AN INVARIANT FOR SUBFACTORS IN THE VONNEUMANN ALGEBRA OF A FREE GROUP

In this Note we are considering a new invariant for subfactors in the von Neumann algebra L(F(k)) of a free group. This invariant is obtained by computing the Connes's chi invariant for the enveloping von Neumann algebra in the iteration of the Jone's basic construction for the given inclusion. In the case of the subfactors considered in [22], [24] this invariant is easily computed as a relative chi invariant, in the form considered in [14]. One considers the inclusion L (F(n)) subset-or-equal-to A = L (F(n)) x (theta)Z(k)2, where theta is an injective homomorphism from Z(k) into Out (L (F(n))) (i. e. a Z(n)-kernel) with minimal period k2 [in Aut (L (F(n)))]. Then there exists a canonical copy theta of Z(k) in chi(A) which can be lifted to Aut(A) [4]. The decomposition of the generator of the dual action of Z(k) on A x (theta)Z(k) as the product of a centrally trivial automorphism and an almost inner automorphism, gives an action of Z(k)2 + Z(k)2 on the algebra A x (theta)Z(k). The algebraic invariants [9] for this last action give a more subtle invariant for theta. As an application we show that, contrary to the case of finite group actions (or more general G-kernels) on the hyperfine II1 factor (settled in [2], [9], [18]), there exists non outer conjugate, injective homomorphisms (i.e. two Z2-kernels) from Z, into Out(L(F(k))), with non-trivial obstruction to lifting to an action on L (F(k)). Moreover the algebraic invariants [3] do not distinguish between these two Z2-kernels. Also, there exists two non-outer conjugate, outer actions of Z2 on L(F(k)) x R that are neither almost inner or centrally trivial.