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von Neumann algebra factor

Definition

For $A$ a von Neumann algebra write $A\prime$ for its commutant in the ambient algebra $B(\mathscr{H})$ of bounded operators.

Definition

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

$$Z(A):=A\cap A\prime =ℂ1.$$Z(A) := A \cap A' = \mathbb{C}1 \,.

Equivalently: if $A$ and its commutant $A\prime$ generate the full algebra of bounded operators $B(\mathscr{H})$.

Properties

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

Classification

Type I

(…)

Type II

(…)

Type III

(…)

Related concepts

• C-star algebra

• uniformly hyperfinite algebra

References

General

The original sources are

• Murray, John von Neumann, …

• John von Neumann, On rings of operators, reduction theory, Annals of Mathematics Second Series, Vol. 50, No. 2 (1949) (jstor)

• Alain Connes, …

Lecture notes include

• V.S. Sunder, von Neumann algebras, ${\mathrm{II}}_1$-factors, and their subfactors (pdf)

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

Subfactors

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)