Relative dimension, towers of projections and commuting squares of subfactors
type: Item
instance of: scholarly article
title: Relative dimension, towers of projections and commuting squares of subfactors
zbMATH Open document ID: 0699.46042
published in: Pacific Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: If \(N\subset M\) is a subfactor of the type \(II_ 1\) factor M the index of N in M, [M:N], introduced by V. Jones, can only take the values \(\{4 \cos^ 2\pi /(n+2):\) \(n\geq 0\}\cup [4,\infty]\). It can also be defined as the dimension of the smallest nonzero projection in M which expected on N is a scalar multiple of the identity. This paper deals with the set \(\Lambda\) (M,N) of nonzero scalars appearing this way. The main result is the following: Let \(N\subset M\) be a subfactor of finite index. (i) If \([M:N]=4 \cos^ 2\pi /(n+2)\) then \(\Lambda (M,N)=\{0\}\cup \{P_{k-1}([M:N]^{-1})/[M:N]P_ k([M:N]^{- 1\quad}):\quad 0\leq k\leq n-1\},\) (ii) If \([M:N]\geq 4\) and \(t\leq 1/2\) is so that \(t(1-t)=[M:N]^{-1}\) then \(\Lambda (M,N)\cap (0,t)=\{P_{k- 1}([M:N]^{-1})/[M:N]P_ k([M:N]^{-1\quad}):\quad k\geq 0\}.\) Where \(P_ k(x)\) is the Jones' polynomial defined by \(P_{-1}\equiv 1\), \(P_ 0\equiv 1\), \(P_{n+1}(x)=P_ n(x)-xP_{n-1}(x)\).
Mathematics Subject Classification ID: 46L35
zbMATH DE Number: 4146866
zbMATH Keywords: relative dimension
zbMATH Keywords: towers of projections
zbMATH Keywords: commuting squares of
zbMATH Keywords: subfactors
zbMATH Keywords: Jones index
zbMATH Keywords: index
zbMATH Keywords: subfactor of finite index
zbMATH Keywords: Jones' polynomial
author: Sorin Popa
reviewed by: Gustavo Corach
MaRDI profile type: Publication
full work available at URL: pjm.1989.137.181
OpenAlex ID: W1998125204
DOI: 10.2140/PJM.1989.137.181
description: scientific article; zbMATH DE number 4146866