Fernando Lledó

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.

EXISTENCE OF SPECTRAL GAPS, COVERING MANIFOLDS AND RESIDUALLY FINITE GROUPS

In the present paper we consider Riemannian coverings (X,g) → (M,g) with residually finite covering group Γ and compact base space (M,g). In particular, we give two general procedures resulting in a family of deformed coverings (X,ge) → (M,ge) such that the spectrum of the Laplacian Δ(Xe,ge) has at least a prescribed finite number of spectral gaps provided e is small enough. If Γ has a positive Kadison constant, then we can apply results by Bruning and Sunada to deduce that spec Δ(X,ge) has, in addition, band-structure and there is an asymptotic estimate for the number of components of spec Δ(X,ge) that intersect the interval [0,λ]. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results.

Self-adjoint extensions of the Laplace–Beltrami operator and unitaries at the boundary

Abstract We construct in this article a class of closed semi-bounded quadratic forms on the space of square integrable functions over a smooth Riemannian manifold with smooth compact boundary. Each of these quadratic forms specifies a semi-bounded self-adjoint extension of the Laplace–Beltrami operator. These quadratic forms are based on the Lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible. The corresponding quadratic forms are semi-bounded below and closable. Finally, the representing operators correspond to semi-bounded self-adjoint extensions of the Laplace–Beltrami operator. This family of extensions is compared with results existing in the literature and various examples and applications are discussed.

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 .

On Self-Adjoint Extensions and Symmetries in Quantum Mechanics

Given a unitary representation of a Lie group G on a Hilbert space

On spectral approximation, Følner sequences and crossed products

{\mathcal H}

Følner sequences and finite operators

H, we develop the theory of G-invariant self-adjoint extensions of symmetric operators using both von Neumann’s theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the G-invariant unbounded operator. We also prove a G-invariant version of the representation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of G-invariant self-adjoint extensions of the Laplace–Beltrami operator on a smooth Riemannian manifold with boundary on which the group G acts. These extensions are labeled by admissible unitaries U acting on the L2-space at the boundary and having spectral gap at −1. It is shown that if the unitary representation V of the symmetry group G is traceable, then the self-adjoint extension of the Laplace–Beltrami operator determined by U is G-invariant if U and V commute at the boundary. Various significant examples are discussed at the end.