Minimal index and braided subfactors

Abstract

The minimal index is shown to be multiplicative: if M 1 ⊂ M 2 ⊂ M 3 are factors, then Ind( M 1 , M 3 ) = Ind( M 1 , M 2 ) Ind( M 2 , M 3 ), extending a previous result ( H. Kosaki and R. Longo, J. Funct. Anal. 107 (1992), 458–470 ). If M is an infinite factor, it follows that the dimension (the square root of the index) is an involutive homomorphism d: Sect O (M) → R + of the semiring of sectors with finite index. The result is applied to the study of the class of endomorphisms with a braid group symmetry that satisfies the relation between index and statistics in Quantum Field Theory ( 18. , 19. ); the analysis is generalized to this case. For these endomorphisms, the set of possible index values has several gaps besides the Jones restriction, for example, the index does not lie in the interval (4, 3 √2). As a consequence, subfactors arising in low-dimensional Quantum Field Theory cannot be arbitrary.