3.4. We note that Theorem 3.1 is in large contained in [Reference Evans and Sims14] and we have structured the proof accordingly.
Theorem 3.1 Let $\Lambda$
be a finite, locally convex $k$
-graph. Then $sr(C^{*}(\Lambda ))=1$
if and only if $C^{*}(\Lambda )$
is (stably) finite and $\max _{\mu \in IC(\Lambda )} \ell _{\mu }=1$
.
Proof. Suppose that $sr(C^{*}(\Lambda ))=1$
. Then $sr(M_n(C^{*}(\Lambda )))=1$
for each positive integer $n$
[Reference Rieffel34, Theorem 6.1]. Hence each $M_n(C^{*}(\Lambda ))$
is finite [Reference Blackadar2, V.3.1.5], which implies that $C^{*}(\Lambda )$
is stably finite. Since $C^{*}(\Lambda )$
is finite, it does not contain any infinite projections [Reference Rørdam, Larsen and Laustsen37, Lemma 5.1.2]. Hence by [Reference Evans and Sims14, Proposition 5.9], there exist $n \ge 1$
and $l_1,\, \ldots ,\, l_n \in \{0,\, \ldots ,\, k\}$
such that $C^{*}(\Lambda )$
is stably isomorphic to $\bigoplus ^{n}_{i=1} C(\mathbb {T}^{l_i})$
. Since $sr(C^{*}(\Lambda ))=1$
, we deduce that $sr(\bigoplus _{i=1}^{n} C(\mathbb {T}^{l_i}))=1$
, because stable rank 1 for unital $C^{*}$
-algebras is preserved my stable isomorphism [Reference Rieffel34, Theorem 3.6]. By property 4 in § 1.1, we have $sr(\bigoplus _{i=1}^{n} C(\mathbb {T}^{l_i}))=\max _{i=1}^{n} C(\mathbb {T}^{l_i})$
. For each $i=1,\,\ldots ,\,n$
we use [Reference Rieffel34, Proposition 1.7] to deduce that $sr(C(\mathbb {T}^{l_i}))=\lfloor l_i/2\rfloor +1$
, where $\lfloor \cdot \rfloor$
denotes ‘integer part of’. Hence $\max _{i=1}^{n} l_i=1$
.
By inspection of the proof of [Reference Evans and Sims14, Proposition 5.9], it is clear that each of the integers $l_i$
is the rank of $\mu$
for some $\mu \in IC(\Lambda )$
, so $\max _{\mu \in IC(\Lambda )} \ell _{\mu }\geq \max _{i=1}^{n} l_i=1$
. For each $\mu ,\,\nu \in IC(\Lambda )$
define $P_\nu {:=} \sum \nolimits _{v\in (\nu ^{\infty })^{0}}s_v$
and $\mu \sim \nu \Leftrightarrow (\mu ^{\infty })^{0}=(\nu ^{\infty })^{0}$
. Since $P_\nu =P_\mu$
whenever $\mu \sim \nu$
, the proof of [Reference Evans and Sims14, Proposition 5.9] implies that for each $\mu \in IC(\Lambda )$
, we have $\ell _\mu =l_i$
for some $i\in \{1,\,\ldots ,\,n\}$
. Consequently, $\max _{\mu \in IC(\Lambda )} \ell _{\mu }= \max _{i=1}^{n} l_i$
.
Conversely, suppose $C^{*}(\Lambda )$
is finite and $\max _{\mu \in IC(\Lambda )} \ell _{\mu }=1$
. By [Reference Rørdam, Larsen and Laustsen37, Lemma 5.1.2], $C^{*}(\Lambda )$
has no infinite projections. So [Reference Evans and Sims14, Corollary 5.7] implies that $C^{*}(\Lambda )$
is stably isomorphic to $\bigoplus _{i=1}^{n} C(\mathbb {T}^{l_i})$
for some $n\geq 1$
and $l_1,\,\ldots ,\,l_n\in \{0,\,\ldots ,\,k\}$
such that $\max _{\mu \in IC(\Lambda )} \ell _{\mu }= \max _{i=1}^{n} l_i$
. By the properties in § 1.1, it follows that
Remark 3.2 It turns out that $C^{*}$
-algebras of finite $k$
-graphs with $k>1$
rarely have stable rank one: the condition $\max _{\mu \in IC(\Lambda )} \ell _{\mu }=1$
is rather strict. As Proposition 3.3 indicates, if $\Lambda ^{0}$
is finite and $sr(C^{*}(\Lambda ))=1$
(hence stably finite), then any initial cycle in $\Lambda$
has at most one colour. Using Lemma 2.2(2) and the factorization property, it follows that any cycle in $\Lambda$
has at most one colour.
