K graph C* algebra - Alchetron, The Free Social Encyclopedia

In mathematics, a k-graph (or higher-rank graph, graph of rank k) is a countable category Λ with domain and codomain maps r and s , together with a functor d : Λ → N k which satisfies the following factorisation property: if d ( λ ) = m + n then there are unique μ , ν ∈ Λ with d ( μ ) = m , d ( ν ) = n such that λ = μ ν .

Aside from its category theory definition, one can think of k-graphs as 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, k-graph is just a regular 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*-algebra 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 graph theory perspective, yet just complicated enough to reveal different interesting properties in 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 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 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 → N k defined d ( f 1 n 1 . . . f k n k ) = ( n 1 , … , n k ) , is a k-graph.

Let Ω k = { ( m , n ) : m , n ∈ Z k , m ≤ n } then Ω 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 ∈ N k let Λ n = d − 1 ( n ) .

By the factorisation property it follows that Λ 0 = Obj ⁡ ( Λ ) .

For v , w ∈ Λ 0 and X ⊆ Λ we have v X = { λ ∈ X : r ( λ ) = v } , X w = { λ ∈ X : s ( λ ) = w } and v X w = v X ∩ X w .

If 0 < # v Λ n < ∞ for all v ∈ Λ 0 and n ∈ N k then Λ is said to be row-finite with no sources.

Visualisation - Skeletons

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 = Λ 0 , E 1 = ∪ i = 1 k Λ e i , r , s inherited from Λ and c : E 1 → { 1 , … , k } defined by c ( e ) = i if and only if e ∈ Λ e i where e 1 , … , e n are the canonical generators for N k . The factorisation property in Λ for elements of degree e i + e j where i ≠ j gives rise to relations between the edges of E .

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 ∈ Λ } of operators in B such that

s λ s μ = s λ μ if λ , μ , λ μ ∈ Λ ;
{ s v : v ∈ Λ 0 } are mutually orthogonal projections;
if d ( μ ) = d ( ν ) then s μ ∗ s ν = δ μ , ν s s ( μ ) ;
s v = ∑ λ ∈ v Λ n s λ s λ ∗ for all n ∈ N k and v ∈ Λ 0 .

C ∗ ( Λ ) is then the universal C*-algebra generated by a Cuntz–Krieger Λ -family.

References

K-graph C*-algebra Wikipedia

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