The late merging phase of a galaxy cluster : XMM EPIC Observations of A3266

We present a mosaic of five XMM-Newton observations of the nearby ($z=0.0594$) merging galaxy cluster Abell 3266. We use the spectro-imaging capabilities of \xmm to build precise (projected) temperature, entropy, pressure and Fe abundance maps. The temperature map exhibits a curved, large-scale hot region, associated with elevated entropy levels, very similar to that foreseen in numerical simulations. The pressure distribution is disturbed in the central region but is remarkably regular on large scales. The Fe abundance map indicates that metals are inhomogeneously distributed across the cluster. Using simple physical calculations and comparison with numerical simulations, we discuss in detail merging scenarios that can reconcile the observed gas density, temperature and entropy structure, and the galaxy density distribution

Amenability and subexponential spectral growth rate of Dirichlet forms on von Neumann algebras

In this work we apply Noncommutative Potential Theory to prove (relative) amenability and the (relative) Haagerup Property $(H)$ of von Neumann algebras in terms of the spectral growth of Dirichlet forms. Examples deal with (inclusions of) countable discrete groups and free orthogonal compact quantum groups

Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket

We provide a definition of integral, along paths in the Sierpinski gasket K, for differential smooth 1-forms associated to the standard Dirichlet form K. We show how this tool can be used to study the potential theory on K. In particular, we prove: i) a de Rham reconstruction of a 1-form from its periods around lacunas in K; ii) a Hodge decomposition of 1-forms with respect to the Hilbertian energy norm; iii) the existence of potentials of smooth 1-forms on a suitable covering space of K. We finally show that this framework provides versions of the de Rham duality theorem for the fractal K.Comment: Some proofs have been clarified, reference to previous literature is now more accurate, 33 pages, 6 figure

The warm interstellar medium around the Cygnus Loop

Observations of the oxygen lines [OII]3729 and [OIII]5007 in the medium immediately beyond the Cygnus Loop supernova remnant were carried out with the scanning Fabry-P\'erot spectrophotometer ESOP. Both lines were detected in three different directions - east, northeast and southwest - and up to a distance of 15 pc from the shock front. The ionized medium is in the immediate vicinity of the remnant, as evinced by the smooth brightening of both lines as the adiabatic shock transition (defined by the X-ray perimeter) is crossed. These lines are usually brighter around the Cygnus Loop than in the general background in directions where the galactic latitude is above 5 degrees. There is also marginal (but significant) evidence that the degree of ionization is somewhat larger around the Cygnus Loop. We conclude that the energy necessary to ionize this large bubble of gas could have been supplied by an O8 or O9 type progenitor or the particles heated by the expanding shock front. The second possibility, though highly atractive, would have to be assessed by extensive modelling.Comment: 18 pages, 8 figures, ApJ 512 in pres

Fredholm Modules on P.C.F. Self-Similar Fractals and their Conformal Geometry

The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct Fredholm modules on post critically finite fractals by regular harmonic structures. The modules are d-summable, the summability exponent d coinciding with the spectral dimension of the generalized laplacian operator associated with the regular harmonic structures. The characteristic tools of the noncommutative infinitesimal calculus allow to define a d-energy functional which is shown to be a self-similar conformal invariant.Comment: 16 page

Spectral triples for the Sierpinski Gasket

We construct a family of spectral triples for the Sierpinski Gasket $K$. For suitable values of the parameters, we determine the dimensional spectrum and recover the Hausdorff measure of $K$ in terms of the residue of the volume functional $a\to$ tr$(a\,|D|^{-s})$ at its abscissa of convergence $d_D$, which coincides with the Hausdorff dimension $d_H$ of the fractal. We determine the associated Connes' distance showing that it is bi-Lipschitz equivalent to the distance on $K$ induced by the Euclidean metric of the plane, and show that the pairing of the associated Fredholm module with (odd) $K$-theory is non-trivial. When the parameters belong to a suitable range, the abscissa of convergence $\delta_D$ of the energy functional $a\to$ tr$(|D|^{-s/2}|[D,a]|^2\,|D|^{-s/2})$ takes the value $d_E=\frac{\log(12/5)}{\log 2}$, which we call energy dimension, and the corresponding residue gives the standard Dirichlet form on $K$.Comment: 48 pages, 9 figures. Final version, to appear in J.Funct.Ana

Le problème de Dirichlet dans les C∗-algèbres

AbstractA C∗-algebra A, a closed ideal I, and the generator Δ of a Markov semigroup of completely positive contractions from A into itself, are given. Under the same kind of assumptions as in the classical case, a completely positive lifting Λ from the quotient algebra AI into the algebra M(A) of multipliers of A is constructed, such that, for any α in AI, its lifting Λ(α) is harmonic on I, i.e., belongs to the kernel of the generator Δ localized on I