In mathematics, an ultragraph C*-algebra is a universal C*-algebra generated by partial isometries on a collection of Hilbert spaces constructed from ultragraphs.<sup>pp. 6-7</sup>. These C*-algebras were created in order to simultaneously generalize the classes of graph C*-algebras and ExelâÂÂLaca algebras, giving a unified framework for studying these objects. This is because every graph can be encoded as an ultragraph, and similarly, every infinite graph giving an Exel-Laca algebras can also be encoded as an ultragraph.
An ultragraph consists of a set of vertices , a set of edges , a source map , and a range map taking values in the power set collection of nonempty subsets of the vertex set. A directed graph is the special case of an ultragraph in which the range of each edge is a singleton, and ultragraphs may be thought of as generalized directed graph in which each edges starts at a single vertex and points to a nonempty subset of vertices.
An easy way to visualize an ultragraph is to consider a directed graph with a set of labelled vertices, where each label corresponds to a subset in the image of an element of the range map. For example, given an ultragraph with vertices and edge labels<blockquote>, </blockquote>with source an range maps<blockquote></blockquote>can be visualized as the image on the right.
Given an ultragraph , we define to be the smallest subset of containing the singleton sets , containing the range sets , and closed under intersections, unions, and relative complements. A CuntzâÂÂKrieger -family is a collection of projections together with a collection of partial isometries with mutually orthogonal ranges satisfying
The ultragraph C*-algebra is the universal C*-algebra generated by a CuntzâÂÂKrieger -family.
Every graph C*-algebra is seen to be an ultragraph algebra by simply considering the graph as a special case of an ultragraph, and realizing that is the collection of all finite subsets of and for each . Every ExelâÂÂLaca algebras is also an ultragraph C*-algebra: If is an infinite square matrix with index set and entries in , one can define an ultragraph by , , , and . It can be shown that is isomorphic to the ExelâÂÂLaca algebra .
Ultragraph C*-algebras are useful tools for studying both graph C*-algebras and ExelâÂÂLaca algebras. Among other benefits, modeling an ExelâÂÂLaca algebra as ultragraph C*-algebra allows one to use the ultragraph as a tool to study the associated C*-algebras, thereby providing the option to use graph-theoretic techniques, rather than matrix techniques, when studying the ExelâÂÂLaca algebra. Ultragraph C*-algebras have been used to show that every simple AF-algebra is isomorphic to either a graph C*-algebra or an ExelâÂÂLaca algebra. They have also been used to prove that every AF-algebra with no (nonzero) finite-dimensional quotient is isomorphic to an ExelâÂÂLaca algebra.
While the classes of graph C*-algebras, ExelâÂÂLaca algebras, and ultragraph C*-algebras each contain C*-algebras not isomorphic to any C*-algebra in the other two classes, the three classes have been shown to coincide up to Morita equivalence.