K-graph C*-algebra

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Background

The finitegraph theory in a directed graph form a mathematics category under concatenation called the free object category (which is generated by a graph). The length of a path in $E$ gives a functor from this category into the natural numbers ${\mathbb {N}}$ . A k-graph is a natural generalistion of this concept which was introduced in^[1] by Alex Kumjian and David Pask.

Definition

In mathematics, a k-graph (or higher-rank graph, graph of rank k) is a countable category $\Lambda$ with domain and codomain maps $r$ and $s$ , together with a functor $d:\Lambda \to {\mathbb {N}}^{k}$ which satisfies the following factorisation property: if $d(\lambda )=m+n$ then there are unique $\mu ,\nu \in \Lambda$ with $d(\mu )=m,d(\nu )=n$ such that $\lambda =\mu \nu$ .

Examples

It can be shown that a 1-graph is precisely the path category of a directed graph.
The category $T^{k}$ consisting of a single object and k commuting morphisms ${f_{1},...,f_{k}}$ , together with the map $d:T^{k}\to {\mathbb {N}}^{k}$ defined $d(f_{1}^{{n_{1}}}...f_{k}^{{n_{k}}})=(n_{1},\ldots ,n_{k})$ , is a k-graph.
Let $\Omega _{k}=\{(m,n):m,n\in {\mathbb {Z}}^{k},m\leq n\}$ then $\Omega _{k}$ is a k-graph when gifted with the structure maps $r(m,n)=(m,m)$ , $s(m,n)=(n,n)$ , $(m,n)(n,p)=(m,p)$ and $d(m,n)=n-m$ .

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 $\Lambda ^{0}=\operatorname {Obj}(\Lambda )$ .
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.

Visualisation - Skeletons

A k-graph is best visualised by drawing its 1-skeleton as a k-coloured graph $E=(E^{0},E^{1},r,s,c)$ where $E^{0}=\Lambda ^{0}$ , $E^{1}=\cup _{{i=1}}^{k}\Lambda ^{{e_{i}}}$ , $r,s$ inherited from $\Lambda$ and $c:E^{1}\to \{1,\ldots ,k\}$ defined by $c(e)=i$ if and only if $e\in \Lambda ^{{e_{i}}}$ where $e_{1},\ldots ,e_{n}$ are the canonical generators for ${\mathbb {N}}^{k}$ . The factorisation property in $\Lambda$ for elements of degree $e_{i}+e_{j}$ where $i\neq j$ gives rise to relations between the edges of $E$ .

C*-algebra

As with graph-algebras one may associate a C*-algebra to a k=graph:

Let $\Lambda$ be a row-finite k-graph with no sources then a Cuntz–Kriger $\Lambda$ family in a C*-algebra B is a collection $\{s_{\lambda }:\lambda \in \Lambda \}$ of operators in B such that

$s_{\lambda }s_{\mu }=s_{{\lambda \mu }}$ if $\lambda ,\mu ,\lambda \mu \in \Lambda$ ;
$\{s_{v}:v\in \Lambda ^{0}\}$ are mutually orthogonal projections;
if $d(\mu )=d(\nu )$ then $s_{\mu }^{*}s_{\nu }=\delta _{{\mu ,\nu }}s_{{s(\mu )}}$ ;
$s_{v}=\sum _{{\lambda \in v\Lambda ^{n}}}s_{\lambda }s_{\lambda }^{*}$ for all $n\in {\mathbb {N}}^{k}$ and $v\in \Lambda ^{0}$ .

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

References

↑ Kumjian, A.; Pask, D.A. (2000), "Higher rank graph C*-algebras", The New York Journal of Mathematics 6: 1–20

Raeburn, I., Graph algebras, CBMS Regional Conference Series in Mathematics 103, American Mathematical Society

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