Shifts on the HyperfiniteII1Factor

Geoffrey L Price
1998 Journal of Functional Analysis  
R. T. Powers has constructed a family of unital endomorphisms of the hyperfinite II 1 factor R. The range of each Powers shift _ is a subfactor of index 2 in R. A cocycle conjugacy invariant for a Powers shift is its commutant index, i.e., the first index k for which the range of _ k has a nontrivial relative commutant in R. Previously we have shown that all Powers shifts of commutant index 2 are cocycle conjugate. In this paper results are obtained on the classification of the cocycle
more » ... classes of Powers shifts of higher commutant index. Definition 1.1. A pair : and ; of unital V-endomorphisms of R are conjugate if there exists an automorphism # of R such that # b :=; b #. Definition 1.2. A pair : and ; of unital V-endomorphisms on R are cocycle conjugate if there exists a unitary element W in R such that : b Ad(W) and ; are conjugate. Some care must be exercised in the use of the second definition. In some instances for example, there may exist unitary elements W and U and a shift _ such that the elements Proposition 1.1. The Jones subfactor index [R: _(R)] is a conjugacy invariant. Proof. Suppose : and ; are conjugate shifts. Then the subfactors :(R) and ;(R) of R are conjugate. But then [J, Prop. 2.1.7] :(R) and ;(R) have the same subfactor index in R. K The proof of the following is similar. Proposition 1.2. The Jones subfactor index [R: _(R)] is a cocycle conjugacy invariant. There is another numerical index for shifts which is both a conjugacy and a cocycle conjugacy invariant. This is the least positive integer k such that _ k (R)$ & R is nontrivial, which we shall call the commutant index of _. We shall return to this notion in Section 5 where we give a detailed account of the relationship between the commutant index and a periodicity condition for our main object of study, the binary shifts on R (Theorem 5.8). Next we describe some of the notions which occur in the theory of index for subfactors and which are relevant to our study of shifts of index 2. The reader is referred to [J] (cf. also [Gd]). If N/R is a subfactor of index 2 in R, then there is a hermitian unitary element S in R such that R=[A+BS: A, B # N]. In particular, Ad(S) restricts to a period 2 (outer) automorphism of N. Note that S is not unique: if U is unitary and S$=USU*, for example, then S$ and N also generate R. Let 8 be the conditional expectation from R to N. Then the linear mapping %=28&I is an automorphism of R. Note that %(S)=&S and that % fixes N. Hence % is itself a period 2 automorphism, which satisfies %(A)=A if and only if A # N. If P/N
doi:10.1006/jfan.1997.3225 fatcat:nrrwijfdxbbt3kwnzuafukn5cm