Hellmut Baumgärtel

DUALITY OF COMPACT GROUPS AND HILBERT C*-SYSTEMS FOR C*-ALGEBRAS WITH A NONTRIVIAL CENTER

In this paper we present duality theory for compact groups in the case when the C*-algebra , the fixed point algebra of the corresponding Hilbert C*-system , has a nontrivial center and the relative commutant satisfies the minimality condition as well as a technical condition called regularity. The abstract characterization of the mentioned Hilbert C*-system is expressed by means of an inclusion of C*-categories , where is a suitable DR-category and a full subcategory of the category of endomorphisms of . Both categories have the same objects and the arrows of can be generated from the arrows of and the center . A crucial new element that appears in the present analysis is an abelian group , which we call the chain group of , and that can be constructed from certain equivalence relation defined on , the dual object of . The chain group, which is isomorphic to the character group of the center of , determines the action of irreducible endomorphisms of when restricted to . Moreover, encodes the possibility of defining a symmetry ∊ also for the larger category of the previous inclusion.

Twisted duality of the CAR-algebra

We give a complete proof of the twisted duality property M(q)′=ZM(q⊥)Z* of the (self-dual) CAR-Algebra in any Fock representation. The proof is based on the natural Halmos decomposition of the (reference) Hilbert space when two suitable closed subspaces have been distinguished. We use modular theory and techniques developed by Kato concerning pairs of projections in some essential steps of the proof. As a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator. This formula can be also applied to fermionic free nets, hence giving a formula of the modular operator for any double cone.

Superselection structures for C*-algebras with nontrivial center

We present and prove some results within the framework of Hilbert C*-systems with a compact group . We assume that the fixed point algebra of has a nontrivial center and its relative commutant w.r.t. ℱ coincides with , i.e. we have . In this context we propose a generalization of the notion of an irreducible endomorphism and study the behaviour of such irreducibles w.r.t. . Finally, we give several characterizations of the stabilizer of .

ADDENDUM: "GENERALIZED EIGENVECTORS FOR RESONANCES IN THE FRIEDRICHS MODEL AND THEIR ASSOCIATED GAMOV VECTORS"

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on R with multiplicity space K,dim K < ∞ is constructed such that exactly the resonances (poles of the inverse of the Livšic-matrix) are (generalized) eigenvalues of H. The corresponding eigen(anti-)linearforms are calculated explicitly. Using the wave matrices for the wave (Möller) operators the corresponding eigen(anti-)linearforms on the Schwartz space S for the unperturbed Hamiltonian H0 are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector λ → k/(ζ0−λ) −1, ζ0 resonance, k ∈ K, which is uniquely determined by restriction of S to S ∩H2 +, where H2 + denotes the Hardy space of the upper half plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup for t ≥ 0 of the Toeplitz type on H2 +. That is: exactly those pre-Gamov vectors λ → k/(ζ − λ)−1, ζ from the lower half plane, k ∈ K, have an extension to a generalized eigenvector of H if ζ is a resonance and if k is from that subspace of K which is uniquely determined by its corresponding Dirac type anti-linearform.

On free nets over Minkowski space

Abstract Using standard results on CAR- and CCR-theory and on representation theory of the Poincare group a direct way to construct nets of local C ∗ -algebras satisfying Haag-Kastlers axioms is given. No explicite use of any field operator or of any concrete representation of the algebra is made. The nets are associated to models of mass m ⩾ 0 and arbitrary spin or helicity. Finally, Fock states satisfying the spectrality condition are specified.

Resonances of quantum mechanical scattering systems and Lax–Phillips scattering theory

For selected classes of quantum mechanical scattering systems a canonical association of a decay semigroup is presented. The spectrum of the generator of this semigroup is a pure eigenvalue spectrum and it coincides with the set of all resonances. The essential condition for the results is the meromorphic continuability of the scattering matrix onto C∖(−∞,0] and the rims R−±i0. Further finite multiplicity is assumed. The approach is based on an adaption of the Lax–Phillips scattering theory to semibounded Hamiltonians. It is applied to trace class perturbations with analyticity conditions. A further example is the potential scattering for central-symmetric potentials with compact support and angular momentum 0.

Lax-Phillips evolutions in quantum mechanics and two-space scattering

Canonical extensions of “physical Hamiltonians” are introduced. They are generators of Lax-Phillips (LP-)evolutions. Restricted to the outgoing/incoming subspace an LP-evolution is a contractive semigroup. Spectral properties of the generator and its adjoint are mentioned. Characteristic conditions for the identification operator of a two-space scattering system, consisting of a unitary strongly continuous evolution U(R) on a Hilbert space H and the so-called reference LP-evolution on L2(R, dx, K), are presented in such a way that {U(R), H} is an LP-evolution.

A Modified Approach to the Doplicher/Roberts Theorem on the Construction of the Field Algebra and the Symmetry Group in Superselection Theory

Let be a unital C*-algebra with trivial center . Let denote a tensorial category of unital endomorphisms of equipped with several properties to be explained in the text. Doplicher and Roberts have shown, among other things, that there is a C*-algebra and a compact group of automorphisms of ℱ such that ℱ is a Hilbert C*-system over w.r.t. , where is the fixed point algebra w.r.t. , and the objects are characterized as the canonical endomorphisms of certain algebraic -invariant Hilbert spaces ℋρ⊂ℱ, see Doplicher/Roberts [1, 2, 3]. The starting point of the approach presented in this paper to point out the mentioned result is an -leftmodule ℱ0:={∑ρ,jAρ,jΦρ,j}. ρ runs through a full system of irreducible and mutually disjoint objects of , j=1,2,…,d(ρ), where d(ρ) denotes the statistical dimension of is an orthonormal basis of a d(ρ)-dimensional Hilbert space. The system {Φρj}ρj forms a leftmodule basis of ℱ0, the coefficients Aρj are members of . The strategy is to equip successively ℱ0 with a bimodule structure, a product and a *-structure and finally with a C*-norm ||.||*. The symmetry group appears as the group of all automorphisms of the *-algebra ℱ0 leaving the -scalar product invariant, where F=∑ρ,jAρjΦρj, G=∑ρ,jBρjΦρj. The field algebra is then given by ℱ:= clo||.||*ℱ0.

The Resonance-Decay Problem in Quantum Mechanics

In the paper the so-called “Resonance-Decay Problem in Quantum Mechanics” is solved for a selected class of Hamiltonians: The absolutely continuous part of the Hamiltonian is unitarily equivalent to a selfadjoint operator \( H \) on the Hilbert space \( \mathcal{H}_+\,\,:= L^2(\mathbb{R}_+,\,\mathcal{K},\,d\lambda),\,\mathcal{K} \) the multiplicity space, such that \( H \) together with the multiplication operator on \( \mathcal{H}_+ \) forms an asymptotic complete scatterings ystem such that the scattering matrix \( S(.) \) is holomorphic in the upper half-plane and satisfies certain conditions at 0, at infinity and on the rim \( \mathbb{R}\_\,\,+i0 \). The proof uses methods of the Lax-Phillips scatteringtheo ry.

On a critical radiation density in the Friedmann equation

The paper presents a classification of the basic types of admissible solutions of the general Friedmann equation with non-vanishing cosmological constant and for the case that radiation and matter do not couple. There are four distinct types. The classification uses first the discriminant of a polynomial of the third degree, closely related to the right hand side of the Friedmann equation. The decisive term is then a critical radiation density which can be calculated explicitly.