Shifts of integer index on the hyperfinite \(II_ 1\) factor
For a fixed integer \(n>1\), let \(\Omega_ n\) be the nth roots of unity. Let \(n=p_ 1^{r_ 1}...p_ s^{r_ s}\) be the prime factorization of n. Let \(\sigma\) be a map of \({\mathbb{Z}}\) into \(\Omega_ n\) such that \(\sigma (0)=1\) and \(\sigma (-j)=\sigma (j)^{-1}\). Sets of unitary operators \(\{u_ j|\) \(j\in {\mathbb{N}}\}\) on a Hilbert space that satisfy \(u^ n_ j=1\) for all j and \(u_ iu_ j=\sigma (i-j)u_ ju_ i\) for all i and j have been considered by \textit{R. T. Powers} [Contemp. Math. 62, 447-460 (1987; Zbl 0654.46057), Am. Math. Soc., Providence, R. I., 1987] and \textit{M. Choda} [J. Operator Theory 17, 223- 235 (1987; Zbl 0621.46052)]. Now let \(\sigma\) be given and let A(\(\sigma)\) be the *-algebra generated by all words in the \(u_ i\) and let tr be the trace on A(\(\sigma)\) given by \(tr(1)=1\) and \(tr(w)=0\) for all words in the \(u_ i\) not equal to scalar multiples of 1. Let A(\(\sigma)\) be identified with its image under the representation of A(\(\sigma)\) induced by tr. Then the *-endomorphism \(\alpha\) of \(R=A(\sigma)''\) induced \(\alpha (u_ i)=u_{i+1}\) for all \(i=1,2,..\). is a shift in the sense that \(\cap_{m\geq 1}\alpha^ m(R)={\mathbb{C}}1.\) Now let \(\{k_ j|\) \(j\in {\mathbb{Z}}\}\) be a sequence in \({\mathbb{Z}}/n{\mathbb{Z}}\) such that \(k_{-j}=-k_ j\) and let \(\sigma\) be the maps of \({\mathbb{Z}}\) into \(\Omega_ n\) given by \(\sigma (j)=\exp (2\pi ik_ j/n)\). Let \(\phi\) be the map of \({\mathbb{Z}}/n{\mathbb{Z}}\) into \({\mathbb{Z}}/p_ 1^{r_ 1}\oplus \cdot \cdot \cdot \oplus {\mathbb{Z}}/p_ s^{r_ s}{\mathbb{Z}}\) given by \(\phi (k)=(kn_ 1P_ 1,...,kn_ sP_ s)\) where \(P_ q=n/p_ q^{r_ q}\) and \(\sum n_ qP_ q=1\). For \(1\leq q\leq s\), the \(u_{jq}=u_ j^{n_ qP_ q}\) satisfy \(u_{iq}u_{jq}=\sigma (i-j)u_{jq}u_{iq}\) and \(u_{iq}u_{jt}=u_{jt}u_{iq}\) for \(q\neq t\) and the operators \(\{u_{jq}| j\}\) generate commuting subalgebras of R. The author shows that R is a type \(II_ 1\) hyperfinite factor if and only if the sequence \[ (...,\phi (k_{-2})_ q,\phi (k_{-1})_ q,\phi (k_ 0)_ q,\phi (k_ 1)_ q,(k_ 2)_ q,...) \] is aperiodic for all \(1\leq q\leq s\). Furthermore, the author shows that the Jones index [R: \(\alpha\) (R)] is n and that \(\sigma (R)'\cap R={\mathbb{C}}1\). Finally, the author identifies the normalizer N(\(\alpha)\) of \(\alpha\) (i.e., unitaries \(V\in R\) with \(V\alpha^ k(R)V^*=\alpha^ k(R)\) for all k), as being scalar multiples of words in A(\(\sigma)\) and shows that two shifts \(\alpha\) and \(\beta\) are conjugate if and only if they correspond to the same \(\sigma\). So there is an uncountable number of nonconjugate shifts corresponding to every prime numberp.
uncountable number of nonconjugate shifts corresponding to every prime number
scientific article; zbMATH DE number 4084360