K-graph C*-algebra

Template:Lowercase title 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 $\mathrm{\Lambda }$ together with a functor $d:\mathrm{\Lambda }\to {\mathbb{N}}^k$, called the degree map, which satisfy the following factorization property:

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

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. 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 $S{U}_q(3)$ can be realised as the $C^{\ast }$-algebras of $k$-graphs. 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 ${\mathrm{\Lambda }}^n=d^{-1}(n)$. By the factorisation property it follows that ${\mathrm{\Lambda }}^0=\mathrm{Obj}(\mathrm{\Lambda })$.

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

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

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

Skeletons

A $k$-graph $\mathrm{\Lambda }$ can be visualized via its skeleton. Let $e_1,\dots ,e_n$ be the canonical generators for ${\mathbb{N}}^k$. The idea is to think of morphisms in ${\mathrm{\Lambda }}^{e_i}=d^{-1}(e_i)$ as being edges in a directed graph of a color indexed by $i$.

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

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. 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 $e^{\prime },f^{\prime }\in E^1$ with $c(e^{\prime })=i$, $c(f^{\prime })=j$, and $ef=f^{\prime }{e}^{\prime }$ in $\mathrm{\Lambda }$. 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 ${\mathrm{\Omega }}_k=\{(m,n):m,n\in {\mathbb{Z}}^k,m\le n\}$. Then ${\mathrm{\Omega }}_k$ 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 $\mathrm{\Lambda }$ be a row-finite $k$-graph with no sources then a Cuntz–Krieger $\mathrm{\Lambda }$-family or a represenentaion of $\mathrm{\Lambda }$ in a C*-algebra B is a map $S:\mathrm{\Lambda }\to B$ such that

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

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

See also

• Graph C*-algebra

References

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