×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bisch D., Invent. Math. 128 pp 89–
[2] Fröhlich J., Commun. Math. Phys. 155 pp 569–
[3] Goodman R., J. Reine Angew. Math. 347 pp 69–
[4] DOI: 10.1007/978-1-4613-9641-3 · doi:10.1007/978-1-4613-9641-3
[5] DOI: 10.1006/jfan.1997.3228 · Zbl 0915.46051 · doi:10.1006/jfan.1997.3228
[6] DOI: 10.1007/BF01389127 · Zbl 0508.46040 · doi:10.1007/BF01389127
[7] DOI: 10.1090/S0273-0979-1985-15304-2 · Zbl 0564.57006 · doi:10.1090/S0273-0979-1985-15304-2
[8] DOI: 10.1007/PL00005565 · Zbl 1016.81031 · doi:10.1007/PL00005565
[9] DOI: 10.1016/0022-1236(92)90013-9 · Zbl 0798.46047 · doi:10.1016/0022-1236(92)90013-9
[10] DOI: 10.1007/s00220-003-0814-8 · Zbl 1042.46035 · doi:10.1007/s00220-003-0814-8
[11] DOI: 10.1142/S0129055X95000232 · Zbl 0836.46055 · doi:10.1142/S0129055X95000232
[12] Pimsner M., Ann. Sci. École Norm. Sup. 19 pp 57–
[13] Pressley A., Loop Groups (1986)
[14] DOI: 10.1016/0022-1236(77)90068-4 · Zbl 0347.46070 · doi:10.1016/0022-1236(77)90068-4
[15] Sano T., J. Oper. Theory 32 pp 209–
[16] DOI: 10.1007/978-1-4612-6188-9 · doi:10.1007/978-1-4612-6188-9
[17] DOI: 10.1007/s002220050253 · Zbl 0944.46059 · doi:10.1007/s002220050253
[18] DOI: 10.1007/s002200050800 · Zbl 1040.81085 · doi:10.1007/s002200050800
[19] Xu F., Commun. Contemp. Math. 2 pp 307–
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.