On continuity of the index of subfactors of a finite factor
type: Item
instance of: scholarly article
title: On continuity of the index of subfactors of a finite factor
zbMATH Open document ID: 0641.46035
DOI: 10.1016/0022-1236(88)90048-1
author: Bahman Mashhood
author: Keith F. Taylor
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: The authors consider the behaviour of the index of N in L as N varies, where the von Neumann algebra L is a type \(II_ 1\) factor with normalized trace \(tr_ L\) and N is subfactor of L. First the authors set up their notations. According to \textit{V. F. R. Jones} [Invent. Math. 72, 1-25 (1983; Zbl 0508.46040)] the index of N in L is defined as follows: if M is a finite factor acting on a Hilbert space H with M' also finite and \(tr_ M\) resp. \(tr_{M'}\) is the normalized trace on M resp. M', then the coupling constant \(\dim_ M(H)\) is \(tr_ M(E_ f^{M'})/tr_{M'}(E_ f^ M)\), where f is a nonzero vector in H and \(E_ f^ M\) resp. \(E_ f^{M'}\) is the orthogonal projection of H onto \(\overline{Mf}\) resp. \(\overline{M'f}\). If L is acting on a Hilbert space H then the index of N in L is \[ [L:N]:= \begin{cases} \dim_ N(H)/\dim_ L(H) &\text{ if \(N'\) is finite, } \\ \infty &\text{ if \(N'\) is infinite.}\end{cases} \] If M, N are two subfactors of L, then the distance d(M,N) between M and N is defined as the Hausdorff distance between the unit balls of M and N using the norm \(\| \cdot \|_ 2\) on L induced by \(tr_ L\). Let SF(L) denote the set of all subfactors of L with same identity and let SFT(L) denote the subset of SF(L) consisting of subfactors with trivial relative commutant in L, then (SF(L),d) is a complete metric space and SFT(L) is a closed subset of SF(L). Next the authors define a map index by \[ \text{index}(\cdot):SF(L)\to [1,\infty],\quad M\mapsto \text{index}(M):=[L:M]. \] After these prelimaries the authors prove their main results, which can be stated as follows: 1. The map index is a lower semicontinuous function on SF(L), but 2. in general it is not continuous on SF(L), because for any hyperfinite type \(II_ 1\) factor \(M\in SF(L)\) with index\((M)<\infty\), there exists a sequence \(\{M_ n\}_{n\in {\mathbb{N}}}\) in SF(L) with \(M_ n\subseteq M\), (n\(\in {\mathbb{N}})\), and \(d(M_ n,M)\to 0\) but index \((M_ n)\to \infty\), as \(n\to \inf…
Mathematics Subject Classification ID: 46L35
zbMATH DE Number: 4044486
zbMATH Keywords: index
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: factor with normalized trace
zbMATH Keywords: Hausdorff distance
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90048-1
OpenAlex ID: W2007050488
cites work: Subalgebras of a finite algebra
cites work: Index for subfactors
cites work: A polynomial invariant for knots via von Neumann algebras
cites work: Entropy and index for subfactors
cites work: On a problem of R.V. Kadison on maximal Abelian *-subalgebras in factors
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description: scientific article; zbMATH DE number 4044486