k-graph C*-algebra

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, a k-graph (or higher-rank graph, graph of rank k) is a countable category with domain and codomain maps and , together with a functor which satisfies the following factorisation property: if then there are unique with such that .

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[edit]

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 gives a functor from this category into the natural numbers . A k-graph is a natural generalisation of this concept which was introduced in 2000 by Alex Kumjian and David Pask.[1]


Examples[edit]

  • It can be shown that a 1-graph is precisely the path category of a directed graph.
  • The category consisting of a single object and k commuting morphisms , together with the map defined , is a k-graph.
  • Let then is a k-graph when gifted with the structure maps , , and .

Notation[edit]

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

  • For let .
  • By the factorisation property it follows that .
  • For and we have , and .
  • If for all and then is said to be row-finite with no sources.

Visualisation - Skeletons[edit]

A k-graph is best visualised by drawing its 1-skeleton as a k-coloured graph where , , inherited from and defined by if and only if where are the canonical generators for . The factorisation property in for elements of degree where gives rise to relations between the edges of .

C*-algebra[edit]

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

Let be a row-finite k-graph with no sources then a Cuntz–Krieger family in a C*-algebra B is a collection of operators in B such that

  1. if ;
  2. are mutually orthogonal projections;
  3. if then ;
  4. for all and .

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

References[edit]

  1. ^ Kumjian, A.; Pask, D.A. (2000), "Higher rank graph C*-algebras", The New York Journal of Mathematics, 6: 1–20