K-graph C*-algebra

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 ).

• There are maps s : \Lambda \to \Lambda^0 and r : \Lambda \to \Lambda^0 which take a morphism \lambda \in \Lambda to its source s(\lambda) and its range r(\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 for all v \in \Lambda^0 and n \in \mathbb{N}^k then \Lambda is said to be row-finite with no sources.

Skeletons

A k-graph \Lambda can be visualized via its skeleton. Let e_1 , \ldots , e_n be the canonical generators for \mathbb{N}^k. The idea is to think of morphisms in \Lambda^{e_i} = d^{-1}(e_i) as being edges in a directed graph of a color indexed by i.

To be more precise, the skeleton of a k-graph \Lambda is a k-colored directed graph E=(E^0,E^1,r,s,c) with vertices E^0 = \Lambda^0, edges E^1 = \cup_{i=1}^k \Lambda^{e_i}, range and source maps inherited from \Lambda, and a color map c: E^1 \to { 1 , \ldots , k } defined by c (e) = i if and only if e \in \Lambda^{e_i}.

The skeleton of a k-graph alone is not enough to recover the k-graph. The extra information about factorization can be encoded in a complete and associative collection of commuting squares. In particular, for each i \ne j and e,f \in E^1 with c(e) = i and c(f) = j, there must exist unique e',f' \in E^1 with c(e') = i, c(f') = j, and ef = f'e' in \Lambda. A different choice of commuting squares can yield a distinct k-graph with the same skeleton.

Examples

• A 1-graph is precisely the path category of a directed graph. If \lambda is a path in the directed graph, then d(\lambda) is its length. The factorization condition is trivial: if \lambda is a path of length m+n then let \mu be the initial subpath of length m and let \nu be the final subpath of length n .

• The monoid \mathbb{N}^k can be considered as a category with one object. The identity on \mathbb{N}^k give a degree map making \mathbb{N}^k into a k-graph.

• Let \Omega_k = { (m,n) : m,n \in \mathbb{Z}^k , m \le n }. Then \Omega_k is a category with range map r(m,n)=(m,m), source map s(m,n)=(n,n), and composition (m,n)(n,p)=(m,p). Setting d(m,n) = n-m gives a degree map. The factorization rule is given as follows: if d(m,n) = p + q for some p,q \in \mathbb{N}^k , then (m,n) = (m,m+q) (m+q, n) is the unique factorization.

C*-algebras of k-graphs

Just as a graph C*-algebra can be associated to a directed graph, a universal C*-algebra can be associated to a k-graph.

Let \Lambda be a row-finite k-graph with no sources then a Cuntz–Krieger \Lambda-family or a represenentaion of \Lambda in a C*-algebra B is a map S \colon \Lambda \to B such that

1. { S_v : v \in \Lambda^0 } is a collection of mutually orthogonal projections;
2. S_\lambda S_\mu = S_{\lambda \mu} for all \lambda,\mu \in \Lambda with s(\lambda) =r(\mu) ;
3. S_\mu^* S_\mu = S_{s ( \mu )} for all \mu \in \Lambda ; and
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.

The algebra C^* ( \Lambda ) is the universal C*-algebra generated by a Cuntz–Krieger \Lambda-family.

References

References

1. (2000). "''Higher rank graph C*-algebras''". [[The New York Journal of Mathematics]].
2. (2023). "Quantum SU(3) as the C*-algebra of a 2-Graph".
3. "''Lecture notes on higher-rank graphs and their C*-algebras''".