In mathematics, an ultragraph C*-algebra is a universal C*-algebra generated by partial isometries on a collection of Hilbert spaces constructed from ultragraphs.[1]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

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Ultragraphs

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An ultragraph   consists of a set of vertices  , a set of edges  , a source map  , and a range map   taking values in the power set collection   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

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Ultragraph  

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

 ,  

with source an range maps

 

can be visualized as the image on the right.

Ultragraph algebras

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Given an ultragraph  , we define   to be the smallest subset of   containing the singleton sets  , containing the range sets  , and closed under intersections, unions, and relative complements. A Cuntz–Krieger  -family is a collection of projections   together with a collection of partial isometries   with mutually orthogonal ranges satisfying

  1.  ,  ,   for all  ,
  2.   for all  ,
  3.   whenever   is a vertex that emits a finite number of edges, and
  4.   for all  .

The ultragraph C*-algebra   is the universal C*-algebra generated by a Cuntz–Krieger  -family.

Properties

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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   is the collection of all finite subsets of   and   for each  . Every Exel–Laca algebras is also an ultragraph C*-algebra: If   is an infinite square matrix with index set   and entries in  , one can define an ultragraph by  ,  ,  , and  . It can be shown that   is isomorphic to the Exel–Laca algebra  .[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

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Notes

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