The Daugavet property of C∗C^*-algebras, JB∗JB^*-triples, and of their isometric preduals

A Banach space $X$ is said to have the Daugavet property if every rank-one operator $T:X\longrightarrow X$ satisfies $\|Id + T\| = 1 + \|T\|$. We give geometric characterizations of this property in the settings of $C^*$-algebras, $JB^*$-triples and their isometric preduals. We also show that, in these settings, the Daugavet property passes to ultrapowers, and thus, it is equivalent to an stronger property called the uniform Daugavet property.Comment: To appear in J. Funct. Anal., final form, 19 page

Kn 26, a New Quadrupolar Planetary Nebula

Once classified as an emission line source, the planetary nebula (PN) nature of the source Kn 26 has been only recently recognized in digital sky surveys. To investigate the spectral properties and spatio-kinematical structure of Kn 26, we have obtained high spatial-resolution optical and near-IR narrow-band images, high-dispersion long-slit echelle spectra, and intermediate-resolution spectroscopic observations. The new data reveal an hourglass morphology typical of bipolar PNe. A detailed analysis of its morphology and kinematics discloses the presence of a second pair of bipolar lobes, making Kn 26 a new member of the subclass of quadrupolar PNe. The time-lap between the ejection of the two pairs of bipolar lobes is much smaller than their dynamical ages, implying a rapid change of the preferential direction of the central engine. The chemical composition of Kn 26 is particularly unusual among PNe, with a low N/O ratio (as of type II PNe) and a high helium abundance (as of type I PNe), although not atypical among symbiotic stars. Such an anomalous chemical composition may have resulted from the curtail of the time in the Asymptotic Giant Branch by the evolution of the progenitor star through a common envelope phase.Comment: 9 pages, 5 figures, accepted for publication in A&