Tensor C*-categories arising as bimodule categories of II_1 factors

Abstract

We prove that if C is a tensor C*-category in a certain class, then there exists an uncountable family of pairwise non stably isomorphic II_1 factors (M_i) such that the bimodule category of M_i is equivalent to C for all i. In particular, we prove that every finite tensor C*-category is the bimodule category of a II_1 factor. As an application we prove the existence of a II_1 factor for which the set of indices of finite index irreducible subfactors is {1, \frac{5 + \sqrt{13}}{2}, 12 + 3\sqrt{13}, 4 + \sqrt{13}, \frac{11 + 3\sqrt{13}}{2}, \frac{13 + 3\sqrt{13}}{2}, \frac{19 + 5\sqrt{13}}{2}, \frac{7 + \sqrt{13}}{2}}. We also give the first example of a II_1 factor M such that Bimod(M) is explicitly calculated and has an uncountable number of isomorphism classes of irreducible objects.

30 pages. v2: the main result was improved and an application was added, the title changed. v3: typos corrected, bibliography updated, journal reference added