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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Formulae and continuity for the index
of subfactors


Authors: Sergey Dorofeev and Klaus Thomsen
Journal: Proc. Amer. Math. Soc. 125 (1997), 2007-2011
MSC (1991): Primary 46L37
MathSciNet review: 1376757
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Abstract: Let $N \subset M$ be an inclusion of $II_{1}$-factors, $\tau $ the trace state of $M$, and $\mathcal {P}(M)$, $\mathcal {P}(N)$ the set of projections in $M$ and $N$, respectively. We prove that the Jones index for the inclusion is

\begin{equation*}\begin {split} [M : N ]&= sup_{e \in \mathcal {P}(M) \backslash \{0\}} \ inf_{p \in \mathcal {P}(N) \backslash \{0\}} \ \frac {\tau (p)}{\tau (ep)}\\ &=sup_{e \in \mathcal {P}(M) \backslash \{0\}} \ inf \{ \frac {\tau (p)}{\tau (ep)} : p \in \mathcal {P}(N), \ e \preceq p \} \ . \end {split}\end{equation*}

This formula is exploited to obtain continuity results for the index. In particular, we obtain a formula for the index which expresses $[M:N]$ in terms of the positions of $N_{i}$ and $M_{j} , \ i,j \in \mathbb {N}$, in $M$, when $N_{1} \subset N_{2} \subset N_{3} \subset \cdots $ and $M_{1} \subset M_{2} \subset M_{3} \subset \cdots $ are finite-dimensional $C^{\ast }$-subalgebras with dense union in $N$ and $M$, respectively.


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Additional Information

Sergey Dorofeev
Affiliation: Matematisk Institut, Ny Munkegade, 8000 Aarhus C, Denmark
Email: dorofeev@mi.aau.dk

Klaus Thomsen
Affiliation: Matematisk Institut, Ny Munkegade, 8000 Aarhus C, Denmark
Email: matkt@mi.aau.dk

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03797-0
PII: S 0002-9939(97)03797-0
Received by editor(s): January 16, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society