Figure 5 illustrates two examples of $2$
-graphs $\Lambda$
with $C^{*}(\Lambda )$
of stable rank one. The first example, illustrated on the left, has two vertices $v_1,\,v_2$
, a single edge red (dashed) loop based at $v_1$
, and single edge blue (solid) loop based at $v_2$
. The second example, shown on the right in Figure 5, is different in that it is connected and contains no loops.
Figure 5. Two examples of $2$
-graphs with a $C^{*}$
-algebra of stable rank one.
Following [Reference Evans and Sims14], for $n\in \mathbb {N}^{k}$
there is a shift map $\sigma ^{n}: \{x \in W_\Lambda : n\leq d(x) \} \to W_\Lambda$
such that $d(\sigma ^{n}(x))=d(x)-n$
and $\sigma ^{n}(x)(p,\,q)=x(n+p,\,n+q)$
for $0\leq p \leq q \leq d(x)-n$
where we use the convention $\infty -a=\infty$
for $a \in \mathbb {N}$
. For $x \in W_\Lambda$
and $n\leq d(x)$
, we then have $x(0,\,n)\sigma ^{n}(x) = x$
. We now show an easy way to compute $\ell _\mu$
, using only the degree of $\mu$
.
Proposition 3.3 Let $\Lambda$
be a finite, locally convex $k$
-graph such that $\Lambda$
has no cycle with an entrance. Then for each $\mu \in \operatorname {IC}(\Lambda ),$
Proof. Let $I=\{i\leq k: d(\mu )_i> 0\}$
. We must show that $\ell _\mu = | I |$
. If $I=\emptyset$
then $\ell _\mu = 0= | I |$
, so assume that $I$
is non-empty. By (1.1),
Since each $n,\,m\leq d(\mu ^{\infty })$
satisfy $n,\,m\in \operatorname {span}_{\mathbb {N}}\{e_i : i\in I\}$
, the rank of $G_\mu$
is at most $| I |$
. Consequently, it suffices to show $G_\mu$
contains a subgroup of rank $| I |$
.
Let $v{:=} \mu ^{\infty }(0)$
. We claim that for each colour $i\in I$
, there exists a positive integer $m_i$
such that $\mu ^{\infty }(0,\,m_ie_i)=v$
. Indeed, since $\Lambda ^{0}$
is finite there exists $m< n$
such that $\mu ^{\infty }(m e_i)=\mu ^{\infty }(n e_i)$
. Now using that $\mu ^{\infty }\in \Lambda ^{\leq \infty }$
(see Lemma 2.1) and that for every vertex $w$
on $\mu$
there is a unique path in $w\Lambda ^{\leq \infty }$
(see § 1.7) we get that $\sigma ^{m e_i}(\mu ^{\infty })=\sigma ^{n e_i}(\mu ^{\infty })$
. Now for $N{:=}md(\mu )$
it follows that $\sigma ^{N}(\mu ^{\infty })=\mu ^{\infty }$
. Since
we get $(\mu ^{\infty })((n-m)e_i)=v$
. Hence $\mu ^{\infty }$
contains a cycle of degree $(n-m)e_i$
based at $v$
. In particular, we can use $m_i{:=}n-m$
.
By the preceding paragraph, $\{m_ie_i : i\in I\}\subseteq G_\mu$
is a $\mathbb {Z}$
-linearly independent set generating a rank-$|I|$
subgroup of $G_\mu$
. So the rank of $G_\mu$
is $| I |$
.
Corollary 3.4 Let $\Lambda$
be a finite, locally convex $k$
-graph. Then $sr(C^{*}(\Lambda ))=1$
if and only if $\Lambda$
has no cycle with an entrance and no initial cycle with more than one colour.
