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 ...

Abstract. We construct ergodic actions of compact quantum groups on C ∗-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of different nature from ergodic actions of compact groups. In particular, we construct: (1). an ergodic action of the compact quantum Au(Q) on the type IIIλ Powers factor Rλ for an appropriate positive Q ∈ GL(2, R); (2). an ergodic action of the compact quantum group Au(n) on the hyperfinite II1 factor R; (3). an ergodic action of the compact quantum group Au(Q) on the Cuntz algebra On for each positive matrix Q ∈ GL(n, C); (4). ergodic actions of compact quantum groups on the their homogeneous spaces, and an example of a non-homogeneous classical space that admits an ergodic action of a compact quantum group. 1.

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.

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.

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 ⊃ C and the relative commutant satisfies the minimality condition A ′ ∩ 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 TC < T, where TC is 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 TC and 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 ǫ also for the

Dedicated to R.V. Kadison on the occasion of his seventieth birthday Abstract. In which a theory of dimension related to the Jones index and based on the notion of conjugation is developed. An elementary proof of the additivity and multiplicativity of the dimension is given and there is an associated trace. Applications are given to a class of endomorphisms of factors and to the theory of subfactors. An important role is played by a notion of amenability inspired by the work of Popa.

Abstract. Let Od be the Cuntz algebra on generators S1,..., Sd, 2 ≤ d < ∞, and let Dd ⊆ Od be the abelian subalgebra generated by monomials S α S ∗ α = S α1 · · · S α k S ∗ α k · · · S ∗ α1 where α = (α1... αk) ranges over all multi-indices formed from {1,..., d}. In any representation of Od, Dd may be simultaneously diagonalized. Using Si (SαS ∗ α) = ( SiαS ∗) iα Si, we show that the operators Si from a general representation of Od may be expressed directly in terms of the spectral representation of Dd. We use this in describing a class of type III representations of Od and corresponding endomorphisms, and the heart of the memoir is a description of an associated family of AF-algebras arising as the fixed-point algebras of the associated modular automorphism groups.

We show that there exists a non-cocommutative comultiplication ∆ϕ and a counit ε on the direct sum of Cuntz algebras

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