Shifts of integer index on the hyperfinite \(II_ 1\) factor
type: Item
instance of: scholarly article
title: Shifts of integer index on the hyperfinite \(II_ 1\) factor
zbMATH Open document ID: 0663.46054
author: Geoffrey L. Price
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 4084360
zbMATH Keywords: prime factorization
zbMATH Keywords: Jones index
zbMATH Keywords: uncountable number of nonconjugate shifts corresponding to every prime number
Wikidata QID: Q30039758
reviewed by: Herbert Halpern
MaRDI profile type: Publication
full work available at URL: pjm.1988.132.379
OpenAlex ID: W2077888598
DOI: 10.2140/PJM.1988.132.379
description: scientific article; zbMATH DE number 4084360