Remark 3.5 A graph trace on a locally convex row-finite $k$
-graph $\Lambda$
is a function $g\colon \Lambda ^{0}\to \mathbb {R}^{+}$
satisfying the graph trace property, $g(v)=\sum \nolimits _{\lambda \in v\Lambda ^{\leq n}} g(s(\lambda ))$
for all $v\in \Lambda ^{0}$
and $n\in \mathbb {N}^{k}$
. It is faithful if it is non-zero on every vertex in $\Lambda$
[Reference Pask, Rennie and Sims27, Reference Tomforde45].
It can be shown that Corollary 3.4 remains valid if we replace ‘has no cycle with an entrance’ by ‘admits a faithful graph trace’. Indeed, the $C^{*}$
-algebra of a row-finite and cofinal $k$
-graph $\Lambda$
with no sources is stably finite if and only if $\Lambda$
admits a faithful graph trace [Reference Clark, an Huef and Sims7, Theorem 1.1], and for $\Lambda ^{0}$
finite, this remains true without ‘cofinal’ and with ‘locally convex’ instead of ‘no sources’ (by virtue of Theorem 2.5 and [Reference Pask, Sierakowski and Sims28, Lemma 7.1]).
4. Stable rank in the simple and cofinal case
In this section, we focus on stable rank of $k$
-graph $C^{*}$
-algebras for which the $k$
-graph is cofinal, corresponding to boxes 1 and 3 in Figure 1. Since simple $k$
-graph $C^{*}$
-algebras constitute a sub-case of this situation (as illustrated below), we consider those first.
Let $\Lambda$
be a row-finite, locally convex $k$
-graph. Following [Reference Sims42], $\Lambda$
is cofinal if for all pairs $v,\,w\in \Lambda ^{0}$
there exists $n\in \mathbb {N}^{k}$
such that $s(w\Lambda ^{\leq n})\subseteq s(v\Lambda )$
. Following [Reference Robertson and Sims35], $\Lambda$
has local periodicity $m,\,n$
at $v$
if for every $x\in v\Lambda ^{\leq \infty }$
, we have $m - (m \wedge d(x)) = n - (n \wedge d(x))$
and $\sigma ^{m \wedge d(x)}(x) =\sigma ^{n \wedge d(x)}(x)$
. If $\Lambda$
fails to have local periodicity $m,\,n$
at $v$
for all $m\neq n\in \mathbb {N}^{k}$
and $v\in \Lambda ^{0}$
, we say that $\Lambda$
has no local periodicity. By [Reference Robertson and Sims35, Theorem 3.4],
The stable rank of $1$
-graph $C^{*}$
-algebras is well understood (see [Reference Deicke, Hong and Szymański10, Theorem 3.4], [Reference Jeong, Park and Shin18, Theorem 3.3] and [Reference Jeong17, Theorem 3.1]), but the following is new for $k>1$
. Recall that a cycle is a path $\lambda \in \Lambda \setminus \Lambda ^{0}$
such that $r(\lambda ) = s(\lambda )$
.
Proposition 4.1 Let $\Lambda$
be a finite, locally convex $k$
-graph. Suppose that $\Lambda$
is cofinal and has no local periodicity (i.e., $C^{*}(\Lambda )$
is simple). Then
Proof. If $\Lambda$
contains no cycles then [Reference Evans and Sims14, Corollary 5.7] gives $C^{*}(\Lambda )\cong M_{\Lambda v}(\mathbb {C})$
for some vertex $v \in \Lambda ^{0}$
. Using [Reference Rieffel34, Proposition 1.7 and Theorem 3.6], we obtain that $sr(C^{*}(\Lambda ))=1$
.
If $\Lambda$
contains a cycle, then another application of [Reference Evans and Sims14, Corollary 5.7] (see also [Reference Brown, Clark and an Huef4, Remark 5.8]) gives that $C^{*}(\Lambda )$
is purely infinite. Since $C^{*}(\Lambda )$
is unital, simple and purely infinite, it contains two isometries with orthogonal ranges, so [Reference Rieffel34, Proposition 6.5] gives $sr(C^{*}(\Lambda ))=\infty$
.
In conclusion, the stable rank of a unital simple $k$
-graph $C^{*}$
-algebra is completely determined by the presence or absence of a cycle in the $k$
-graph.
4.1. The cofinal case
We now consider the cofinal case. We start by recalling a result of Jeong, Park and Shin about directed graphs (or $1$
-graphs). We refer to [Reference Jeong, Park and Shin18] for the terminology involved.
