An Invariant for Subfactors in the Von Neumann Algebra of a Free Group
نویسنده
چکیده
In this paper we are considering a new invariant for subfac-tors in the von Neumann algebra L(F k) of a free group. This invariant is obtained by computing the Connes' 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 in Po2], Ra1] this invariant is easily computed as a relative invariant, in the form considered in Kaw]. As an application we show that, contrary to the case of nite group actions (or more general G-kernels) on the hyperrnite II 1 factor (settled in Co1], Oc2], Jo3]), there exist non outer conjugate, injective homomor-phisms (i.e two Z 2-kernels) from Z 2 into Out(L(F k)), with non-trivial obstruction to lifting to an action on L(F k). Moreover, the algebraic in-variants ((Co2]) do not distinguish between these two Z 2-kernels. Also, there exists two non-outer conjugate, outer actions of Z 2 on L(F k)R that are neither almost inner or centrally trivial. The aim of the present paper is to propose a new invariant for subfactors in the von Neumann algebra of a free group. The invariant. Given A B an inclusion of type II 1 factors, of nite index, we let B 1 be the enveloping algebra for the tower of algebras in the (iterated) Jones's basic construction ((Jo1]) for A B. Then (B 1) Out (B 1) is a conjugacy invariant for A B.
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