arXiv subjects

Danny Crytser

Publications and source records attributed to Danny Crytser.

4 recordsLinked to original sources

arXiv2018

Simplicity criteria for etale groupoid $C^*$-algebras

We develop a framework suitable for obtaining simplicity criteria for reduced $C^*$-algebras of Hausdorff etale groupoids. This is based on the study of certain non-degenerate $C^*$-subalgebras (in the case of groupoids, the $C^*$-algebra of the interior isotropy bundle), for which one can control (non-unique) state extensions to the ambient C*-algebra. As an application, we give simplicity criteria for reduced crossed products of abelian $C^*$-algebras by discrete groups.

math.OA↗

arXiv2016

Traces arising from regular inclusions

We study the problem of extending a state on an abelian $C^*$- subalgebra to a tracial state on the ambient $C^*$-algebra. We propose an approach that is well-suited to the case of regular inclusions, in which there is a large supply of normalizers of the subalgebra. Conditional expectations onto the subalgebra give natural extensions of a state to the ambient $C^*$-algebra; we prove that these extensions are tracial states if and only if certain invariance properties of both the state and conditional expectations are satisfied. In the example of a groupoid $C^*$-algebra, these invariance properties correspond to invariance of associated measures on the unit space under the action of bisections. Using our framework, we are able to completely describe the tracial state space of a Cuntz-Krieger graph algebra. Along the way we introduce certain operations called graph tightenings, which both streamline our description and provides connections to related finiteness questions in graph $C^*$-algebras. Our investigation has close connections with the so-called unique state extension property and its variants.

math.OA↗

arXiv2015

Continous-trace $k$-graph $C^*$-algebras

A characterization is given for directed graphs that yield graph $C^*$-algebras with continuous trace. This is established for row-finite graphs with no sources first using a groupoid approach, and extended to the general case via the Drinen-Tomforde desingularization. Partial results are given to characterize higher-rank graphs that yield $C^*$-algebras with continuous trace.

math.OA↗

arXiv2014

Containment: A Variation of Cops and Robbers

We consider "Containment": a variation of the graph pursuit game of Cops and Robber in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop), and the cops win by "containing" the robber---that is, by occupying all $\deg(v)$ of the edges incident with a vertex $v$ while the robber is at $v$. We develop bounds that relate the minimal number of cops, $\xi(G)$, required to contain a robber to the well-known "cop-number" $c(G)$ in the original game: in particular, $c(G) {\le} \xi(G) {\le} \gamma(G) \Delta(G)$. We note that $\xi(G) {\geq} \delta(G)$ for all graphs $G$, and analyze several families of graphs in which equality holds, as well as several in which the inequality is strict. We also give examples of graphs which require an unbounded number of cops in order to contain a robber, and note that there exist cubic graphs with $\xi(G) \geq \Omega(n^{1/6})$.

math.CO↗