Proposition 4.2 Jeong et al. [Reference Jeong, Park and Shin18, Proposition 3.7]
Let $E$
be a locally finite directed graph. If $E$
is cofinal then either $sr(C^{*}(E))= 1$
or $C^{*}(E)$
is purely infinite simple.
Remark 4.3 We illustrate why for $k$
-graphs we can not hope for a result similar to Proposition 4.2. Consider the $2$
-graph $\Lambda$
in Figure 6 with two blue edges $a,\,b\in \Lambda ^{e_1}$
and one red edge $e\in \Lambda ^{e_2}$
and the factorization property $ae=ea$
, $be=eb$
. Since $\Lambda$
has only one vertex, it is automatically cofinal. However, $C^{*}(\Lambda )$
neither has stable rank one nor is purely infinite simple as the following discussion shows:
Figure 6. Example of $2$
-graph with a $C^{*}$
-algebra of stable rank infinity.
The $C^{*}$
-algebra $C^{*}(\Lambda )$
fails to have stable rank one because it is not stably finite (containing a cycle with an entrance). It is not simple, so, in particular, not purely infinite simple because
so $\Lambda$
has local periodicity $p=e_2,\, q=0$
at $v$
and $C^{*}(\Lambda )$
is non-simple.
Because of our particular choice of factorization rules $ae=ea$
, $be=eb$
Lemma 1.7 implies that $C^{*}(\Lambda )\cong \mathcal {O}_2\otimes C(\mathbb {T})$
. If we instead used the factorization $ae=eb$
, $be=ea$
, then Lemma 1.7 would not apply but we would still have $\sigma ^{2e_2}(x)=x$
for each $x\in v\Lambda ^{\leq \infty }$
making $C^{*}(\Lambda )$
non-simple.
Remark 4.3 notwithstanding, we are able to provide a characterization of stable rank in the cofinal case. Given a $C^{*}$
-algebra $A$
, we write $a\oplus b$
for the diagonal matrix $\textrm {diag}(a,\,b)$
in $M_2(A)$
and write $\sim$
for the von Neumann equivalence relation between elements in matrix algebras over $A$
. A unital $C^{*}$
-algebra $A$
is properly infinite if $1\oplus 1\oplus r \sim 1$
for some projection $r$
in some matrix algebra over $A$
(for more details, see [Reference Rørdam and Størmer38]).
Theorem 4.4 Let $\Lambda$
be a cofinal, finite, locally convex $k$
-graph. Suppose that $\Lambda$
contains a cycle with an entrance. Then $C^{*}(\Lambda )$
is properly infinite and has stable rank $\infty$
.
Proof. Let $\mu$
be a cycle with an entrance $\tau$
, that is
Fix $n\geq 1$
such that $m:=d(\mu )n\geq d(\tau )$
. Since $\tau \neq \mu ^{\infty }(0,\,d(\tau ))=\mu ^{n}(0,\,d(\tau ))$
, there exists $\beta \in s(\tau )\Lambda$
such that $\mu ^{n}$
and $\tau \beta$
are distinct elements of $r(\mu )\Lambda ^{\leq m}$
. Write $r(\mu )\Lambda ^{{\leq m}}=\{\nu _1,\, \ldots ,\, \nu _N\}$
with $\nu _1=\mu ^{n}$
and $\nu _2=\tau \beta$
. For each $i=1,\,\ldots ,\,N$
set $v_i=s(\nu _i)$
and let $x=(s_{\nu _1},\, \ldots ,\, s_{\nu _N})$
. Then $xx^{*}= \sum \nolimits _{\lambda \in {v_1}\Lambda ^{{\leq m}}} s_\lambda s_\lambda ^{*}=s_{v_1}$
. Moreover for $i\neq j$
, $s_{\nu _i}^{*}s_{\nu _j}=0$
, so
We claim that for any pair of vertices $u,\,v \in \Lambda ^{0}$
there exist a constant $M_{u,v}$
and a projection $p_{u,v}$
in some matrix algebra over $C^{*}(\Lambda )$
such that
To see this, fix $u,\,v\in \Lambda ^{0}$
. Since $\Lambda$
is cofinal there exists $n\in \mathbb {N}^{k}$
such that $s(v\Lambda ^{\leq n})\subseteq s(u\Lambda )$
. Writing $v\Lambda ^{\leq n}=\{\mu _1,\, \ldots ,\, \mu _{M_{u,v}}\}$
and $u_i=s(\mu _i)$
, we have $s_{v}\sim s_{u_1}\oplus \cdots \oplus s_{u_{{M_{u,v}}}}$
. Since $s(v\Lambda ^{\leq n})=\{u_i: i\leq {M_{u,v}}\}\subseteq s(u\Lambda )$
, for each $i\leq M_{u,v}$
there exists $\lambda _i\in u\Lambda$
such that $s(\lambda _i)=u_i$
. Let $m_i=d(\lambda _i)$
for each $i$
. Then for each $i=1,\,\ldots ,\,{M_{u,v}}$
,
for some projection $p_i$
in a matrix algebra over $C^{*}(\Lambda )$
. With $p_{u,v}=\bigoplus _i p_i$
, we obtain
which establishes the claim.
