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.{{citation |first1=A.|last1=Kumjian |first2=D.A. |last2=Pask |title=Higher rank graph C*-algebras |journal=The New York Journal of Mathematics | volume=6 |pages=1–20 | year=2000 |url=http://nyjm.albany.edu/j/2000/6-1.html}} 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.{{cite arXiv |mode=cs2 |first1=O.|last1=Giselsson |title=Quantum SU(3) as the C*-algebra of a 2-Graph | year = 2023|class=math.OA |eprint=2307.12878}}

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 $v X = \{ \lambda \in X : r ( \lambda ) = v \}$ , $X w = \{ \lambda \in X : s ( \lambda ) = w \}$ and $v X w = v X \cap X w$ .

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.{{citation |first1=A.|last1=Sims |title=Lecture notes on higher-rank graphs and their C*-algebras|url=https://www.aidansims.com/files/k-graphNotes2010.pdf}} In particular, for each $i \ne j$ 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 $\Omega_k = \{ (m,n) : m,n \in \mathbb{Z}^k , m \le n \}$ . 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 $v \in \Lambda^0$ .

The algebra $C^* ( \Lambda )$ is the universal C*-algebra generated by a Cuntz–Krieger $\Lambda$ -family.

References

{{citation|title=Graph algebras |volume=103 |series=CBMS Regional Conference Series in Mathematics |first1=I. |last1=Raeburn |publisher=American Mathematical Society }}

Category:C*-algebras