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 obtaine d by computing the Connes's chi invariant for the enveloping von Neuma nn algebra in the iteration of the Jone's basic construction for the g iven 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 h omomorphism from Z(k) into Out (L (F(n))) (i. e. a Z(n)-kernel) with m inimal period k2 [in Aut (L (F(n)))]. Then there exists a canonical co py theta of Z(k) in chi(A) which can be lifted to Aut(A) [4]. The deco mposition 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 in ner 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, contr ary to the case of finite group actions (or more general G-kernels) on the hyperfine II1 factor (settled in [2], [9], [18]), there exists no n outer conjugate, injective homomorphisms (i.e. two Z2-kernels) from Z, into Out(L(F(k))), with non-trivial obstruction to lifting to an ac tion on L (F(k)). Moreover the algebraic invariants [3] do not disting uish between these two Z2-kernels. Also, there exists two non-outer co njugate, outer actions of Z2 on L(F(k)) x R that are neither almost in ner or centrally trivial.