Applying (4.2) to $u=v_2$
and $v=v_1$
, we get
Recall that $s_{v_1}\sim s_{v_1}\oplus s_{v_2}\oplus (\bigoplus _{i=3}^{N} s_{v_i})$
. Let $q{:=}p_{v_2,v_1}\oplus (\bigoplus _{l=1}^{M_{v_2,v_1}}\bigoplus _{i=3}^{N} s_{v_i})$
, meaning that if $N=2$
then $q= p_{v_2,v_1} \oplus 0$
. Then
Applying (4.2) to $u=v_1$
and to each $v\in \Lambda ^{0}\setminus \{v_1\}$
at the second equality, and putting $L := 2 + \sum \nolimits _{v \in \Lambda ^{0}\setminus \{v_1\}} M_{v_1,v}$
, we calculate:
Using (4.3), we have $s_{v_1}\sim s_{v_1}\oplus (\bigoplus _{j=1}^{L-1} s_{v_1}) \oplus (\bigoplus _{j=1}^{L-1} q)$
, so $r= (\bigoplus _{v\in \Lambda ^{0}\setminus \{v_1\}} p_{v_1,v} ) \oplus (\bigoplus _{j=1}^{L-1} q)$
satisfies
Hence $1$
is properly infinite. Now [Reference Rieffel34, Proposition 6.5] gives $sr(C^{*}(\Lambda ))=\infty$
.
With Proposition 4.2 in mind, the following is a dichotomy for the $C^{*}$
-algebras associated with cofinal finite $k$
-graphs.
Corollary 4.5 Let $\Lambda$
be a cofinal, finite, locally convex $k$
-graph. Then either $C^{*}(\Lambda )$
is stably finite and $sr(C^{*}(\Lambda ))$
is given by Corollary 2.9, or $C^{*}(\Lambda )$
is properly infinite and $sr(C^{*}(\Lambda ))=\infty$
.
Proof. If $C^{*}(\Lambda )$
is not stably finite then $\Lambda$
contains a cycle with an entrance by Proposition 2.7. Hence $C^{*}(\Lambda )$
is properly infinite and $sr(C^{*}(\Lambda ))=\infty$
by Theorem 4.4. Conversely, if $C^{*}(\Lambda )$
is properly infinite, then it is also infinite, so $C^{*}(\Lambda )$
is not finite and hence not stably finite. If $C^{*}(\Lambda )$
is stably finite then Corollary 2.9 applies.
Remark 4.6 Theorem 4.4 includes cases not covered by any of our preceding results. Consider, for example, the $2$
-graph $\Lambda$
in Figure 6. By Theorem 4.4, the associated $C^{*}$
-algebra has stable rank infinity.
Example 4.7 By Corollary 4.5, we can compute the stable rank of $k$
-graph $C^{*}$
-algebras in boxes 1 to 3 in Figure 1. It, therefore, makes sense to consider the range of stable rank achieve by these $C^{*}$
-algebra. In box 3, stable rank infinity can be obtained as in Remark 4.3. For finite stable rank, Table 1 lists a few $4n$
-graphs $\Lambda$
together with their associated $C^{*}$
-algebra and its stable rank (we use a multiple of 4 because it makes the formulas for the stable rank simpler).
Table 1. A few examples of $4n$
-graphs.
