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 $$\Omega_k = \{ (m,n) : m,n \in \mathbb{Z}^k, m \le n \}$$, 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 $$v X = \{ \lambda \in X : r ( \lambda ) = v \}$$, $$X w = \{ \lambda \in X : s ( \lambda ) = w \}$$ and $$ v X w = v X \cap X w$$.
 * 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.