Bimodules, higher relative commutants and the fusion algebra associated to a subfactor.

*(English)*Zbl 0894.46046
Fillmore, Peter A. (ed.) et al., Operator algebras and their applications. Providence, RI: American Mathematical Society. Fields Inst. Commun. 13, 13-63 (1997).

Summary: We prove in this paper that the tensor product of reduced bimodules associated to a subfactor can be recovered as a product of certain projections in the higher relative commutants associated to the subfactor. After giving an elementary introduction to bimodules of \(\text{II}_1\) factors and their relative tensor product, we prove various formulas relating the representations of the Jones tower coming from different \(k\)-step basic constructions and show that the natural shift on the higher relative commutants, defined by two consecutive modular conjugations of the tower, can be computed in terms of orthonormal bases and the Jones projections \(e_i\). We give a detailed account of how the principal graphs of a subfactor can be recovered by calculating dimensions of intertwiner spaces of certain (reduced) bimodules and show that each vertex of the principal graphs represents a unique reduced bimodule. Then we define the (full) fusion algebra associated to a subfactor and prove that this fusion algebra can be calculated by computing products of certain projections in the higher relative commutants of the subfactor. Explicit formulas for these products are given. Finally, we discuss reduced subfactors and give a procedure to compute the fusion algebra of a subfactor in those situations, when the principal graphs are simple. We show the relation to reduced subfactors and discuss in detail the example of a subfactor with principal graph \(E_6\) to illustrate the general algorithm.

For the entire collection see [Zbl 0855.00023].

For the entire collection see [Zbl 0855.00023].

##### MSC:

46L37 | Subfactors and their classification |

46L10 | General theory of von Neumann algebras |

46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |