A Dirac operator D on the standard Podleś sphere S 2 q is defined and investigated. It yields a real spectral triple such that |D| −z is of trace class for Re z > 0. Commutators with the Dirac operator give the distinguished 2dimensional covariant differential calculus on S 2 q . The twisted cyclic cocycle associated with the volume form of the differential calculus is expressed by means of the Dirac operator.

A definition of frames in Krein spaces is stated and a complete characterization is given by comparing them to frames in the associated Hilbert space. The basic tools of frame theory are described in the formalism of Krein spaces. It is shown how to transfer a frame for Hilbert spaces to Krein spaces given by a W -metric, where the Gram operator W is not necessarily regular and possibly unbounded.1 Proof. Since P commutes with J, the subspaces P K and (1 − P )K of K are Krein spaces with fundamental symmetry P J and (1−P )J, respectively. Given a frame {k n } n∈N for K with frame bounds A ≤ B, we have for all k ∈ P Khence {P k n } n∈N is a frame for P K with frame bounds A ≤ B. The same remains true for P replaced by 1 − P . This completes the prove of i). Now let {k + n } n∈N + and {k − n } n∈N − be two frames satisfying the assumptions stated in the proposition. For k ∈ K, set k + := P k and k − := (1 − P )k. Note 8 that [k, k + n ] = [k, P k + n ] = [P k, k + n ] = [k + , k + n ] and, similarly, [k, k − n ] = [k − , k − n ]. From P J = JP , it follows that k 2 J = k + 2 J + k − 2 J . Therefore A k 2 J = A k + 2 J + A k − 2 J

Oxygen and carbon isotopic compositions of tree ring cellulose (delta13Ccell and delta18Ocell) were measured for pines growing at four sites in east Germany. Three sites differed markedly in soil water availability within a short distance and the fourth site served as a reference. The choice of the sites was guided by the desire to detect effects of air pollution on the long-term trend of isotopic compositions and to examine the influence of soil water availability on the relationship between the carbon and oxygen isotope ratios. Locations in east Germany are particularly well suited for the study of pollution effects because there was a steady increase in environmental contamination until the German Reunification in 1990, followed by a sharp decline due to the implementation of stricter environmental standards. The long-term trend of delta13Ccell showed an extraordinary increase in the period 1945-1990 and a rapid decrease after 1990, whereas delta18Ocell remained nearly constant. The increase of delta13Ccell is explained by secondary fractionation caused by phytotoxicity of SO2. Two effects are mainly responsible for the secondary fractionation under SO2 exposure: increase of dark respiration, and changes in photosynthate allocation and partitioning. Both effects do not influence delta18Ocell. Furthermore, a significant positive correlation between the year-to-year variations of carbon and oxygen isotope ratios (delta13Cresid and delta18Oresid) has been found for all sites. The slopes of the relationship between delta13Cresid and delta18Oresid differ insignificantly. It is concluded that this relationship is not influenced by soil water availability but by climatic variables.

An operator-theoretic approach to invariant integrals on non-compact quantum spaces is introduced on the examples of quantum ball algebras. In order to describe an invariant integral, operator algebras are associated to the quantum space which allow an interpretation as "rapidly decreasing" functions and as functions with compact support. If an operator representation of a first order differential calculus over the quantum space is known, then it can be extended to the operator algebras of integrable functions. The important feature of the approach is that these operator algebras are topological spaces in a natural way. For suitable representations and with respect to the bounded and weak operator topologies, it is shown that the algebra of functions with compact support is dense in the algebra of closeable operators used to define these algebras of functions and that the infinitesimal action of the quantum symmetry group is continuous. §1. IntroductionThe development of quantum mechanics at the beginning of the past century resulted in the discovery that nuclear physics is governed by noncommutative quantities. Recently, there have been made various suggestions that spacetime may be described by non-commutative structures at Planck scale. Within this approach, quantum groups might play a fundamental role.

Abstract. We construct spectral triples on all Podleś quantum spheres S 2 qt . These noncommutative geometries are equivariant for a left action of U q .su.2// and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the sphere S 2 . There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.