A class of shifts on the hyperfinite ${\rm II}\sb 1$ factor

Abstract

We construct and classify up to conjugacy certain shifts on the hyperfinite II,-factor, each being a shift of Jones index n which fails to be an n-shift. In particular for each prime n we construct uncountably many such shifts. A shift a on a von Neumann algebra R is an isomorphism of R into itself such that nF o ak (R) = C. Shifts were introduced by R. T. Powers in [8] and studied further in [4], [9], and [10]. In [2] and [3] we introduced the notion of group shift, thereby generalizing and unifying many of the earlier results. An n-shift is the simplest kind of shift, coming from the group eD=_Z(k) with its canonical shift. The first example of a shift with index n which is not an nshift was constructed in [9] for n = 2. Recently, in [5], Choda, Enomoto, and Watatani, using essentially the methods of [2], constructed uncountably many shifts of index 2 which are not 2-shifts. To construct a group shift, we need to specify a group G, a shift s on G, and a nondegenerate s-invariant 2-cocycle co on G. We fix G to be e= _ Z(k) and s to be given by s(e1) = e +e1+, for all integers j, where eJ is a generator of the j th copy of Zn . By [2], provided s is a shift on G and co is nondegenerate, the data G, s, co give rise to a shift a = a(G, s, co) on the hyperfinite II,-factor R realized as the twisted group von Neumann algebra W* (G, co). In concrete terms, R is presented as generated by a family of unitaries (Uk)kEZ of order n with commutation relations u u1u U* = COw(ei, e1)wt)(e1, ei). The shift a takes uk to w-)(ek, ek+1)Ukuk+l We note that it is easy to see that a is a shift; the difficulty lies in proving that any particular cocycle is nondegenerate. From there it is relatively easy using [2] to see which cocycles give rise to conjugate shifts. Hence we choose a class of s-invariant cocycles which are relatively easy to show nondegenerate. It would be possible to carry out the same kind of calculations for the shift el -fei + el+I + + el+m for fixed m, but the details of specifying an invariant Received by the editors September 6, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L10. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page