A DICHOTOMY FOR THE NUMBER OF ULTRAPOWERS

We prove a strong dichotomy for the number of ultrapowers of a given model of cardinality ≤ 2ℵ0 associated with nonprincipal ultrafilters on ℕ. They are either all isomorphic, or else there are 22ℵ0 many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II1 factors, as well as their relative commutants and include several applications. We also show that the CAF001-algebra [Formula: see text] always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on ℕ.

LOCAL NATURE OF COSET MODELS

The local algebras of the maximal Coset model [Formula: see text] associated with a chiral conformal subtheory [Formula: see text] are shown to coincide with the local relative commutants of [Formula: see text] in ℬ, provided [Formula: see text] possesses a stress-energy tensor.Making the same assumption, the adjoint action of the unique inner-implementing representation [Formula: see text] associated with [Formula: see text] on the local observables in ℬ is found to define net-endomorphisms of ℬ. This property is exploited for constructing from ℬ a conformally covariant holographic image in (1+1) dimensions which proves useful as a geometric picture for the joint inclusion [Formula: see text].Immediate applications to the analysis of current subalgebras are given and the relation to normal canonical tensor product subfactors is clarified. A natural converse of Borchers' theorem on half-sided translations is made accessible.

Multi-Sided Braid Type Subfactors

We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite II1 factor R in [E1]—which arise by considering unitary braid representations with certain properties—to multisided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples—where the braid representations are obtained in connection with the representation theory of Lie algebras of types A, B, C, D. We also compute the (first) relative commutants.