We classify two-dimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of A-D-E Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of one-dimensional local conformal nets, Dynkin diagrams D2n+1 and E7 do not appear, but now they do appear in this classification of two-dimensional local conformal nets. Such nets are also characterized as two-dimensional local conformal nets with µ-index equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for one-dimensional nets, is 2-cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.

Canonical tensor product subfactors (CTPS’s) describe, among other things, the embedding of chiral observables in two-dimensional conformal quantum field theories. A new class of CTPS’s is constructed some of which are associated with certain modular invariants, thereby establishing the expected existence of the corresponding two-dimensional theories. 1 Introduction and

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.

We construct the crossed product A ⋊ E ̂ρ of a C*-algebra A with centre Z by an endomorphism ρ, which is special in a weaker sense w.r.t. the notion introduced by Doplicher and Roberts. The notation ̂ρ denotes the tensor C*-category of powers of ρ. We assign to ρ a geometrical invariant, representing a cohomological obstruction to be special in the usual sense, and determining rank and first Chern class of the vector bundle E whose module of sections (contained in A ⋊ E ̂ρ) induces ρ on A. We prove that A is the fixed point C*-algebra w.r.t. a G-action on A ⋊ E ̂ρ, and characterize ̂ρ as the category of tensor powers of a suitable G-Hilbert Z-bimodule (a so-called ’noncommutative pullback ’ of E). G is interpreted as the generally noncompact space of sections of a group bundle.

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.

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

Abstract. Monoidal functors U: C → M with left adjoints determine, in a universal way, monoids T in the category of oplax monoidal endofunctors on ”quantum groupoids ” we derive Tannaka duality between left adjointable monoidal functors and bimonads. Bialgebroids, i.e., Takeuchi’s ×R-bialgebras, appear as the special case when T has also a right adjoint. Street’s 2-category of monads then leads to a natural definition of the 2-category of bialgebroids. Contents

The notion of extension of a given C*-category C by a C*-algebra A is introduced. In the commutative case A = C(Ω), the objects of the extension category are interpreted as fiber bundles over Ω of objects belonging

We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the c = 1 Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge c is in a certain subset of (1, ∞), including [2, ∞), and h ≥ (c − 1)/24, the irreducible representation with lowest weight h of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge c is in the above set and satisfies c ≤ 25 then the corresponding Virasoro net has no proper local extensions of compact type. 1