Graph C*-algebra

Graph C*-algebras are direct generalizations of the Cuntz algebras and Cuntz-Krieger algebras, but the class of graph C*-algebras has been shown to also include several other widely studied classes of C*-algebras.

As a result, graph C*-algebras provide a common framework for investigating many well-known classes of C*-algebras that were previously studied independently.

Among other benefits, this provides a context in which one can formulate theorems that apply simultaneously to all of these subclasses and contain specific results for each subclass as special cases.

Although graph C*-algebras include numerous examples, they provide a class of C*-algebras that are surprisingly amenable to study and much more manageable than general C*-algebras.

The graph not only determines the associated C*-algebra by specifying relations for generators, it also provides a useful tool for describing and visualizing properties of the C*-algebra.

This visual quality has led to graph C*-algebras being referred to as "operator algebras we can see.

"[1][2] Another advantage of graph C*-algebras is that much of their structure and many of their invariants can be readily computed.

Using data coming from the graph, one can determine whether the associated C*-algebra has particular properties, describe the lattice of ideals, and compute K-theoretic invariants.

are partial isometries with mutually orthogonal ranges, the elements of

It is important to be aware that there are competing conventions regarding the "direction of the edges" in the Cuntz-Krieger relations.

Throughout this article, and in the way that the relations are stated above, we use the convention first established in the seminal papers on graph C*-algebras.

[3][4] The alternate convention, which is used in Raeburn's CBMS book on Graph Algebras,[5] interchanges the roles of the range map

The fact that the Cuntz-Krieger relations take a simpler form for row-finite graphs has technical consequences for many results in the subject.

Historically, much of the early work on graph C*-algebras was done exclusively in the row-finite case.

Even in modern work, where infinite emitters are allowed and C*-algebras of general graphs are considered, it is common to state the row-finite case of a theorem separately or as a corollary, since results are often more intuitive and transparent in this situation.

We use the convention that a double arrow drawn from one vertex to another and labeled

The following table provides a short list of some of the more well-known equivalences.

The universal property produces a natural action of the circle group

It is important to note that the canonical gauge action depends on the choice of the generating Cuntz-Krieger

The canonical gauge action is a fundamental tool in the study of

It appears in statements of theorems, and it is also used behind the scenes as a technical device in proofs.

The uniqueness theorems are fundamental results in the study of graph C*-algebras, and they serve as cornerstones of the theory.

Consequently, the uniqueness theorems can be used to determine when a C*-algebra generated by a Cuntz-Krieger

are partially ordered by inclusion, and they form a lattice with meet

The gauge-invariant ideals are partially ordered by inclusion and form a lattice with meet

The following theorem shows that gauge-invariant ideals correspond to saturated hereditary subsets.

Then the following hold: The Drinen-Tomforde Desingularization, often simply called desingularization, is a technique used to extend results for C*-algebras of row-finite graphs to C*-algebras of countable graphs.

[7] Drinen and Tomforde described a method for constructing a desingularization from any countable graph: If

is given by the graph Desingularization has become a standard tool in the theory of graph C*-algebras,[8] and it can simplify proofs of results by allowing one to first prove the result in the (typically much easier) row-finite case, and then extend the result to countable graphs via desingularization, often with little additional effort.

The technique of desingularization may not work for graphs containing a vertex that emits an uncountable number of edges.