Approximately Inner and Centrally Free Commuting Squares of Type II1Factors and Their Classification

We investigate in detail the concepts of approximate innerness and central freeness for commuting squares of type II1factors, with finite horizontal Jones index, introduced in [Po2], and prove a classification result showing that if[formula]is a nondegenerate approximately inner and centrally free commuting square of type II1factors withQ⊂Nstable andΓQ, Pstrongly amenable, then the commuting square is isomorphic to[formula], whereQst⊂Pstis the standard model of the inclusionQ⊂P.