k-graph C*-algebra

In mathematics, for $k\in \mathbb {N}$ , a $k$ -graph (also known as a higher-rank graph or graph of rank $k$ ) is a countable category $\Lambda$ together with a functor $d:\Lambda \to \mathbb {N} ^{k}$ , called the degree map, which satisfy the following factorization property:

if $\lambda \in \Lambda$ and $m,n\in \mathbb {N} ^{k}$ are such that $d(\lambda )=m+n$ , then there exist unique $\mu ,\nu \in \Lambda$ such that $d(\mu )=m$ , $d(\nu )=n$ , and $\lambda =\mu \nu$ .

An immediate consequence of the factorization property is that morphisms in a $k$ -graph can be factored in multiple ways: there are also unique $\mu ',\nu '\in \Lambda$ such that $d(\mu ')=m$ , $d(\nu ')=n$ , and $\mu \nu =\lambda =\nu '\mu '$ .

A 1-graph is just the path category of a directed graph. In this case the degree map takes a path to its length. By extension, $k$ -graphs can be considered higher-dimensional analogs of directed graphs.

Another way to think about a $k$ -graph is as a $k$ -colored directed graph together with additional information to record the factorization property. The $k$ -colored graph underlying a $k$ -graph is referred to as its skeleton. Two $k$ -graphs can have the same skeleton but different factorization rules.

Kumjian and Pask originally introduced $k$ -graphs as a generalization of a construction of Robertson and Steger.^[1] By considering representations of $k$ -graphs as bounded operators on Hilbert space, they have since become a tool for constructing interesting C*-algebras whose structure reflects the factorization rules. Some compact quantum groups like $SU_{q}(3)$ can be realised as the $C^{*}$ -algebras of $k$ -graphs.^[2] There is also a close relationship between $k$ -graphs and strict factorization systems in category theory.

Notation

The notation for $k$ -graphs is borrowed extensively from the corresponding notation for categories:

For $n\in \mathbb {N} ^{k}$ let $\Lambda ^{n}=d^{-1}(n)$ . By the factorisation property it follows that $\Lambda ^{0}=\operatorname {Obj} (\Lambda )$ .

There are maps $s:\Lambda \to \Lambda ^{0}$ and $r:\Lambda \to \Lambda ^{0}$ which take a morphism $\lambda \in \Lambda$ to its source $s(\lambda )$ and its range $r(\lambda )$ .

For $v,w\in \Lambda ^{0}$ and $X\subseteq \Lambda$ we have $vX=\{\lambda \in X:r(\lambda )=v\}$ , $Xw=\{\lambda \in X:s(\lambda )=w\}$ and $vXw=vX\cap Xw$ .

If $0<\#v\Lambda ^{n}<\infty$ for all $v\in \Lambda ^{0}$ and $n\in \mathbb {N} ^{k}$ then $\Lambda$ is said to be row-finite with no sources.

Skeletons

A $k$ -graph $\Lambda$ can be visualized via its skeleton. Let $e_{1},\ldots ,e_{n}$ be the canonical generators for $\mathbb {N} ^{k}$ . The idea is to think of morphisms in $\Lambda ^{e_{i}}=d^{-1}(e_{i})$ as being edges in a directed graph of a color indexed by $i$ .

To be more precise, the skeleton of a $k$ -graph $\Lambda$ is a k-colored directed graph $E=(E^{0},E^{1},r,s,c)$ with vertices $E^{0}=\Lambda ^{0}$ , edges $E^{1}=\cup _{i=1}^{k}\Lambda ^{e_{i}}$ , range and source maps inherited from $\Lambda$ , and a color map $c:E^{1}\to \{1,\ldots ,k\}$ defined by $c(e)=i$ if and only if $e\in \Lambda ^{e_{i}}$ .

The skeleton of a $k$ -graph alone is not enough to recover the $k$ -graph. The extra information about factorization can be encoded in a complete and associative collection of commuting squares.^[3] In particular, for each $i\neq j$ and $e,f\in E^{1}$ with $c(e)=i$ and $c(f)=j$ , there must exist unique $e',f'\in E^{1}$ with $c(e')=i$ , $c(f')=j$ , and $ef=f'e'$ in $\Lambda$ . A different choice of commuting squares can yield a distinct $k$ -graph with the same skeleton.

Examples

A 1-graph is precisely the path category of a directed graph. If $\lambda$ is a path in the directed graph, then $d(\lambda )$ is its length. The factorization condition is trivial: if $\lambda$ is a path of length $m+n$ then let $\mu$ be the initial subpath of length $m$ and let $\nu$ be the final subpath of length $n$ .

The monoid $\mathbb {N} ^{k}$ can be considered as a category with one object. The identity on $\mathbb {N} ^{k}$ give a degree map making $\mathbb {N} ^{k}$ into a $k$ -graph.

Let $\Omega _{k}=\{(m,n):m,n\in \mathbb {Z} ^{k},m\leq n\}$ . Then $\Omega _{k}$ is a category with range map $r(m,n)=(m,m)$ , source map $s(m,n)=(n,n)$ , and composition $(m,n)(n,p)=(m,p)$ . Setting $d(m,n)=n-m$ gives a degree map. The factorization rule is given as follows: if $d(m,n)=p+q$ for some $p,q\in \mathbb {N} ^{k}$ , then $(m,n)=(m,m+q)(m+q,n)$ is the unique factorization.

C*-algebras of k-graphs

Just as a graph C*-algebra can be associated to a directed graph, a universal C*-algebra can be associated to a $k$ -graph.

Let $\Lambda$ be a row-finite $k$ -graph with no sources then a Cuntz–Krieger $\Lambda$ -family or a represenentaion of $\Lambda$ in a C*-algebra B is a map $S\colon \Lambda \to B$ such that

$\{S_{v}:v\in \Lambda ^{0}\}$ is a collection of mutually orthogonal projections;
$S_{\lambda }S_{\mu }=S_{\lambda \mu }$ for all $\lambda ,\mu \in \Lambda$ with $s(\lambda )=r(\mu )$ ;
$S_{\mu }^{*}S_{\mu }=S_{s(\mu )}$ for all $\mu \in \Lambda$ ; and
$S_{v}=\sum _{\lambda \in v\Lambda ^{n}}S_{\lambda }S_{\lambda }^{*}$ for all $n\in \mathbb {N} ^{k}$ and $v\in \Lambda ^{0}$ .