Ultragraph C*-algebra
In mathematics, an ultragraph C*-algebra is a universal C*-algebra generated by partial isometries on a collection of Hilbert spaces constructed from the ultragraph. These C*-algebras were created in order to simultaneously generalize the classes of graph C*-algebras and Exel–Laca 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. Definitions. Ultragraphs. An ultragraph formula_1 consists of a set of vertices formula_2, a set of edges formula_3, a source map formula_4 , and a range map formula_5 taking values in the power set collection formula_6 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. Example. 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 labelsformula_7, formula_8with source an range mapsformula_9can be visualized as the image on the right. Ultragraph algebras. Given an ultragraph formula_1, we define formula_11 to be the smallest subset of formula_12 containing the singleton sets formula_13, containing the range sets formula_14, and closed under intersections, unions, and relative complements. A Cuntz–Krieger formula_15-family is a collection of projections formula_16 together with a collection of partial isometries formula_17 with mutually orthogonal ranges satisfying The ultragraph C*-algebra formula_28 is the universal C*-algebra generated by a Cuntz–Krieger formula_15-family . Properties. 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 formula_11 is the collection of all finite subsets of formula_2 and formula_32 for each formula_21. Every Exel–Laca algebras is also an ultragraph C*-algebra: If formula_34 is an infinite square matrix with index set formula_35 and entries in formula_36, one can define an ultragraph by formula_37, formula_38, formula_39, and formula_40. It can be shown that formula_28 is isomorphic to the Exel–Laca algebra formula_42. Ultragraph C*-algebras are useful tools for studying both graph C*-algebras  and Exel–Laca algebras. Among other benefits, modeling an Exel–Laca 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 Exel–Laca algebra. Ultragraph C*-algebras have been used to show that every simple AF-algebra is isomorphic to either a graph C*-algebra or an Exel–Laca algebra. They have also been used to prove that every AF-algebra with no (nonzero) finite-dimensional quotient is isomorphic to an Exel–Laca algebra. While the classes of  graph C*-algebras, Exel–Laca 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.