K-graph C*-algebra explained

Λ with domain and codomain maps r and s , together with a functor d : Λ \to N ^{k} which satisfies the following factorisation property: if d( λ ) = m + n then there are unique \mu, \nu \in Λ with d( \mu ) = m , d( \nu ) = n such that λ = \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 afunctor from this category into the natural numbers 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\toN} ^{k} defined by d(f ^{n _{1}} _{1} ...f ^{n _{k}} _{k} ) = (n _{1} , \ldots , n _{k)} is a k-graph.

• Let

\Omega _{k} = \{ (m,n) : m,n \in 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 N ^{k} let Λ ^{n} = d ^{-1} (n) .

• By the factorisation property it follows that

Λ ^{0} = \operatorname{Obj} ( 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.

Visualisation - Skeletons

E = (E ^{0,E} ^{1,r,s,c)} whereE ^{0} = Λ ^{0} , E ^{1} = \cup ^{k} _{i =1} Λ ^{e _{i}} , r,s inheritedfrom Λ and c: E ^{1} \to \{ 1 , \ldots , k \} defined by c (e) = i if and only if e \in Λ ^{e _{i}} where e _{1} , \ldots , e _{n} are the canonicalgenerators for N ^{k} . The factorisation property in Λ for elementsof degree e _{i+e} _{j} where i ≠ j gives rise to relations between the edges ofE .

C*-algebra

As with graph-algebras one may associate a C*-algebra to a k-graph:

Let

Λ be a row-finite k-graph with no sources then a Cuntz–Krieger Λ family in a C*-algebra B is a collection \{ s _{λ} : λ \in Λ \} of operators in B such thats _{λ} s _{\mu} = s _{λ} if λ , \mu , λ \mu \in Λ ;\{ s _{v} : v \in Λ ^{0} \} are mutually orthogonal projections;

1. if

d ( \mu ) = d ( \nu ) then s ^{*} _{\mu} s _{\nu} = \delta _{\mu} s _{s} ;s _{v} = \sum _{λ \in v Λ ^{n}} s _{λ} s ^{*} _{λ} for all n \in N ^{k} and v \in Λ ^{0} .C ^{*} ( Λ ) is then the universal C*-algebra generated by a Cuntz–Krieger Λ -family