Markus Haase

In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky-Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the decomposition of Jacobs, de Leeuw and Glicksberg). The proof rests heavily on the operator theory on Kaplansky-Hilbert modules, in particular the spectral theorem for Hilbert-Schmidt homomorphisms on such modules. As an application, a generalization of the celebrated Furstenberg-Zimmer structure theorem to the case of measure-preserving actions of arbitrary groups on arbitrary probability spaces is established.

In this paper, the abstract multiplier theorems for 0 0 -sectorial and 0 0 -strip type operators by Kriegler and Weis [Math. Z. 289 (2018), pp. 405–444] are refined and generalized to arbitrary sectorial and strip-type operators. To this end, holomorphic Hörmander-type functions on sectors and strips are introduced, with even a finer scale of smoothness than the classical polynomial scale. Moreover, we establish alternative descriptions of these spaces involving Schwartz and “holomorphic Schwartz” functions. Finally, the abstract results are combined with a result by Carbonaro and Dragičević [Duke Math. J. 166 (2017), pp. 937–974] to obtain an improvement—with respect to the smoothness condition—of the known Hörmander-type multiplier theorem for general symmetric contraction semigroups.

The structural parameters of the rare-earth diacetate halide trihydrates, RE(OAc)₂Hal·3H₂O with RE = Ce − (Pm) − Lu and Hal = Cl, Br, have been determined by low temperature, high-resolution SCXRD in order to examine the effect of lanthanide contraction on the coordination geometry in this series of isomorphous compounds consisting of cationic, acetate-bridged, non-linear, one-dimensional coordination polymers of composition [RE(H₂O)₃(OAc)₂]⁺ and laterally hydrogen bonded halide ions, Hal⁻. Although the shrinkage of the unit cell volume follows lanthanide contraction very well over the complete range of investigated RE elements, many other parameters (i.e., lattice constants, angles and distances in the RE··· RE alignment, RE-O bond lengths, etc.) exhibit a more complex response on lanthanide contraction often expressed by sigmoid curves that can be ascribed to a continuous transition from CN9 (RE = Ce) to CN8 (RE = Lu) as one acetate group loses the chelate function, an effect accompanied by significant structural changes of the carboxylate group. Therefore, data are best analyzed by use of two subsets represented by the two different structure types of Ce and Lu, the structural features of which change with decreasing/increasing the size of RE³⁺, up to the borderline between both subsets.

Here we present an approach to functionalize silanized single-walled carbon nanotubes (SWNTs) through copper-free click chemistry for the assembly of inorganic and biological nanohybrids. The nanotube functionalization route involves silanization and strain-promoted azide–alkyne cycloaddition reactions (SPACC). This was characterized by X-ray photoelectron spectroscopy, scanning electron microscopy, transmission electron microscopy, Raman spectroscopy and Fourier transform infra-red spectroscopy. Silane–azide-functionalized SWNTs were immobilized from solution onto patterned substrates through dielectrophoresis (DEP). We demonstrate the general applicability of our strategy for the functionalization of SWNTs with metal nanoparticles (gold nanoparticles), fluorescent dyes (Alexa Fluor 647) and biomolecules (aptamers). In this regard, dopamine-binding aptamers were conjugated to the functionalized SWNTs to perform real-time detection of dopamine at different concentrations. Additionally, the chemical route is shown to selectively functionalize individual nanotubes grown on the surface of silicon substrates, contributing towards future nano electronic device applications.

High‐temperature treatment of functional nanomaterials, through postsynthesis calcination, often represents an important step to unlock their full potential. However, such calcination steps usually severely limit the preparation of colloidal solutions of the nanoparticles due to the formation of sintered agglomerates. Herein, a simple route is reported to obtain colloidal solutions of calcined n‐conductive antimony doped tin oxide (ATO) as well as titanium dioxide (TiO₂) nanoparticles without the need for additional sacrificial materials. This is achieved by making use of the reduced contact between individual nanoparticles when they are assembled into aerogels. Following the calcination of the aerogels at 500 °C, redispersion of the nanoparticles into stable colloidal solutions with various solvents can be achieved. Although a slight degree of sintering is inevitable, the size of the resulting aggregates in solution is still remarkably small with values below 30 nm.

Charge storage layers are an important component of electrochromic devices, which are expected to exhibit high storage capacity and transparency as well as fast electron transfer rates. However, these layers often rely on the (de)intercalation of ions into the crystal lattice of the material and therefore require optimization to be compatible with non‐intercalating electrolytes. In this report, the post‐modification of mesoporous antimony‐doped tin oxide (ATO) nanoparticle layers with a redox‐active cerium compound is described. In particular, the switching of the Ce³⁺/Ce⁴⁺ couple on the conductive nanoparticle scaffold is demonstrated using tetrabutylammonium perchlorate as a non‐intercalating electrolyte. Remarkably, high storage capacities of up to 27 mC cm⁻² and transmittance values of ≈90% are achieved. Variation of the antimony doping concentration revealed that nanoparticle layers doped with 15% Sb exhibit the highest capacity, which can be attributed to increased conductivity in the potential range where the Ce³⁺ ions are oxidized. Finally, the cerium‐modified ATO films show promising performance as charge storage layers in an electrochromic device with a viologen‐anchored ATO layer as the electrochromic working electrode. Switching times of ≈0.4 s highlight the fast electron transfer capability of the cerium‐decorated ATO layer, even when a non‐intercalating electrolyte is used.