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.

Definitions

Ultragraphs

An ultragraph ${𝒢}=(G^0,{{𝒢}}^1,r,s)$ consists of a set of vertices $G^0$, a set of edges ${{𝒢}}^1$, a source map $s:{{𝒢}}^1\to G^0$, and a range map $r:{{𝒢}}^1\to P(G^0)\setminus \{\mathrm{\varnothing }\}$ taking values in the power set collection $P(G^0)\setminus \{\mathrm{\varnothing }\}$ 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

Ultragraph $(\{v,w,x\},\{e,f,g\},s,r)$

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

$G^0=\{v,w,x\}$

,

${{𝒢}}^1=\{e,f,g\}$

with source an range maps

$\begin{array}{ccc}s(e)=v& s(f)=w& s(g)=x\\ r(e)=\{v,w,x\}& r(f)=\{x\}& r(g)=\{v,w\}\end{array}$

can be visualized as the image on the right.

Ultragraph algebras

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

1. $p_{\mathrm{\varnothing }}$, $p_A{p}_B=p_{A\cap B}$, $p_A+p_B-p_{A\cap B}=p_{A\cup B}$ for all $A\in {{𝒢}}^0$,
2. $s_e^{*}{s}_e=p_{r(e)}$ for all $e\in {{𝒢}}^1$,
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 $e\in {{𝒢}}^1$.

The ultragraph C*-algebra $C^{*}({𝒢})$ is the universal C*-algebra generated by a Cuntz–Krieger ${𝒢}$-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 ${{𝒢}}^0$ is the collection of all finite subsets of $G^0$ and $p_A=\sum _{v\in A}{p}_v$ for each $A\in {{𝒢}}^0$. 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 $C^{*}({𝒢})$ is isomorphic to the Exel–Laca algebra ${{𝒪}}_A$.

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.

See also

• Leavitt path algebra
• Exel–Laca algebras
• Infinite matrix
• Infinite graph