Ultragraph C*-algebra explained

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.^[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

l{G} = (G ^{0,} l{G} ^{1,} r, s) consists of a set of vertices G ^{0} , a set of edges l{G} ^{1} , a source map s: l{G} ^{1} \to G ^{0} , and a range map r : l{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

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

l{G} = (G ^{0,} l{G} ^{1,} r, s) , we define l{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 l{G} ^{1} \} , and closed under intersections, unions, and relative complements. A Cuntz–Krieger l{G} -family is a collection of projections \{ p _{A} : A \in l{G} ^{0} \} together with a collection of partial isometries \{ s _{e} : e \in l{G} ^{1} \} with mutually orthogonal ranges satisfyingp _{\emptyset} , p _{A} p _{B} = p _{A} , p _{A} + p _{B} - p _{A} = p _{A} for all A \in l{G} ^{0} ,s ^{*s} _{e} = p _{r(e)} for all e \in l{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, ands _{e} s ^{*} _{e} \le p _{s(e)} for all e \in l{G} ^{1} .

The ultragraph C*-algebra

C ^{*(l{G})} is the universal C*-algebra generated by a Cuntz–Krieger l{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

l{G} ^{0} is the collection of all finite subsets of G ^{0} and p _{A} = \sum _{v} p _{v} for each A \in l{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 ^{*(l{G})} is isomorphic to the Exel–Laca algebra l{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.^[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.

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]

See also

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

Notes and References

1. [arxiv:math/0106161|A unified approach to Exel–Laca algebras and C*-algebras associated to graphs]
2. https://www.sciencedirect.com/science/article/pii/S0022123609002158 Realization of AF-algebras as graph algebras, Exel–Laca algebras, and ultragraph algebras
3. [arxiv:0809.0164|Graph algebras, Exel–Laca algebras, and ultragraph algebras coincide up to Morita equivalence]