Except for the last $4n$
-graph, each black edge represents exactly $4n$
edges of different colours, one of each colour; the last $4n$
-graph has $2n$
loops at $v_2$
, one each of the first $2n$
colours and $2n$
edges from $v_1$
to $v_2$
, one each of the remaining $2n$
colours. Each example admits a unique factorization rule, so each illustration in Table 1 represents a unique $4n$
-graph.
5. Stable rank in the non-stably finite, non-cofinal case
So far we have looked at the stably finite case (including stable rank one) and the cofinal case (including the simple case). Here, we study the remaining case corresponding to box 4 in Figure 1.
We start by revisiting the cofinality condition for row-finite locally convex $k$
-graphs. Following [Reference Robertson and Sims35], a subset $H\subseteq \Lambda ^{0}$
is hereditary if $s(H\Lambda )\subseteq H$
. We say $H$
is saturated if for all $v\in \Lambda ^{0}$
,
or equivalently, if $v\not \in H$
implies that for each $n\in \mathbb {N}^{k}$
, $s(v\Lambda ^{\leq n})\not \subseteq H$
(see Lemma 5.1). The relevant characterization of cofinal is included in Lemma 5.2 below with a short proof based on [Reference Lewin and Sims23, Reference Sims42]. Since this paper focuses on unital $k$
-graph $C^{*}$
-algebras, it is worth pointing out that Lemmas 5.1 and 5.2 do not assume that $|\Lambda ^{0}|<\infty$
.
Lemma 5.1 Let $\Lambda$
be a row-finite locally convex $k$
-graph. Then $H\subseteq \Lambda ^{0}$
is saturated if and only if for all $v\in \Lambda ^{0},$
$v\not \in H$
implies that for each $n\in \mathbb {N}^{k},$
$s(v\Lambda ^{\leq n})\not \subseteq H$
.
Proof. Fix $v\in \Lambda ^{0}$
. Suppose $v\not \in H$
. Since $\Lambda$
is saturated, for all $i\leq k$
, $\{s(\lambda ):\lambda \in v\Lambda ^{\leq e_i}\}\not \subseteq H$
. Clearly $s(v\Lambda ^{\leq m})\not \subseteq H$
for $m=0$
. Fix any $m\in \mathbb {N}^{k}\setminus \{0\}$
. Set $(n^{(0)},\,v^{(0)},\,\lambda ^{(0)})=(m,\,v,\,v)$
. Choose $i$
such that $n^{(0)}_i\neq 0$
. Since $v^{(0)}\not \in H$
, there exists $\mu ^{(1)}\in v^{(0)}\Lambda ^{\leq e_i}\setminus \Lambda H$
. Set $(n^{(1)},\,v^{(1)},\,\lambda ^{(1)})=(n^{(0)}-e_i,\,s(\mu ^{(1)}),\,\lambda ^{(0)}\mu ^{(1)})$
. Choose $i$
such that $n^{(1)}_i\neq 0$
. Since $v^{(1)}\not \in H$
, there exists $\mu ^{(2)}\in v^{(1)}\Lambda ^{\leq e_i}\setminus \Lambda H$
. Set $(n^{(2)},\,v^{(2)},\,\lambda ^{(2)})=(n^{(1)}-e_i,\,s(\mu ^{(2)}),\,\lambda ^{(1)}\mu ^{(2)})$
.
For each step, $|n^{(i)}|=|m|-i$
, so $l=|m|$
satisfies $n^{(l)}=0$
. Notice that $\lambda ^{(0)}\in v\Lambda ^{\leq (m-n^{(0)})},\, \lambda ^{(1)}\in v\Lambda ^{\leq (m-n^{(1)})},\, \ldots ,\, \lambda ^{(l)}\in v\Lambda ^{\leq (m-n^{(l)})}$
. Hence $\lambda ^{(l)}\in v\Lambda ^{\leq m}$
and $s(\lambda ^{(l)})\not \in H$
so $s(v\Lambda ^{\leq m})\not \subseteq H$
.
Lemma 5.2 [Reference Lewin and Sims23, Reference Sims42]
Let $\Lambda$
be a row-finite locally convex $k$
-graph. Then the following are equivalent:
(1) $\Lambda$
is cofinal;(2) for all $v\in \Lambda ^{0}$
, and $(\lambda _i)$ with $\lambda _i\in \Lambda ^{\leq (1,\ldots , 1)}$, and $s(\lambda _i)=r(\lambda _{i+1})$ there exist $i\in \mathbb {N}$ and $n\leq d(\lambda _i)$ such that $v\Lambda \lambda _i(n) \neq \emptyset ;$ and(3) $\Lambda ^{0}$
contains no non-trivial hereditary saturated subsets.
