Operator algebras and a theorem of Dieudonne

LetA be aC*-algebra with second dualA″. Let (φ _n)(n=1,...) be a sequence in the dual ofA such that limφ _n(a) exists for eacha εA. In general, this does not imply that limφ _n(x) exists for eachx εA″. But if limφ _n(p) exists whenever p is the range projection of a positive self-adjoint element of the unit ball ofA, then it is shown that limφ _n(x) does exist for eachx inA″. This is a non-commutative generalisation of a celebrated theorem of Dieudonné. A new proof of Dieudonné’s theorem, for positive measures, is given here. The proof of the main result makes use of Dieudonné’s original theorem.     

Suzuki coupling catalyzed by chloro({2-[mesityl(quinolin-8-yl-κN)boryl]-3,5-dimethylphenyl}methyl-κC)palladium(II)

The title compound was evaluated as catalyst for Suzuki coupling of aryl halides (X?=?I, Br) with phenylboronic acid at room temperature. The rates of the reactions increase with increasing Hammet sigma constant of m-substituents on the arylhalide. The presence of ortho groups on the arylhalides had the effect of lowering the reaction rates. Density functional theory study of the mechanism for the formation of a Pd⁽⁰⁾ complex from the title molecule was performed. Reductive elimination of chloride and the η⁴-BCCC moiety was endergonic and ruled out as a likely pathway. Instead, conversion of the title molecule to phenyl({2-[mesityl(quinolin-8-yl-κN)boryl]-3,5-dimethylphenyl}methyl-κC)palladium(II) complex is thermodynamically favored and predicted to further react with the solvent and the Suzuki coupling reagents to undergo reductive elimination of diphenyl to form a reactive Pd⁽⁰⁾ complex.     

Hydrothermal growth of mesoporous SBA-15 silica in the presence of PVP-stabilized Pt nanoparticles: synthesis, characterization, and catalytic properties

A novel high surface area heterogeneous catalyst based on solution phase colloidal nanoparticle chemistry has been developed. Monodisperse platinum nanoparticles of 1.7-7.1 nm have been synthesized by alcohol reduction methods and incorporated into mesoporous SBA-15 silica during hydrothermal synthesis. Characterization of the Pt/SBA-15 catalysts suggests that Pt particles are located within the surfactant micelles during silica formation leading to their dispersion throughout the silica structure. After removal of the templating polymer from the nanoparticle surface, Pt particle sizes were determined from monolayer gas adsorption measurements. Infrared studies of CO adsorption revealed that CO exclusively adsorbs to atop sites and red-shifts as the particle size decreases suggesting surface roughness increases with decreasing particle size. Ethylene hydrogenation rates were invariant with particle size and consistent with a clean Pt surface. Ethane hydrogenolysis displayed significant structure sensitivity over the size range of 1-7 nm, while the apparent activation energy increased linearly up to a Pt particle size of approximately 4 nm and then remained constant. The observed rate dependence with particle size is attributed to a higher reactivity of coordinatively unsaturated surface atoms in small particles compared to low-index surface atoms prevalent in large particles. The most reactive of these unsaturated surface atoms are responsible for ethane decomposition to surface carbon. The ability to design catalytic structures with tunable properties by rational synthetic methods is a major advance in the field of catalyst synthesis and for the development of accurate structure-function relationships in heterogeneous reaction kinetics.     

Adsorption and co-adsorption of ethylene and carbon monoxide on silica-supported monodisperse Pt nanoparticles: volumetric adsorption and infrared spectroscopy studies

