k-graph C*-algebra

For C*-algebra 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 $\lambda =\mu \nu$.

Aside from its category theory definition, one can think of k-graphs as a higher-dimensional analogue of directed graphs (digraphs). k- here signifies the number of "colors" of edges that are involved in the graph. If k=1, a k-graph is just an ordinary directed graph. If k=2, there are two different colors of edges involved in the graph and additional factorization rules of 2-color equivalent classes should be defined. The factorization rule on k-graph skeleton is what distinguishes one k-graph defined on the same skeleton from another k-graph. k can be any natural number greater than or equal to 1.

The reason k-graphs were first introduced by Kumjian, Pask et al. was to create examples of C*-algebras from them. k-graphs consist of two parts: skeleton and factorization rules defined on the given skeleton. Once k-graph is well-defined, one can define functions called 2-cocycles on each graph, and C*-algebras can be built from k-graphs and 2-cocycles. k-graphs are relatively simple to understand from a graph theory perspective, yet just complicated enough to reveal different interesting properties at the C*-algebra level. The properties such as homotopy and cohomology on the 2-cocycles defined on k-graphs have implications to C*-algebra and K-theory research efforts. No other known use of k-graphs exist to this day; k-graphs are studied solely for the purpose of creating C*-algebras from them.

Background

The finite graph theory in a directed graph form a category under concatenation called the free object category (generated by the 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 generalisation of this concept which was introduced in 2000 by Alex Kumjian and David Pask.

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 by $d(f_{1}^{n_{1}}...f_{k}^{n_{k}})=(n_{1},\ldots ,n_{k})$ 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 visualized 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–Krieger $\Lambda$ family in a C*-algebra B is a collection $\{s_{\lambda }:\lambda \in \Lambda \}$ of operators in B such that

1. $s_{\lambda }s_{\mu }=s_{\lambda \mu }$ if $\lambda ,\mu ,\lambda \mu \in \Lambda$;
2. $\{s_{v}:v\in \Lambda ^{0}\}$ are mutually orthogonal projections;
3. if $d(\mu )=d(\nu )$ then $s_{\mu }^{*}s_{\nu }=\delta _{\mu ,\nu }s_{s(\mu )}$;
4. $s_{v}=\sum _{\lambda \in v\Lambda ^{n}}s_{\lambda }s_{\lambda }^{*}$ 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

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