Proof. Firstly, we show that (1)$\Rightarrow$
(3). Suppose (1) and suppose that $H\subseteq \Lambda ^{0}$
is a non-empty hereditary, saturated set. We show that $H=\Lambda ^{0}$
. Fix $v\in \Lambda ^{0}$
. Since $H$
is non-empty, there exists $w\in H$
. By (1) there exists $n\in \mathbb {N}^{k}$
such that $s(v\Lambda ^{\leq n})\subseteq s(w\Lambda )$
. Since $H$
is hereditary, $s(v\Lambda ^{\leq n})\subseteq s(H\Lambda ) \subseteq H$
. Hence Lemma 5.1 gives $v\in H$
.
Now we show that (3)$\Rightarrow$
(2). Suppose that (2) fails, that is, there exist $v\in \Lambda ^{0}$
, and a sequence $(\lambda _i)$
with $\lambda _i\in \Lambda ^{\leq (1,\ldots , 1)},\, s(\lambda _i)=r(\lambda _{i+1})$
for all $i$
such that for all $i\in \mathbb {N}$
and all $n\leq d(\lambda _i)$
, we have $v\Lambda \lambda _i(n) = \emptyset$
. Let
Then $H$
is non-trivial as $v\in H$
and hereditary because if $u\Lambda w\neq \emptyset$
then $s(w\Lambda )\subseteq s(u\Lambda )$
. To show that $H$
is saturated take $u \in \Lambda ^{0}$
and $j\leq k$
such that $s(u{\Lambda }^{\le e_j})\subseteq H$
. We must show that $u\in H$
. Assume otherwise for contradiction. We have $u\not \in s(u{\Lambda }^{\le e_j})$
because otherwise $u=s(u)$
belongs to $H$
, so $u{\Lambda }^{e_j}\neq \emptyset$
. Since $u\not \in H$
, there exists $\lambda \in u\Lambda$
such that $s(\lambda )=\lambda _i(n)$
for some $i,\,n$
. We claim that $d(\lambda )_j=0$
. Indeed, if not, then $\lambda =\mu \mu '$
for some $\mu \in u{\Lambda }^{e_j}$
. We then have $s(\mu )\in s(u{\Lambda }^{e_j})\subseteq H$
, so $s(\mu )\Lambda \lambda _i(n) = \emptyset$
contradicting $\mu '\in s(\mu )\Lambda \lambda _i(n)$
. Since $\Lambda$
is locally convex and $d(\lambda )_j=0$
and since $u\Lambda ^{e_j}\neq \emptyset$
, we have $\lambda _i(n)\Lambda ^{e_j}=s(\lambda )\Lambda ^{e_j}\neq \emptyset$
. Let $\beta =\lambda _i(n,\, d(\lambda _i))$
. Since $\Lambda$
is locally convex, either $d(\beta )_j\neq 0$
or $s(\lambda _i)\Lambda ^{e_j}\neq \emptyset$
. Since $\lambda _{i+1}\in \Lambda ^{\leq (1,\ldots ,1)}$
it follows that $d(\beta \lambda _{i+1})\geq e_j$
. Now $\lambda ':=\lambda \beta \lambda _{i+1} \in u\Lambda$
satisfies $s(\lambda ')=\lambda _{i'}({n'})$
for some ${i'},\,{n'}$
. But then, just as we got $d(\lambda )_j=0$
, we deduce $d(\lambda ')_j=0$
, a contradiction. So $H$
is saturated, so (3) does not hold.
