On a Subfactor Construction of a Factor Non Anti-Isomorphic to Itself

We give a subfactor construction for a $II_{1}$ factor M which is not anti-isomorphic to itself. The $II_{1}$ factor we consider is essentially the same as the example previously given by Connes. However, our construction uses the recently developed theory of free group factors. We show that there exists an inclusion of $II_{1}$ factors $A\subset B$ which by iteration of the Jones basic construction produces $M$ as the enveloping algebra. Here A is a free group factor and B is isomorphic to the crossed product of A by an action of a finite group. By using a Connes' argument involving the invariant $\chi (M)$, we verify that $M$ is not anti--isomorphic to itself. Publication of this manuscript is funded in part by the National Science Foundation. This material is based upon work supported by the National Science Foundation under Grant No. DMS--9810361.