k-graph C*-algebra

In mathematics, a k-graph (or higher-rank graph, graph of rank k) is a countable category $\mathrm{\Lambda }$ with domain and codomain maps $r$ and $s$, together with a functor $d:\mathrm{\Lambda }\to {\mathbb{\mathbb{N}}}^k$ which satisfies the following factorisation property: if $d(\lambda )=m+n$ then there are unique $\mu ,\nu \in \mathrm{\Lambda }$ 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 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, k-graph is just a regular 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*-algebra 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 graph theory perspective, yet just complicated enough to reveal different interesting properties in 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 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 $\mathbb{\mathbb{N}}$. 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 $d:T^k\to {\mathbb{\mathbb{N}}}^k$ defined $d(f_1^{n_1}...f_k^{n_k})=(n_1,\dots ,n_k)$, is a k-graph.
• Let ${\mathrm{\Omega }}_k=\{(m,n):m,n\in {\mathbb{\mathbb{Z}}}^k,m\le n\}$ then ${\mathrm{\Omega }}_k$ 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 $n\in {\mathbb{\mathbb{N}}}^k$ let ${\mathrm{\Lambda }}^n=d^{-1}(n)$.
• By the factorisation property it follows that ${\mathrm{\Lambda }}^0=\mathrm{Obj}(\mathrm{\Lambda })$.
• 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 $n\in {\mathbb{\mathbb{N}}}^k$ then $\mathrm{\Lambda }$ 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 $E^0={\mathrm{\Lambda }}^0$, $E^1=\underset{i=1}{\overset{k}{\cup }}{\mathrm{\Lambda }}^{e_i}$, $r,s$ inherited from $\mathrm{\Lambda }$ and $c:E^1\to \{1,\dots ,k\}$ defined by $c(e)=i$ if and only if $e\in {\mathrm{\Lambda }}^{e_i}$ where $e_1,\dots ,e_n$ are the canonical generators for ${\mathbb{\mathbb{N}}}^k$. The factorisation property in $\mathrm{\Lambda }$ for elements of degree $e_i+e_j$ where $i\ne j$ 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 $\mathrm{\Lambda }$ be a row-finite k-graph with no sources then a Cuntz–Krieger $\mathrm{\Lambda }$ family in a C*-algebra B is a collection $\{s_{\lambda }:\lambda \in \mathrm{\Lambda }\}$ of operators in B such that

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

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

References

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