Research Statement - Mathematics and Computer Science

Sarah Wright - Research Statement 2
In Figure 1(b), we can see that there is a path from vm to vn if m > n and a path from v1 to any
other vertex. These two observations together give a path from any vertex to any other vertex, so
the graph is cofinal.
v1
v2
(a) Cycle Without Entry
(b) A cofinal graph
(c) The Graph En
Figure 1: Three Examples of Directed Graphs
The existence of entries for each cycle and the cofinality of the graph relate to the structure of the
C ∗ -algebra by the following theorem of Kumjian, Pask, Raeburn, and Renault. The requirement
of a graph to be row-finite (each vertex receives only finitely many edges) was later dealt with by
Drinen and Tomforde yielding a similar theorem in [4].
Theorem. [15, Theorem 4.14] Suppose E is a row-finite graph in which every cycle has an entry.
Then C ∗ (E) is simple if and only if E is cofinal.
In particular, this theorem tells us the C ∗ -algebra associated to the graph En, Figure 1(c), which is
the Cuntz algebra On, is simple when n ≥ 2. As the Cuntz Algebra is well-studied, its simplicity is
not new upon the proof of this theorem, but the ease with which the theorem is applied demonstrates
its importance and beauty. There are further theorems in the field that give a description of the
ideal structure of a graph-algebra, see [2, 8]. Similar to the theorem above, the results give a way
to nearly “read-off” the ideal structure of the algebra from properties of the graph.
The study of graph algebras had been particularly fruitful in its provision of a rich class of easily
accessible examples of C ∗ -algebras with various properties. To this end, the idea of a graph has been
generalized in many ways to provide even more examples. The long list of generalizations includes
the labeled graphs of Bates and Pask where edges of the graph are labelled from an alphabet set, the
k-graphs of Kumjian and Pask, where rather than paths of length ℓ ∈ N we work with k-dimensional
paths of shape n ∈ N k , and the topological graphs of Katsura, where instead of discrete sets of
vertices and edges we give these spaces interesting topologies. For definitions of these structures see
[1], [12], and [11] respectively. Most recently the work of Yeend provides a generalization of both
higher-rank graphs and topological graphs, with the unifying theory of topological k-graphs [21].
2 Aperiodicity Conditions and Related Results
The importance of cycles having or not having entries is the idea of being able to “back out of a
cycle” and build an aperiodic infinite path. (The convention to construct paths backward comes
from the methods used to construct the associated C ∗ -algebra.) The relationship between the
periodic paths (or lack there of) in the graph and the simplicity of the associated C ∗ -algebra is
an important result for all types of graphs. The C ∗ -algebras of directed graphs, k-graphs, and
topological k-graphs were all originally constructed using groupoid methods (a well-studied method
of constructing a C ∗ -algebra that generalizes the algebra generated by a group). More elementary
methods for construction have been established since these initial works. As the infinite paths of
the graph play such a central role in the groupoid construction of the C ∗ -algebra, in the case of
higher-rank graphs, the original aperiodicity condition, called Condition (A), was stated in terms
v3
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v5
v6
en
.
e3
e2
e1
v