K-graph C*-algebra

In mathematics, for , a -graph (also known as a higher-rank graph or graph of rank ) is a countable category together with a functor , called the degree map, which satisfy the following factorization property:

<blockquote> if and are such that , then there exist unique such that , , and . </blockquote>

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 CuntzÃ¢ÂÂKrieger -family or a represenentaion of in a C*-algebra B is a map such that

is a collection of mutually orthogonal projections;
for all with ;
for all ; and
for all and

The algebra is the universal C*-algebra generated by a CuntzÃ¢ÂÂKrieger -family.

References