K-graph C*-algebra explained

In mathematics, for

k \in N , a k -graph (also known as a higher-rank graph or graph of rank k ) is a countable category Λ together with a functor d : Λ \to N ^{k} , called the degree map, which satisfy the following factorization property:

if

λ \in Λ and m,n \in N ^{k} are such that d( λ ) = m + n , then there exist unique \mu,\nu \in Λ such that d( \mu ) = m , d( \nu ) = n , and λ = \mu\nu .

An immediate consequence of the factorization property is that morphisms in a

k -graph can be factored in multiple ways: there are also unique \mu',\nu' \in Λ such that d( \mu' ) = m , d( \nu' ) = n , and \mu \nu = λ = \nu' \mu' .

A 1-graph is just the path category of a directed graph. In this case the degree map takes a path to its length. By extension,

k -graphs can be considered higher-dimensional analogs of directed graphs.

Another way to think about a

k -graph is as a k -colored directed graph together with additional information to record the factorization property. The k -colored graph underlying a k -graph is referred to as its skeleton. Two k -graphs can have the same skeleton but different factorization rules.

Kumjian and Pask originally introduced

k -graphs as a generalization of a construction of Robertson and Steger. By considering representations of k -graphs as bounded operators on Hilbert space, they have since become a tool for constructing interesting C*-algebras whose structure reflects the factorization rules. Some compact quantum groups like SU _{q(3)} can be realised as the C ^{*} -algebras of k -graphs.There is also a close relationship between k -graphs and strict factorization systems in category theory.

Notation

The notation for

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

• For

n \in N ^{k} let Λ ^{n} = d ^{-1} (n) . By the factorisation property it follows that Λ ^{0} = \operatorname{Obj} ( Λ ) .

• There are maps

s : Λ \to Λ ^{0} and r : Λ \to Λ ^{0} which take a morphism λ \in Λ to its source s( λ ) and its range r( λ ) .

• For

v,w \in Λ ^{0} and X \subseteq Λ we have v X = \{ λ \in X : r ( λ ) = v \} , X w = \{ λ \in X : s ( λ ) = w \} and v X w = v X \cap X w .

• If

0 < \# v Λ ^{n} < inf ty for all v \in Λ ^{0} and n \in N ^{k} then Λ is said to be row-finite with no sources.

Skeletons

A

k -graph Λ can be visualized via its skeleton. Let e _{1} , \ldots , e _{n} be the canonicalgenerators for N ^{k} . The idea is to think of morphisms in Λ ^{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 Λ is a k-colored directed graph E = (E ^{0,E} ^{1,r,s,c)} with verticesE ^{0} = Λ ^{0} , edges E ^{1} = \cup ^{k} _{i =1} Λ ^{e _{i}} , range and source maps inheritedfrom Λ ,and a color map c: E ^{1} \to \{ 1 , \ldots , k \} defined by c (e) = i if and only if e \in Λ ^{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 Λ . 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

λ is a path in the directed graph, then d( λ ) is its length. The factorization condition is trivial: if λ 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 . N ^{k} can be considered as a category with one object. The identity on N ^{k} give a degree map making N ^{k} into a k -graph.

• Let

\Omega _{k} = \{ (m,n) : m,n \in 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 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

Λ be a row-finite k -graph with no sources then a Cuntz–Krieger Λ -family or a represenentaion of Λ in a C*-algebra B is a map S \colon Λ \to B such that\{ S _{v} : v \in Λ ^{0} \} is a collection of mutually orthogonal projections;S _{λ} S _{\mu} = S _{λ} for all λ ,\mu \in Λ with s( λ ) = r(\mu) ;S ^{*} _{\mu} S _{\mu} = S _{s} for all \mu \in Λ ; andS _{v} = \sum _{λ \in v Λ ^{n}} S _{λ} S ^{*} _{λ} for all n \in N ^{k} and v \in Λ ^{0} .

The algebra

C ^{*} ( Λ ) is the universal C*-algebra generated by a Cuntz–Krieger Λ -family.

See also

• Graph C*-algebra