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[1]^{pg 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.[1] 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 ${\mathcal {G}}=(G^{0},{\mathcal {G}}^{1},r,s)$ 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

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

${\mathcal {G}}^{0}=\{v,w,x\}$ , ${\mathcal {G}}^{1}=\{e,f,g\}$

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

$p_{\emptyset }$ , $p_{A}p_{B}=p_{A\cap B}$ , $p_{A}+p_{B}-p_{A\cap B}=p_{A\cup B}$ for all $A\in {\mathcal {G}}^{0}$ ,
$s_{e}^{*}s_{e}=p_{r(e)}$ for all $e\in {\mathcal {G}}^{1}$ ,
$p_{v}=\sum _{s(e)=v}s_{e}s_{e}^{*}$ whenever $v\in G^{0}$ is a vertex that emits a finite number of edges, and
$s_{e}s_{e}^{*}\leq p_{s(e)}$ 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 ${\mathcal {O}}_{A}$ .[1]

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.[2] They have also been used to prove that every AF-algebra with no (nonzero) finite-dimensional quotient is isomorphic to an Exel–Laca algebra.[2]

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.[3]

Notes

A unified approach to Exel–Laca algebras and C*-algebras associated to graphs, Mark Tomforde, J. Operator Theory 50 (2003), no. 2, 345–368.
Realization of AF-algebras as graph algebras, Exel–Laca algebras, and ultragraph algebras, Takeshi Katsura, Aidan Sims, and Mark Tomforde, J. Funct. Anal. 257 (2009), no. 5, 1589–1620.
Graph algebras, Exel–Laca algebras, and ultragraph algebras coincide up to Morita equivalence, Takeshi Katsura, Paul Muhly, Aidan Sims, and Mark Tomforde, J. Reine Angew. Math. 640 (2010), 135–165.

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[Tom-1] A unified approach to Exel–Laca algebras and C*-algebras associated to graphs, Mark Tomforde, J. Operator Theory 50 (2003), no. 2, 345–368.

[KST-2] Realization of AF-algebras as graph algebras, Exel–Laca algebras, and ultragraph algebras, Takeshi Katsura, Aidan Sims, and Mark Tomforde, J. Funct. Anal. 257 (2009), no. 5, 1589–1620.

[3] Graph algebras, Exel–Laca algebras, and ultragraph algebras coincide up to Morita equivalence, Takeshi Katsura, Paul Muhly, Aidan Sims, and Mark Tomforde, J. Reine Angew. Math. 640 (2010), 135–165.