Geometry in preduals of spaces of 2-homogeneous polynomials on Hilbert spaces

Let H be a (real or complex) Hilbert space. Using spectral theory and properties of the Schatten–Von Neumann operators, we prove that every symmetric tensor of unit norm in HoH is an infinite absolute convex combination of points of the form xox with x in the unit sphere of the Hilbert space. We use this to obtain explicit characterizations of the smooth points of the unit ball of HoH .

The Co-Entrapment of a Homogeneous Catalyst and an Ionic Liquid by a Sol-gel Method: Recyclable Ionogel Hydrogenation Catalysts

Molecular hydrogenation catalysts have been co-entrapped with the ionic liquid [Bmim]NTf(2) inside a silica matrix by a sot-gel method. These catalytic ionogels have been compared to simple catalyst-doped glasses, the parent homogeneous catalysts, commercial heterogeneous catalysts, and Rh-doped mesoporous silica. The most active ionogel has been characterised by transmission electron microscopy, X-ray photoelectron spectroscopy, and solid state NMR before and after catalysis. The ionogel catalysts were found to be remarkably active, recyclable and resistant to chemical change.