The adsorption of carbon monoxide and ethylene, and their sequential adsorption, was studied over a series of Pt/SBA-15 catalysts with monodisperse particle sizes ranging from 1.7 to 7.1 nm by diffuse-reflectance infrared spectroscopy and chemisorption. Gas adsorption was dependent on the Pt particle size, temperature, and sequence of gas exposure. Adsorption of CO at room temperature on Pt/SBA-15 gives rise to a spectroscopic feature assigned to the C-O stretch: nu(CO) = 2075 cm-1 (1.9 nm); 2079 cm-1 (2.9 nm); 2082 cm-1 (3.6 nm); and 2090 cm-1 (7.1 nm). The intensity of the signal decreased in a sigmoidal fashion with increasing temperature, thereby providing semiquantitative surface coverage information. Adsorption of ethylene on Pt/SBA-15 gave rise to spectroscopic features at approximately 1340, approximately 1420, and approximately 1500 cm-1 assigned to ethylidyne, di-sigma-bonded ethylene, and pi-bonded ethylene, respectively. The ratio of these surface species is highly dependent on the Pt particle size. At room temperature, Pt particles stabilize ethylidyne as well as di-sigma- and pi-bonded ethylene; however, ethylidyne predominated on the surfaces of larger particles. Ethylidyne was the only identifiable species at 403 K, with its formation being more facile on larger particles. Co-adsorption experiments reveal that the composition of the surface layer is dependent on the order of exposure to gases. Exposure of a C2H4-covered Pt surface to CO resulted in an approximately 50% decrease in chemisorbed CO compared to a fresh Pt surface. The nu(CO) appeared at 2050 cm-1 on Pt/SBA-15 pretreated with C2H4 at room temperature. The di-sigma-bonded and pi-bonded species are the most susceptible to displacement from the surface by CO. The formation of ethylidyne appeared to be less sensitive to the presence of adsorbed carbon monoxide, especially on larger particles. Upon exposure of C2H4 to a CO-covered Pt surface, little irreversible uptake occurred due to nearly 100% site blocking. These results demonstrate that carbon monoxide competes directly with ethylene for surface sites, which will have direct implications on the poisoning of the heterogeneously catalyzed conversion of hydrocarbons.     

Reverse micelle synthesis of rhodium nanoparticles

Water-in-oil reverse micelles of butyl ammonium laurate in hexanes that contain sodium hexachlororhodate were reduced with sodium borohydride to yield rhodium nanoparticles. The size of the micelle, determined by dynamic light scattering, was from 3 to 20 nm and varied as the water to surfactant ratio (W) was changed. The rhodium nanoparticles exhibited a Gaussian size distribution (sigma=0.35 nm), with average diameters of 1.5, 2.2, and 2.9 nm. The products were characterized with TEM, HRTEM, and X-ray photoemission spectroscopy.     

Radial oscillations of neutron stars in strong magnetic fields

The eigen frequencies of radial pulsations of neutron stars are calculated in a strong magnetic field. At low densities we use the magnetic BPS equation of state (EOS) similar to that obtained by Lai and Shapiro while at high densities the EOS obtained from the relativistic nuclear mean field theory is taken and extended to include strong magnetic field. It is found that magnetized neutron stars support higher maximum mass whereas the effect of magnetic field on radial stability for observed neutron star masses is minimal.     

High-surface-area catalyst design: Synthesis, characterization, and reaction studies of platinum nanoparticles in mesoporous SBA-15 silica

Platinum nanoparticles in the size range of 1.7-7.1 nm were produced by alcohol reduction methods. A polymer (poly(vinylpyrrolidone), PVP) was used to stabilize the particles by capping them in aqueous solution. The particles were characterized by X-ray diffraction (XRD) and transmission electron microscopy (TEM). TEM investigations demonstrate that the particles have a narrow size distribution. Mesoporous SBA-15 silica with 9-nm pores was synthesized by a hydrothermal process and used as a catalyst support. After incorporation into mesoporous SBA-15 silica using low-power sonication, the catalysts were calcined to remove the stabilizing polymer from the nanoparticle surface and reduced by H2. Pt particle sizes determined from selective gas adsorption measurements are larger than those determined by bulk techniques such as XRD and TEM. Room-temperature ethylene hydrogenation was chosen as a model reaction to probe the activity of the Pt/SBA-15 materials. The reaction was shown to be structure insensitive over a series of Pt/SBA-15 materials with particle sizes between 1.7 and 3.6 nm. The hydrogenolysis of ethane on Pt particles from 1.7 to 7.1 nm was weakly structure sensitive with smaller particles demonstrating higher specific activity. Turnover rates for ethane hydrogenolysis increased monotonically with increasing metal dispersion, suggesting that coordinatively unsaturated metal atoms present in small particles are more active for C2H6 hydrogenolysis than the low index planes that dominate in large particles. An explanation for the structure sensitivity is suggested, and the potential applications of these novel supported nanocatalysts for further studies of structure-activity and structure-selectivity relationships are discussed.