Ultragraph C*-algebra

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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 ${\displaystyle {\mathcal {G}}=(G^{0},{\mathcal {G}}^{1},r,s)}$ consists of a set of vertices ${\displaystyle G^{0}}$, a set of edges ${\displaystyle {\mathcal {G}}^{1}}$, a source map ${\displaystyle s:{\mathcal {G}}^{1}\to G^{0}}$, and a range map ${\displaystyle r:{\mathcal {G}}^{1}\to P(G^{0})\setminus \{\emptyset \}}$ taking values in the power set collection ${\displaystyle P(G^{0})\setminus \{\emptyset \}}$ 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 ${\displaystyle (\{v,w,x\},\{e,f,g\},s,r)}$

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

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

with source an range maps

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

can be visualized as the image on the right.

Ultragraph algebras[]

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

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

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

See also[]

• Leavitt path algebra
• Infinite matrix
• Infinite graph

Notes[]