Duality of compact groups and Hilbert C*-systems for C*-algebras with a nontrivial center

In the present paper we prove a duality theory for compact groups in the case when the C*-algebra A, the fixed point algebra of the corresponding Hilbert C*-system (F,G), has a nontrivial center Z and the relative commutant satisfies the minimality condition A.'\cap F = Z 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 T_\c < T, where T_\c{i}s a suitable DR-category and T a full subcategory of the category of endomorphisms of A. Both categories have the same objects and the arrows of T can be generated from the arrows of T_\c{a}nd the center Z. A crucial new element that appears in the present analysis is an abelian group C(G), which we call the chain group of G, and that can be constructed from certain equivalence relation defined on G^, the dual object of G. The chain group, which is isomorphic to the character group of the center of G, determines the action of irreducible endomorphisms of A when restricted to Z. Moreover, C(G) encodes the possibility of defining a symmetry $\epsilon$ also for the larger category T of the previous inclusion.Comment: Final version appeared in Int. J. Math. 15 (2004) 759-812. Minor changes w.r.t. to the previous versio

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 finite-dimensional multiplicity space K, is constructed such that exactly the resonances (poles of the inverse of the Livsic-matrix) are (generalized) eigenvalues of H. The corresponding eigen-antilinearforms are calculated explicitly. Using the wave matrices for the wave (Moller) operators the corresponding eigen-antilinearforms on the Schwartz space S for the unperturbed Hamiltonian are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector, which is uniquely determined by restriction of S to the intersection of S with the Hardy space of the upper half plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup of the Toeplitz type for the positive half line on the Hardy space. That is: exactly those pre-Gamov vectors (eigenvectors of the decay semigroup) have an extension to a generalized eigenvector of H if the eigenvalue is a resonance and if the multiplicity parameter k is from that subspace of K which is uniquely determined by its corresponding Dirac type antilinearform.Comment: 16 page

Superselection structures for C*-algebras with nontrivial center

35 pages, no figures.-- MSC2000 codes: 46L05, 46L60.MR#: MR1475657 (99g:46097)Zbl#: Zbl 0893.46046We present and prove some results within the framework of Hilbert C*-systems $\{{\cal F},{\cal G}\}$ with a compact group ${\cal G}$. We assume that the fixed point algebra ${\cal A}\subset{\cal F}$ of ${\cal G}$ has a nontrivial center ${\cal Z}$ and its relative commutant w.r.t. ${\cal F}$ coincides with ${\cal Z}$, i.e., we have {\cal A}'\cap{\cal F}= {\cal Z}\supset\bbfC\text{\bf 1}. In this context, we propose a generalization of the notion of an irreducible endomorphism and study the behaviour of such irreducibles w.r.t. ${\cal Z}$. Finally, we give several characterizations of the stabilizer of ${\cal A}$.Publicad