von Neumann algebra factor



Functional analysis



For AA a von Neumann algebra write AA' for its commutant in the ambient algebra B()B(\mathcal{H}) of bounded operators.


A von Neumann algebra AA is called a factor if its center is trivial

Z(A):=AA=1. Z(A) := A \cap A' = \mathbb{C}1 \,.

Equivalently: if AA and its commutant AA' generate the full algebra of bounded operators B()B(\mathcal{H}).


Every von Neumann algebra may be written as a direct integral? over factors. (von Neumann 49)


Type I


Type II


Type III




The original sources are

Lecture notes include

  • V.S. Sunder, von Neumann algebras, II 1II_1-factors, and their subfactors (pdf)

  • Hideki Kosaki, Type III factors and index theory (1993) (pdf)


The mathematics of inclusions of subfactors is giving deep structural insights. See also at planar algebra.

  • Vaughan Jones,

    Index for subfactors, Invent. Math. 72, I (I983);

    A polynomial invariant for links via von Neumann algebras, Bull. AMS 12, 103 (1985);

    Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126, 335 (1987)

  • Vaughan Jones, Scott Morrison, Noah Snyder, The classification of subfactors of index at most 5 (arXiv:1304.6141)

  • Vaughan F. R. Jones, David Penneys, Infinite index subfactors and the GICAR categories, arxiv/1410.0856

Relation to quantum field theory

Last revised on June 13, 2016 at 15:31:49. See the history of this page for a list of all contributions to it.