27 research outputs found
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 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 with a compact group . We assume that the fixed point algebra of has a nontrivial center and its relative commutant w.r.t. coincides with , 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. . Finally, we give several characterizations of the stabilizer of .Publicad