Simple purely infinite C*-algebras and n-filling actions

Let $n$ be a positive integer. We introduce a concept, which we call the $n$-filling property, for an action of a group on a separable unital $C^*$-algebra $A$. If $A=C(\Omega)$ is a commutative unital $C^*$-algebra and the action is induced by a group of homeomorphisms of $\Omega$ then the $n$-filling property reduces to a weak version of hyperbolicity. The $n$-filling property is used to prove that certain crossed product $C^*$-algebras are purely infinite and simple. A variety of group actions on boundaries of symmetric spaces and buildings have the $n$-filling property. An explicit example is the action of $\Gamma=SL_n({\bf Z})$ on the projective $n$-space.Comment: 16 page

Property (RD) for Hecke pairs

As the first step towards developing noncommutative geometry over Hecke C*-algebras, we study property (RD) (Rapid Decay) for Hecke pairs. When the subgroup H in a Hecke pair (G,H) is finite, we show that the Hecke pair (G,H) has (RD) if and only if G has (RD). This provides us with a family of examples of Hecke pairs with property (RD). We also adapt Paul Jolissant's works in 1989 to the setting of Hecke C*-algebras and show that when a Hecke pair (G,H) has property (RD), the algebra of rapidly decreasing functions on the set of double cosets is closed under holomorphic functional calculus of the associated (reduced) Hecke C*-algebra. Hence they have the same K_0-groups.Comment: A short note added explaining other methods to prove that the subalgebra of rapidly decreasing functions is smooth. This is the final version as published. The published version is available at: springer.co

Quantum Symmetries and Strong Haagerup Inequalities

In this paper, we consider families of operators $\{x_r\}_{r \in \Lambda}$ in a tracial C$^\ast$-probability space $(\mathcal A, \phi)$, whose joint $\ast$-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups $\{H_n^+\}_{n \in \N}$. We prove a strong form of Haagerup's inequality for the non-self-adjoint operator algebra $\mathcal B$ generated by $\{x_r\}_{r \in \Lambda}$, which generalizes the strong Haagerup inequalities for $\ast$-free R-diagonal families obtained by Kemp-Speicher \cite{KeSp}. As an application of our result, we show that $\mathcal B$ always has the metric approximation property (MAP). We also apply our techniques to study the reduced C$^\ast$-algebra of the free unitary quantum group $U_n^+$. We show that the non-self-adjoint subalgebra $\mathcal B_n$ generated by the matrix elements of the fundamental corepresentation of $U_n^+$ has the MAP. Additionally, we prove a strong Haagerup inequality for $\mathcal B_n$, which improves on the estimates given by Vergnioux's property RD \cite{Ve}

New holomorphically closed subalgebras of C∗C^*-algebras of hyperbolic groups

We construct dense, unconditional subalgebras of the reduced group $C^*$-algebra of a word-hyperbolic group, which are closed under holomorphic functional calculus and possess many bounded traces. Applications to the cyclic cohomology of group $C^*$-algebras and to delocalized $L^2$-invariants of negatively curved manifolds are given

The design of an adaptive optics telescope: the case of DAG

In this paper, we describe in detail the optical design of DAG, a new 4 m telescope for Turkey. DAG is an "adaptive optics friendly" telescope, in a sense that each design decision is taken considering the potential impact on the AO performance (vibrations, static aberrations etc.) The objective is to make this telescope fully ready for AO at first light. It is designed as a Ritchey-Chretien combination, 56 m focal length, with Nasmyth foci only, and active optics. Its total RMS error is expected to be 45 nm up to Zernike mode 78, and 26 nm for the higher, non AO corrected modes. A final design optimization has been done by the telescope manufacturers, demonstrating that our AO-based requirements can be satisfied, without much difficulty.Publisher's Versio

Speckle noise and dynamic range in coronagraphic images

This paper is concerned with the theoretical properties of high contrast coronagraphic images in the context of exoplanet searches. We derive and analyze the statistical properties of the residual starlight in coronagraphic images, and describe the effect of a coronagraph on the speckle and photon noise. Current observations with coronagraphic instruments have shown that the main limitations to high contrast imaging are due to residual quasi-static speckles. We tackle this problem in this paper, and propose a generalization of our statistical model to include the description of static, quasi-static and fast residual atmospheric speckles. The results provide insight into the effects on the dynamic range of wavefront control, coronagraphy, active speckle reduction, and differential speckle calibration. The study is focused on ground-based imaging with extreme adaptive optics, but the approach is general enough to be applicable to space, with different parameters.Comment: 31 pages, 18 figure

On twisted Fourier analysis and convergence of Fourier series on discrete groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.Comment: 35 pages; abridged, revised and update

An E-ELT Case Study: Colour-Magnitude Diagrams of an Old Galaxy in the Virgo Cluster

One of the key science goals for a diffraction limited imager on an Extremely Large Telescope (ELT) is the resolution of individual stars down to faint limits in distant galaxies. The aim of this study is to test the proposed capabilities of a multi-conjugate adaptive optics (MCAO) assisted imager working at the diffraction limit, in IJHK$_s$ filters, on a 42m diameter ELT to carry out accurate stellar photometry in crowded images in an Elliptical-like galaxy at the distance of the Virgo cluster. As the basis for realistic simulations we have used the phase A studies of the European-ELT project, including the MICADO imager (Davies & Genzel 2010) and the MAORY MCAO module (Diolaiti 2010). We convolved a complex resolved stellar population with the telescope and instrument performance expectations to create realistic images. We then tested the ability of the currently available photometric packages STARFINDER and DAOPHOT to handle the simulated images. Our results show that deep Colour-Magnitude Diagrams (photometric error, $\pm$0.25 at I$\ge$27.2; H$\ge$25. and K$_s\ge$24.6) of old stellar populations in galaxies, at the distance of Virgo, are feasible at a maximum surface brightness, $\mu_V \sim$ 17 mag/arcsec$^2$ (down to M$_I > -4$ and M$_H \sim$ M$_K > -6$), and significantly deeper (photometric error, $\pm$0.25 at I$\ge$29.3; H$\ge$26.6 and K$_s\ge$26.2) for $\mu_V \sim$ 21 mag/arcsec$^2$ (down to M$_I \ge -2$ and M$_H \sim$ M$_K \ge -4.5$). The photometric errors, and thus also the depth of the photometry should be improved with photometry packages specifically designed to adapt to an ELT MCAO Point Spread Function. We also make a simple comparison between these simulations and what can be expected from a Single Conjugate Adaptive Optics feed to MICADO and also the James Webb Space Telescope.Comment: 17 pages, 22 figures, accepted on A&

Fundamental limitations on Earth-like planet detection with Extremely Large Telescopes

We analyse the fundamental limitations for the detection of extraterrestrial planets with Extremely Large Telescopes. For this task, a coronagraphic device combined to a very high order wavefront correction system is required but not sufficient to achieve the $10^{-10}$ contrast level needed for detecting an Earth-like planet. The stellar residuals left uncorrected by the wavefront correction system need to be calibrated and subtracted. In this paper, we consider a general model including the dynamic phase aberrations downstream the wavefront correction system, the static phase aberrations of the instrument and some differential aberrations provided by the calibration unit. A rather optimistic case of a filled circular pupil and of a perfect coronagraph is elsewhere assumed. As a result of the analytical study, the limitation mostly comes from the static aberrations. Using numerical simulations we confirm this result and evaluate the requirements in terms of phase aberrations to detect Earth-like planets on Extremely Large Telescopes.Comment: 8 pages, 8 figures, accepted in A&