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Intersections of finite families of finite index subfactors. (English) Zbl 1059.46043
Let $$M\subset B(H)$$ be a factor with separable predual and let $$E:M\to N\subset M$$ be a faithful conditional expectation. It is proved that if $$e=\sum_{1\leq i\leq k}m_iR_i$$ with $$m_i\in M$$, $$R_i\in B(H)$$, such that $$R_im=\rho_i(m)R_i$$ for any $$m\in M$$ and each $$\rho_i\in\text{End}(M)$$, $$1\leq i\leq k$$, has finite index, then $$N\subset M$$ has finite index. This result implies that if the conditional expectations onto a finite family of finite index subfactors generate a finite-dimensional algebra, then the intersection of the subfactors has finite index. Some applications to conformal nets are considered.

##### MSC:
 46L37 Subfactors and their classification 46S99 Other (nonclassical) types of functional analysis 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations
##### Keywords:
finite index subfactor; conditional expectation
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##### References:
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