K-graph C*-algebra


In mathematics, for, a -graph is a countable category together with a functor, called the degree map, which satisfy the following factorization property:
if and are such that, then there exist unique such that,, and.

An immediate consequence of the factorization property is that morphisms in a -graph can be factored in multiple ways: there are also unique such that,, and.
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, -graphs can be considered higher-dimensional analogs of directed graphs.
Another way to think about a -graph is as a -colored directed graph together with additional information to record the factorization property.
The -colored graph underlying a -graph is referred to as its skeleton.
Two -graphs can have the same skeleton but different factorization rules.
Kumjian and Pask originally introduced -graphs as a generalization of a construction of Robertson and Steger. By considering representations of -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 can be realised as the -algebras of -graphs.
There is also a close relationship between -graphs and strict factorization systems in category theory.

Notation

The notation for -graphs is borrowed extensively from the corresponding notation for categories:
  • For let. By the factorisation property it follows that.
  • There are maps and which take a morphism to its source and its range.
  • For and we have, and.
  • If for all and then is said to be row-finite with no sources.

Skeletons

A -graph can be visualized via its skeleton. Let be the canonical
generators for. The idea is to think of morphisms in as being edges in a directed graph of a color indexed by.
To be more precise, the skeleton of a -graph is a k-colored directed graph with vertices
, edges, range and source maps inherited
from,
and a color map defined by
if and only if.
The skeleton of a -graph alone is not enough to recover the -graph. The extra information about factorization can be encoded in a complete and associative collection of commuting squares. In particular, for each and with and, there must exist unique with,, and in. A different choice of commuting squares can yield a distinct -graph with the same skeleton.

Examples

  • A 1-graph is precisely the path category of a directed graph. If is a path in the directed graph, then is its length. The factorization condition is trivial: if is a path of length then let be the initial subpath of length and let be the final subpath of length.
  • The monoid can be considered as a category with one object. The identity on give a degree map making into a -graph.
  • Let. Then is a category with range map, source map, and composition. Setting gives a degree map. The factorization rule is given as follows: if for some, then 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 -graph.
Let be a row-finite -graph with no sources then a Cuntz–Krieger -family or a represenentaion of in a C*-algebra B is a map such that
  1. is a collection of mutually orthogonal projections;
  2. for all with ;
  3. for all ; and
  4. for all and
The algebra is the universal C*-algebra generated by a Cuntz–Krieger -family.