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 ultragraphs. pp. 6-7. 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.

Ultragraphs
An ultragraph $$\mathcal{G} = (G^0, \mathcal{G}^1, r, s)$$ consists of a set of vertices $$G^0$$, a set of edges $$\mathcal{G}^1$$, a source map $$s:\mathcal{G}^1 \to G^0$$, and a range map $$r : \mathcal{G}^1 \to P(G^0) \setminus \{ \emptyset \}$$ taking values in the power set collection $$P(G^0) \setminus \{ \emptyset \}$$ 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 labels"$\mathcal{G}^0 = \{v,w,x \}$, $\mathcal{G}^1 = \{e,f,g \}$"with source an range maps $$\begin{matrix} s(e) = v & s(f) = w & s(g) = x \\ r(e) = \{v,w,x \} & r(f) = \{x \} & r(g) = \{v,w \} \end{matrix}$$ can be visualized as the image on the right.

Ultragraph algebras
Given an ultragraph $$\mathcal{G} = (G^0, \mathcal{G}^1, r, s)$$, we define $$\mathcal{G}^0$$ to be the smallest subset of $$P(G^0)$$ containing the singleton sets $$\{ \{ v \} : v \in G^0 \}$$, containing the range sets $$\{ r(e) : e \in \mathcal{G}^1 \}$$, and closed under intersections, unions, and relative complements. A Cuntz–Krieger $$\mathcal{G}$$-family is a collection of projections $$\{ p_A : A \in \mathcal{G}^0 \}$$ together with a collection of partial isometries $$\{ s_e : e \in \mathcal{G}^1 \}$$ with mutually orthogonal ranges satisfying


 * 1) $$p_{\emptyset}$$, $$ p_A p_B = p_{A \cap B}$$, $$p_A + p_B - p_{A \cap B} = p_{A \cup B}$$ for all $$A \in \mathcal{G}^0$$,
 * 2) $$s_e^*s_e = p_{r(e)}$$ for all $$e \in \mathcal{G}^1$$,
 * 3) $$p_v = \sum_{s(e)=v} s_e s_e^*$$ whenever $$v \in G^0$$ is a vertex that emits a finite number of edges, and
 * 4) $$s_e s_e^* \le p_{s(e)}$$ for all $$e \in \mathcal{G}^1$$.

The ultragraph C*-algebra $$C^*(\mathcal{G})$$ is the universal C*-algebra generated by a Cuntz–Krieger $$\mathcal{G}$$-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 $$\mathcal{G}^0$$ is the collection of all finite subsets of $$G^0$$ and $$p_A = \sum_{v \in A} p_v$$ for each $$A \in \mathcal{G}^0$$. Every Exel–Laca algebras is also an ultragraph C*-algebra: If $$A$$ is an infinite square matrix with index set $$I$$ and entries in $$\{ 0, 1 \}$$, one can define an ultragraph by $$G^0 :=$$, $$G^1 := I$$, $$s(i) = i$$, and $$r(i) = \{ j \in I : A(i,j)=1 \}$$. It can be shown that $$C^*(\mathcal{G})$$ is isomorphic to the Exel–Laca algebra $$\mathcal{O}_A$$.

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.