# nLab von Neumann algebra factor

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

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

###### Definition

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

$Z\left(A\right):=A\cap A\prime =ℂ1\phantom{\rule{thinmathspace}{0ex}}.$Z(A) := A \cap A' = \mathbb{C}1 \,.

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

## Properties

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

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## References

### General

The original sources are

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.

• 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)

Revised on April 24, 2013 17:07:17 by Urs Schreiber (82.169.65.155)