Exact groups and Fell bundles

Kirchberg and Wasserman's Theorem, according to which a group is exact if and only if its reduced C*-algebra is exact, is extended to the context of Fell bundles. Received: 31 December 2000 / Revised version: 7 August 2001 / Published online: 4 April 2002     

Generalized resolvent matrices and spaces of analytic functions

We introduce triplet spaces for symmetric relations with defect index (1, 1) in a Pontryagin space. Representations of Pontryagin spaces by spaces of vector-valued analytic functions are investigated. These concepts are used to study 2×2-matrix valued analytic functions which satisfy a certain kernel condition.     

Scale functions on $p$-adic Lie groups

Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis [14] associates its scale function G : G→ℕ. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over ℚ _p , s(x)=s(h) is determined by the semisimple part h of x∈G. For every finite extension K of ℚ _p , the scale functions of G and H:=G(K) are related by s _H ∣ _G =s _G ^{[ K :ℚ} _p ^]. More generally, we clarify the relations between the scale function of a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients. Received: 20 February 1997; Revised version: 18 May 1998     

Rigidity of the scattering length spectrum

In this paper we consider properties of obstacles satisfying some non-degeneracy conditions that can be recovered from the scattering length spectrum (SLS). Clearly the latter tells us whether the obstacle K is trapping or non-trapping. If the set of trapped points is relatively small, then the SLS also determines the volume of the obstacle, the number of its connected components, and whether its boundary is convex everywhere or it has non-trivial concavities. Under the additional assumption that the curvature of the obstacle does not vanish of infinite order, it is proved that from the SLS one can recover certain information about the number of reflection points of any simply reflecting ray in the exterior of the obstacle. Finally, for some special classes of obstacles (e.g. star-shaped ones), it is shown that the SLS completely determines the obstacle. Received: 2 March 1999 / Revised version: 16 January 2001 / Published online: 5 September 2002     

A high-resolution MAS NMR study on the potential catalysts Mo/HBeta for olefin metathesis: The interaction of Mo species with HBeta zeolite

The interfacial interaction of Mo species with the HBeta zeolite was studied by multinuclear MAS NMR, XRD and N₂ adsorption. As proved by the quantitative ²⁷Al MAS NMR, this interaction is so strong as to dealuminate the framework of HBeta, and leads to a new peak appearing at −14 ppm, which indicates the formation of crystalline Al₂(MoO₄)₃. This can also be detected by XRD measurements when the Mo loading is as high as 9.0 wt.%. The corresponding quantitative ²⁹Si and ¹H MAS NMR spectra show that the amount of silanols and Brønsted acidic sites decrease obviously with increasing Mo loading. This also reveals an interaction between Mo species and HBeta support through an oxygen bridge resulting from condensation with the hydroxyls on the support. At higher Mo loadings, the interaction is so strong that it results in an extraction of aluminum from the zeolite framework, and subsequently appearance of Al₂(MoO₄)₃ and loss of Brønsted acidic sites. These can be correlated to the low catalytic activity of Mo/HBeta in metathesis of ethylene and 2-butylene to propylene.     

Almost maximal spaces

A topological space X is called almost maximal if it is without isolated points and for every x∈X, there are only finitely many ultrafilters on X converging to x. We associate with every countable regular homogeneous almost maximal space X a finite semigroup Ult(X) so that if X and Y are homeomorphic, Ult(X) and Ult(Y) are isomorphic. Semigroups Ult(X) are projectives in the category F of finite semigroups. These are bands decomposing into a certain chain of rectangular components. Under MA, for each projective S in F, there is a countable almost maximal topological group G with Ult(G) isomorphic to S. The existence of a countable almost maximal topological group cannot be established in ZFC. However, there are in ZFC countable regular homogeneous almost maximal spaces X with Ult(X) being a chain of idempotents.     

Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds

We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are given for the lack of local injectivity of the exponential map around its singularities.