An Invariant for Subfactors in the von Neumann Algebra of a Free Group

In this paper we consider a new invariant for subfactors in the von Neumann algebra ℒ(Fk) of a free group. This invariant is obtained by computing the Connes χ invariant for the enveloping von Neumann algebra of the iteration of the Jones basic construction for the given inclusion. In the case of the subfactors considered by S. T. Popa [Invent. Math. 111 (1993), no. 2, 375–405; MR1198815] and by us previously [Invent. Math. 115 (1994), no. 2, 347–389; MR1258909], this invariant is easily computed as a relative χ invariant, in the form considered by Y. Kawahigashi [Duke Math. J. 71 (1993), no. 1, 93–118; MR1230287]. "As an application we show that, in contrast to the case of finite-group actions (or more general G-kernels) on the hyperfinite H1 factor, there exist non-outer-conjugate, injective homomorphisms (i.e., two Z2-kernels) from Z2 into Out(ℒ(Fk)), with non-trivial obstruction to lifting to an action on ℒ(Fk). Moreover, the algebraic invariants do not distinguish between these two Z2-kernels. Also, there exist two non-outer-conjugate, outer actions of Z2 on ℒ(Fk)⊗R that are neither almost inner nor centrally trivial.