arXiv subjects

Charles Starling

Publications and source records attributed to Charles Starling.

24 records · Page 2Linked to original sources

arXiv2014

Boundary quotients of C*-algebras of right LCM semigroups

We study C*-algebras associated to right LCM semigroups, that is, semigroups which are left cancellative and for which any two principal right ideals are either disjoint or intersect in another principal right ideal. If $P$ is such a semigroup, its C*-algebra admits a natural boundary quotient $\mathcal{Q}(P)$. We show that $\mathcal{Q}(P)$ is isomorphic to the tight C*-algebra of a certain inverse semigroup associated to $P$, and thus is isomorphic to the C*-algebra of an étale groupoid. We use this to give conditions on $P$ which guarantee that $\mathcal{Q}(P)$ is simple and purely infinite, and give applications to self-similar groups and Zappa-Szép products of semigroups.

math.OA↗

arXiv2014

Self-Similar Graph C*-Algebras and Partial Crossed Products

In a recent paper, Pardo and the first named author introduced a class of C*-algebras which which are constructed from an action of a group on a graph. This class was shown to include many C*-algebras of interest, including all Kirchberg algebras in the UCT class. In this paper, we study the conditions under which these algebras can be realized as partial crossed products of commutative C*-algebras by groups. In addition, for any $n\geq 2$ we present a large class of groups such that for any group $H$ in this class, the Cuntz algebra $\mathcal{O}_n$ is isomorphic to a partial crossed product of a commutative C*-algebra by $H$.

math.OA↗

arXiv2013

Division Point Measures from Primitive Substitutions

In this note we extend results of Olli concerning limits of point measures arising from substitutions. We consider a general primitive substitution on a finite polygon set in $\mathbb{R}^2$ and show that limits of certain atomic measures each converge to Lebesgue measure.

math.DS↗

arXiv2013

K-theory of crossed products of tiling C*-algebras by rotation groups

Let $Ω$ be a tiling space and let $G$ be the maximal group of rotations which fixes $Ω$. Then the cohomology of $Ω$ and $Ω/G$ are both invariants which give useful geometric information about the tilings in $Ω$. The noncommutative analog of the cohomology of $Ω$ is the K-theory of a C*-algebra associated to $Ω$, and for translationally finite tilings of dimension 2 or less the K-theory is isomorphic to the direct sum of cohomology groups. In this paper we give a prescription for calculating the noncommutative analog of the cohomology of $Ω/G$, that is, the K-theory of the crossed product of the tiling C*-algebra by $G$. We also provide a table with some calculated K-groups for many common examples, including the Penrose and pinwheel tilings.

math.OA↗

arXiv2013

Finite symmetry group actions on substitution tiling C*-algebras

For a finite symmetry group $G$ of an aperiodic substitution tiling system $(\p,ω)$, we show that the crossed product of the tiling C*-algebra $\Aw$ by $G$ has real rank zero, tracial rank one, a unique trace, and that order on its K-theory is determined by the trace. We also show that the action of $G$ on $\Aw$ satisfies the weak Rokhlin property, and that it also satisfies the tracial Rokhlin property provided that $\Aw$ has tracial rank zero. In the course of proving the latter we show that $\Aw$ is finitely generated. We also provide a link between $\Aw$ and the AF algebra Connes associated to the Penrose tilings.

math.OA↗

arXiv2011

The tiling C*-algebra viewed as a tight inverse semigroup algebra

We realize Kellendonk'?s C*-algebra of an aperiodic tiling as the tight C*-algebra of the inverse semigroup associated to the tiling, thus providing further evidence that the tight C*-algebra is a good candidate to be the natural associative algebra to go along with an inverse semigroup.

math.OA↗