Spectral triples for hyperbolic dynamical systems

Spectral triples are defined for C*-algebras associated with hyperbolic dynamical systems known as Smale spaces. The spectral dimension of one of these spectral triples is shown to recover the topological entropy of the Smale space

C*-algebras of tilings with infinite rotational symmetry

A tiling with infinite rotational symmetry, such as the Conway-Radin Pinwheel Tiling, gives rise to a topological dynamical system to which an \'etale equivalence relation is associated. A groupoid C*-algebra for a tiling is produced and a separating dense set is exhibited in the C*-algebra which encodes the structure of the topological dynamical system. In the case of a substitution tiling, natural subsets of this separating dense set are used to define an AT-subalgebra of the C*-algebra. Finally our results are applied to the Pinwheel Tiling

Twisted C*-algebras associated to finitely aligned higher-rank graphs

We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish versions of the usual uniqueness theorems and the classification of gauge-invariant ideals. We show that all twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs are nuclear and satisfy the UCT, and that for twists that lift to real-valued cocycles, the K-theory of a twisted relative Cuntz-Krieger algebra is independent of the twist. In the final section, we identify a sufficient condition for simplicity of twisted Cuntz-Krieger algebras associated to higher-rank graphs which are not aperiodic. Our results indicate that this question is significantly more complicated than in the untwisted setting.Comment: Version 2: This paper has now appeared in Documenta Mathematica. This version on arXiv exactly matches the pagination and format of the published version. Original published version available from http://www.math.uni-bielefeld.de/documenta/vol-19/28.htm

Graph algebras and orbit equivalence

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their C∗C∗-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs EE we construct a groupoid G(C∗(E),D(E))G(C∗(E),D(E)) from the graph algebra C∗(E)C∗(E) and its diagonal subalgebra D(E)D(E) which generalises Renault’s Weyl groupoid construction applied to (C∗(E),D(E))(C∗(E),D(E)). We show that G(C∗(E),D(E))G(C∗(E),D(E)) recovers the graph groupoid GEGE without the assumption that every cycle in EE has an exit, which is required to apply Renault’s results to (C∗(E),D(E))(C∗(E),D(E)). We finish with applications of our results to out-splittings of graphs and to amplified graphs

Fractal spectral triples on Kellendonk's C∗C^*-algebra of a substitution tiling

We introduce a new class of noncommutative spectral triples on Kellendonk's $C^*$-algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using Perron-Frobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is $\theta$-summable, and respects the hierarchy of the substitution system. To elucidate our results, we construct a fractal tree on the Penrose tiling, and explicitly show how it gives rise to a collection of spectral triples.Comment: Updated to agree with published versio

Separating weak lensing and intrinsic alignments using radio observations

We discuss methods for performing weak lensing using radio observations to recover information about the intrinsic structural properties of the source galaxies. Radio surveys provide unique information that can benefit weak lensing studies, such as HI emission, which may be used to construct galaxy velocity maps, and polarized synchrotron radiation; both of which provide information about the unlensed galaxy and can be used to reduce galaxy shape noise and the contribution of intrinsic alignments. Using a proxy for the intrinsic position angle of an observed galaxy, we develop techniques for cleanly separating weak gravitational lensing signals from intrinsic alignment contamination in forthcoming radio surveys. Random errors on the intrinsic orientation estimates introduce biases into the shear and intrinsic alignment estimates. However, we show that these biases can be corrected for if the error distribution is accurately known. We demonstrate our methods using simulations, where we reconstruct the shear and intrinsic alignment auto and cross-power spectra in three overlapping redshift bins. We find that the intrinsic position angle information can be used to successfully reconstruct both the lensing and intrinsic alignment power spectra with negligible residual bias.Comment: 17 pages, 10 figures, submitted to MNRA

Graph algebras and orbit equivalence

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto's notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their $C^*$-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs $E$ we construct a groupoid $\mathcal{G}_{(C^*(E),\mathcal{D}(E))}$ from the graph algebra $C^*(E)$ and its diagonal subalgebra $\mathcal{D}(E)$ which generalises Renault's Weyl groupoid construction applied to $(C^*(E),\mathcal{D}(E))$. We show that $\mathcal{G}_{(C^*(E),\mathcal{D}(E))}$ recovers the graph groupoid $\mathcal{G}_E$ without the assumption that every cycle in $E$ has an exit, which is required to apply Renault's results to $(C^*(E),\mathcal{D}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.Comment: 27 page

K-Theoretic Duality for Hyperbolic Dynamical Systems

The K-theoretic analog of Spanier-Whitehead duality for noncommutative C*-algebras is shown to hold for the Ruelle algebras associated to irreducible Smale spaces. This had previously been proved only for shifts of finite type. Implications of this result as well as relations to the Baum-Connes conjecture and other topics are also considered.Comment: 36 page

Radio-Optical Galaxy Shape and Shear Correlations in the COSMOS Field using 3 GHz VLA Observations

We present a weak lensing analysis of the 3 GHz VLA radio survey of the COSMOS field, which we correlate with overlapping HST-ACS optical observations using both intrinsic galaxy shape and cosmic shear correlation statistics. After cross-matching sources between the two catalogues, we measure the correlations of galaxy position angles and find a Pearson correlation coefficient of $0.14 \pm 0.03$. This is a marked improvement from previous studies which found very weak, or non-existent correlations, and gives insight into the emission processes of radio and optical galaxies. We also extract power spectra of averaged galaxy ellipticities (the primary observable for cosmic shear) from the two catalogues, and produce optical-optical, radio-optical and radio-radio spectra. The optical-optical auto-power spectrum was measured to a detection significance of 9.80$\sigma$ and is consistent with previous observations of the same field. For radio spectra (which we do not calibrate, given the unknown nature of their systematics), although we do not detect significant radio-optical (1.50$\sigma$) or radio-radio (1.45$\sigma$) $E$-mode power spectra, we do find the $E$-mode spectra to be more consistent with the shear signal expected from previous studies than with a null signal, and vice versa for $B$-mode and $EB$ cross-correlation spectra. Our results give promise that future radio weak lensing surveys with larger source number densities over larger areas will have the capability to measure significant weak lensing signals.Comment: 19 pages, 17 figures, accepted for publication in MNRA