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Index of faithful normal conditional expectations


Author: Sze-Kai Tsui
Journal: Proc. Amer. Math. Soc. 111 (1991), 111-118
MSC: Primary 46L37; Secondary 46L10, 46L35
DOI: https://doi.org/10.1090/S0002-9939-1991-1033962-7
MathSciNet review: 1033962
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Abstract: Let $ E$ be a faithful normal conditional expectation of a factor $ M$ onto its subfactor $ N$, and the index of $ E$ be denoted by $ {\operatorname{IND}_E}$. We investigate the question: For two such faithful normal conditional expectations $ {E_1},{E_2}$ of $ M$ onto $ N$, when does $ {\operatorname{IND}}_{{E_1}} = {\operatorname{IND}}_{{E_2}}$ hold? In this paper we answer this question completely for type $ I$ factor $ M$. We also derive a tensor product formula for index, i.e., $ {\operatorname{IND}}_{{E_1} \otimes {E_2}} = ({\operatorname{IND}}_{{E_1}})({\operatorname{IND}}_{{E_2}})$. For any $ \alpha > 9$ we construct uncountable nonisomorphic faithful normal conditional expectations $ E$ of a factor $ M$ onto its subfactor $ N$ with $ {\operatorname{IND}_E} = \alpha $ such that both of $ M$ and $ N$, are of type $ I$ or $ II$ or $ II{I_\lambda },0 \leq \lambda \leq 1$. For each $ \beta \in \{ 4{\cos ^2}\pi /n,\vert n \geq 3\} \cup [4,\infty )$ we exhibit a type $ II{I_\lambda }$ factor $ M$ and its subfactor $ N$ and a faithful normal conditional expectation $ E$ such that $ {\operatorname{IND}}_E = \beta $.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1033962-7
Keywords: Index of faithful normal conditional expectations, index of tensor products, spatial derivatives, relative commutants
Article copyright: © Copyright 1991 American Mathematical Society

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