CB-norm estimates for maps between noncommutative Lp-spaces and quantum channel theory.

In the first part of this work, we show how certain techniques from quantum information theory can be used in order to obtain very sharp embeddings between noncommutative Lp-spaces. Then, we use these estimates to study the classical capacity with restricted assisted entanglement of the quantum erasure channel and the quantum depolarizing channel. In particular, we exactly compute the capacity of the first one and we show that certain nonmultiplicative results hold for the second one.

[C II] and 12^CO(1-0) emission maps in HLSJ091828.6+514223: a strongly lensed interacting system AT z = 5.24

We present Submillimeter Array [C II] 158 μm and Karl G. Jansky Very Large Array 12^CO(1-0) line emission maps for the bright, lensed, submillimeter source at z = 5.2430 behind A 773: HLSJ091828.6+514223 (HLS0918). We combine these measurements with previously reported line profiles, including multiple 12^CO rotational transitions, [C I], water, and [N II], providing some of the best constraints on the properties of the interstellar medium in a galaxy at z > 5. HLS0918 has a total far-infrared (FIR) luminosity L_FIR(8–1000 μm) = (1.6 ± 0.1) × 10^14 L_☉ μ^–1, where the total magnification μ_total = 8.9 ± 1.9, via a new lens model from the [C II] and continuum maps. Despite a HyLIRG luminosity, the FIR continuum shape resembles that of a local LIRG. We simultaneously fit all of the observed spectral line profiles, finding four components that correspond cleanly to discrete spatial structures identified in the maps. The two most redshifted spectral components occupy the nucleus of a massive galaxy, with a source-plane separation <1 kpc. The reddest dominates the continuum map (demagnified L_FIR, component = (1.1 ± 0.2) × 10^13 L_☉) and excites strong water emission in both nuclear components via a powerful FIR radiation field from the intense star formation. A third star-forming component is most likely a region of a merging companion (ΔV ~ 500 km s^–1) exhibiting generally similar gas properties. The bluest component originates from a spatially distinct region and photodissociation region analysis suggests that it is lower density, cooler, and forming stars less vigorously than the other components. Strikingly, it has very strong [N II] emission, which may suggest an ionized, molecular outflow. This comprehensive view of gas properties and morphology in HLS0918 previews the science possible for a large sample of high-redshift galaxies once ALMA attains full sensitivity.

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Let S(M) be the ring of (continuous) semialgebraic functions on a semialgebraic set M and S*(M) its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps Spec(j)1:Spec(S(N))→Spec(S(M)) and Spec(j)2:Spec(S*(N))→Spec(S*(M)) induced by the inclusion j:N M of a semialgebraic subset N of M. The ring S(M) can be understood as the localization of S*(M) at the multiplicative subset WM of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion iM:Spec(S(M)) Spec(S*(M)) that reduces both problems above to an analysis of the fibers of the spectral map Spec(j)2:Spec(S*(N))→Spec(S*(M)). If we denote Z:=ClSpec(S*(M))(M N), it holds that the restriction map Spec(j)2|:Spec(S*(N)) Spec(j)2-1(Z)→Spec(S*(M)) Z is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of Spec(j)2 at the points of Z. The size of the fibers of prime ideals "close" to the complement Y:=M N provides valuable information concerning how N is immersed inside M. If N is dense in M, the map Spec(j)2 is surjective and the generic fiber of a prime ideal p∈Z contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber Spec(j)2-1(p) is a finite set for p∈Z. If such is the case, our procedure allows us to compute the size s of Spec(j)2-1(p). If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in p. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.