We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of one--sided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C --algebras are Morita equivalent; the groupoid C --algebra C (G) is the Cuntz--Krieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C --algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C --algebras using the general theory of Renault, and calculate their K-theory. 1 Introduction Over the past fifteen years many C -algebras and classes of C -algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...

The first part of this paper extends the Doplicher-Haag-Roberts theory of superselection sectors to quantum field theory on arbitrary globally hyperbolic spacetimes. The statistics of a superselection sector may be defined as in flat spacetime and each charge has a conjugate charge when the spacetime possesses non-compact Cauchy surfaces. In this case, the field net and the gauge group can be constructed as in Minkowski spacetime. The second part of this paper derives spin-statistics theorems on spacetimes with appropriate symmetries. Two situations are considered: First, if the spacetime has a bifurcate Killing horizon, as is the case in the presence of black holes, then restricting the observables to the Killing horizon together with “modular covariance” for the Killing flow yields a conformally covariant quantum field theory on the circle and a conformal spin-statistics theorem for charged sectors localizable on the Killing horizon. Secondly, if the spacetime has a rotation and PT symmetry like the Schwarzschild-Kruskal black holes, “geometric modular action” of the rotational symmetry leads to a spin-statistics theorem for charged covariant sectors where the spin is defined via the SU(2)-covering of the spatial rotation group SO(3).

A two-sided coaction ffi : M! G\Omega M\Omega G of a Hopf algebra (G; \Delta; ffl; S) on an associative algebra M is an algebra map of the form ffi = (\Omega id M ) ffi ae = (id M\Omega ae) ffi , where (; ae) is a commuting pair of left and right G-coactions on M, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra G on M by / and . , respectively, we define the diagonal crossed product M ./ G to be the algebra generated by M and G with relations given by 'm = (' (1) . m / S \Gamma1 (' (3) )) ' (2) ; m 2 M; ' 2 G : We give a natural generalization of this construction to the case where G is a quasi--Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e., where the coproduct \Delta is non-unital). In these cases our diagonal crossed product will still be an associative algebra structure on M\Omega G extending M j M\Omega 1, even though the analogue of an ordinary crossed ...

We present and proof some results in the context of superselection theory with the following conditions fixed: the observable algebra A has a nontrivial center Z and its relative commutant w.r.t. the field algebra F coincides with Z , i.e. we have A 0 " F = Z oe C 1l . In this frame we study Hilbert C*-systems with a compact group. We propose a generalization of the notion of an irreducible endomorphism and study the influence of the sector structure on Z . Finally we give several characterizations of the stabilizer of A.

Abstract. In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C ∗-algebra ON. A main tool in our analysis is the infinite-dimensional group of all maps T → U (N) (where U(N) is the group of all unitary N-by-N matrices), and we study the extension problem from low-pass filter to multiresolution filter using this group.

The compact quantum groups are objects which generalise at the same time the compact groups, the duals of discrete groups and the q−deformations (with q> 0) of classical compact Lie groups. A compact quantum group is an abstract object which may be described by (is by definition the dual of) the algebra of “continuous functions

We prove that the complete normal field net with compact symmetry group constructed by Doplicher and Roberts starting from a net of local observables in ≥ 2+1 space time dimensions and its set of localized (DHR) representations does not possess nontrivial DHR sectors. Whereas the superselection structure in 1+1 dimensions typically does not arise from a compact group, the DR construction is applicable to ‘degenerate sectors’, the existence of which (in the rational case) is equivalent to non-invertibility of Verlinde’s S-matrix. We prove Rehren’s conjecture that the enlarged theory is nondegenerate, which implies that every degenerate theory is an ‘orbifold ’ theory. Thus, the symmetry of a generic model ‘factorizes ’ into a group part and a pure quantum part which still must be clarified.

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.

We regard a right Hilbert C ∗ –module X over a C ∗ –algebra A endowed with an isometric ∗ –homomorphism φ: A → LA(X) as an object XA of the C ∗ –category of right Hilbert A–modules. Following [11], we associate to it a C ∗ –algebra OXA containing X as a “Hilbert A–bimodule in OXA ”. If X is full and finite projective OXA is the C ∗ –algebra C ∗ (X) , the generalization of the Cuntz–Krieger algebras introduced by Pimsner [27]. More generally, C ∗ (X) is canonically embedded in OXA as the C ∗ –subalgebra generated by X. Conversely, if X is full OXA is canonically embedded in C ∗ (X) ∗ ∗. Moreover, regarding X as an object AXA of the C ∗ –category of Hilbert A–bimodules, we associate to it a C ∗ –subalgebra OAXA of OXA commuting with A, on which X induces a canonical endomorphism ρ. We discuss conditions under which A and OAXA are the relative commutant of each other and X is precisely the subspace of intertwiners in OXA between the identity and ρ on O AXA. We also discuss conditions which imply the simplicity of C ∗ (X) or of OXA; in particular, if X is finite projective and full, C ∗ (X) will be simple if A is X–simple and the “Connes spectrum ” of X is T. 1

Dedicated to Masamichi Takesaki on the occasion of his 70th birthday Abstract. We show that the fixed point subnet of a strongly additive conformal net under the action of a compact group is strongly additive. Using the idea of the proof we define the notion of strong additivity for a pair of conformal nets and we show that a key result about the induction of the pair which we proved previously under the finite index assumption can be generalized to strongly additive pairs of conformal nets. These results are applied to classify conformal nets of central charge c = 1 which are not necessarily rational and satisfy a spectrum condition. Let A be a conformal net (or precosheaf) (cf. §2.1). Let I1, I2 be two disjoint intervals of the circle. In this paper an interval of the circle is defined to be an open connected proper subset of the circle. By [FJ] A is additive in the sense that if In is a sequence of intervals which cover an interval J, i.e., ∪nIn ⊃ J, then ∨nA(In) ⊃ A(J)