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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.

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
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