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k-graph C*-algebras

Background
The path category of a directed graph forms a category (the free catgory generated by a graph). The length of a path give a functor into $$\mathbb{N}$$. A k-graph is a natural generalistion of this concept.

Definition
A k-graph is a countable category $$\Lambda$$ 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 $$\lambda = \mu \nu$$

Notation

 * 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.

C*-algebra
Let $$\Lambda$$ be a row-finite $$k$$-graph with no sources then a Cuntz-Kriger $$\Lambda$$ 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.