Finally, we prove (2)$\Rightarrow$
(1). Given (2), we suppose that (1) fails, and we derive a contradiction. Since (1) fails, there exist $v,\,w\in \Lambda ^{0}$
such that for all $n\in \mathbb {N}^{k}$
, we have $s(w\Lambda ^{\leq n})\not \subseteq s(v\Lambda )$
. Set
Fix $u\in K$
and $j\leq k$
. We claim that there exists $\mu \in u\Lambda ^{\leq e_j}$
such that $s(\mu )\in K$
. Indeed if $s(u\Lambda ^{\leq e_j})\subseteq \Lambda ^{0}\setminus K$
, then for each $\mu \in u\Lambda ^{\leq e_j}$
there exists $n_\mu \in \mathbb {N}^{k}$
such that $s(\mu \Lambda ^{n_\mu })\subseteq s(v\Lambda )$
. Since $s(v\Lambda )$
is hereditary, it follows that $n=\bigvee _{\mu \in u\Lambda ^{\leq e_j}} n_\mu$
satisfies $s(u\Lambda ^{\leq {n+e_j}})=\bigcup _{\mu \in u\Lambda ^{\leq e_j}}s(\mu \Lambda ^{\leq {n}}) \subseteq s(v\Lambda )$
, contradicting $u\in K$
.
Since $w\in K$
, we can construct a sequence $(\lambda _i)$
such that each $\lambda _i\in \Lambda ^{\leq (1,\ldots , 1)}$
, each $s(\lambda _i)=r(\lambda _{i+1})$
, and for each $n\leq d(\lambda _i)$
, we have $\lambda _i(n)\in K$
. By (2), there exist $i$
and $n\leq d(\lambda _i)$
such that $v\Lambda \lambda _i(n) \neq \emptyset$
, i.e., such that $s(\lambda _i(n)\Lambda ^{\leq 0})\subseteq s(v\Lambda )$
. So $\lambda _i(n)\not \in K$
, a contradiction.
Remark 5.3 When a $k$
-graph $\Lambda$
has only one vertex, it is automatically cofinal, and we deduce that the stable rank of $C^{*}(\Lambda )$
is infinite if there exists $j \le k$
such that $|\Lambda ^{e_j}| \ge 2$
, and is equal to $\lfloor k/2\rfloor + 1$
if each $|\Lambda ^{e_j}| = 1$
.
Remark 5.4 We now present all the $2$
-graphs $\Lambda$
with $|\Lambda ^{0}|=2$
for which we have been unable to compute the stable rank of the associated $C^{*}$
-algebra $C^{*}(\Lambda )$
(see Figure 7). In each case, the $2$
-graph $\Lambda$
fails to be cofinal, because $\Lambda ^{0}$
contains one non-trivial hereditary saturated subset, denoted $H$
.
Figure 7. Example of $2$
-graphs $\Lambda$
with $C^{*}(\Lambda )$
of stable rank two or three.
In Figure 7, for each 2-graph $\Lambda$
the $C^{*}$
-algebra $C^{*}(\Lambda )$
is non-simple with $H=\{u\}$
. In the first case, we have $C^{*}(H\Lambda ) \cong C(\mathbb {T})$
, which has stable rank 1, and so $I_H$
has stable rank 1 because stable rank 1 is preserved by stable isomorphism. In the remaining two cases, if there is one loop of each colour at $u$
then $C^{*}(H\Lambda ) \cong C(\mathbb {T}^{2})$
has stable rank 2, and otherwise, Theorem 4.4 implies that $C^{*}(H\Lambda )$
has stable rank $\infty$
; either way, since $I_H \cong C^{*}(H\Lambda ) \otimes \mathcal {K}$
Footnote 4, we have $sr(I_H) = 2$
as discussed in § 1.1.
In all three cases, the quotient of $C^{*}(\Lambda )$
by $I_H$
is $C^{*}(\Lambda )/I_H\cong C^{*}(\Lambda \setminus \Lambda H)\cong C(\mathbb {T}^{2})$
. Hence, by [Reference Blackadar2, V.3.1.21], we deduce that $sr(C^{*}(\Lambda ))\in \{2,\,3\}$
, but we have been unable to determine the exact value in any of these cases.
Perhaps the easiest-looking case is the $2$
-graph (top left) with one red (dashed) edge from $u$
to $v$
. In this case $C^{*}(\Lambda )\cong \mathcal {T}\otimes C(\mathbb {T})$
, where $\mathcal {T}$
is the Toeplitz algebra generated by the unilateral shift. Despite knowing the stable rank of each of the components ($sr(\mathcal {T})=2$
and $sr(C(\mathbb {T}))=1$
), the stable rank of the tensor product is not known (there is no general formula for stable rank of tensor products).
Acknowledgements
This research was supported by Australian Research Council grant DP180100595.
