k-graph C*-algebra

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

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 [math]\displaystyle{ E }[/math] gives a functor from this category into the natural numbers [math]\displaystyle{ \mathbb{N} }[/math]. 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 [math]\displaystyle{ T^k }[/math] consisting of a single object and k commuting morphisms [math]\displaystyle{ {f_1,...,f_k} }[/math], together with the map [math]\displaystyle{ d:T^k\to\mathbb{N}^k }[/math] defined [math]\displaystyle{ d(f_1^{n_1}...f_k^{n_k})=(n_1 , \ldots , n_k) }[/math], is a k-graph.
• Let [math]\displaystyle{ \Omega_k = \{ (m,n) : m,n \in \mathbb{Z}^k , m \le n \} }[/math] then [math]\displaystyle{ \Omega_k }[/math] is a k-graph when gifted with the structure maps [math]\displaystyle{ r(m,n)=(m,m) }[/math], [math]\displaystyle{ s(m,n)=(n,n) }[/math], [math]\displaystyle{ (m,n)(n,p)=(m,p) }[/math] and [math]\displaystyle{ d(m,n) = n-m }[/math].

Notation

The notation for k-graphs is borrowed extensively from the corresponding notation for categories:

• For [math]\displaystyle{ n \in \mathbb{N}^k }[/math] let [math]\displaystyle{ \Lambda^n = d^{-1} (n) }[/math].
• By the factorisation property it follows that [math]\displaystyle{ \Lambda^0 = \operatorname{Obj} ( \Lambda ) }[/math].
• For [math]\displaystyle{ v,w \in \Lambda^0 }[/math] and [math]\displaystyle{ X \subseteq \Lambda }[/math] we have [math]\displaystyle{ v X = \{ \lambda \in X : r ( \lambda ) = v \} }[/math], [math]\displaystyle{ X w = \{ \lambda \in X : s ( \lambda ) = w \} }[/math] and [math]\displaystyle{ v X w = v X \cap X w }[/math].
• If [math]\displaystyle{ 0 \lt \# v \Lambda^n \lt \infty }[/math] for all [math]\displaystyle{ v \in \Lambda^0 }[/math] and [math]\displaystyle{ n \in \mathbb{N}^k }[/math] then [math]\displaystyle{ \Lambda }[/math] 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 [math]\displaystyle{ E=(E^0,E^1,r,s,c) }[/math] where [math]\displaystyle{ E^0 = \Lambda^0 }[/math], [math]\displaystyle{ E^1 = \cup_{i=1}^k \Lambda^{e_i} }[/math], [math]\displaystyle{ r,s }[/math] inherited from [math]\displaystyle{ \Lambda }[/math] and [math]\displaystyle{ c: E^1 \to \{ 1 , \ldots , k \} }[/math] defined by [math]\displaystyle{ c (e) = i }[/math] if and only if [math]\displaystyle{ e \in \Lambda^{e_i} }[/math] where [math]\displaystyle{ e_1 , \ldots , e_n }[/math] are the canonical generators for [math]\displaystyle{ \mathbb{N}^k }[/math]. The factorisation property in [math]\displaystyle{ \Lambda }[/math] for elements of degree [math]\displaystyle{ e_i+e_j }[/math] where [math]\displaystyle{ i \neq j }[/math] gives rise to relations between the edges of [math]\displaystyle{ E }[/math].

C*-algebra

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

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

1. [math]\displaystyle{ s_\lambda s_\mu = s_{\lambda \mu} }[/math] if [math]\displaystyle{ \lambda , \mu , \lambda \mu \in \Lambda }[/math];
2. [math]\displaystyle{ \{ s_v : v \in \Lambda^0 \} }[/math] are mutually orthogonal projections;
3. if [math]\displaystyle{ d ( \mu ) = d ( \nu ) }[/math] then [math]\displaystyle{ s_\mu^* s_\nu = \delta_{\mu , \nu} s_{s ( \mu )} }[/math];
4. [math]\displaystyle{ s_v = \sum_{\lambda \in v \Lambda^n} s_\lambda s_\lambda^* }[/math] for all [math]\displaystyle{ n \in \mathbb{N}^k }[/math] and [math]\displaystyle{ v \in \Lambda^0 }[/math].

[math]\displaystyle{ C^* ( \Lambda ) }[/math] is then the universal C*-algebra generated by a Cuntz–Krieger [math]\displaystyle{ \Lambda }[/math]-family.

References

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