K-graph C*-algebra

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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.


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{N}. A k-graph is a natural generalisation of this concept which was introduced in 2000 by Alex Kumjian and David Pask.[1]


  • 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 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.


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[edit]

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 = \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.


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.


  1. ^ Kumjian, A.; Pask, D.A. (2000), "Higher rank graph C*-algebras", The New York Journal of Mathematics 6: 1–20