Von Neumann factor - Imported Knowlpedia Content

Definition

A von Neumann factor, or simply a factor, is a von Neumann algebra $M$ whose center is

Z(M)=\mathbb C1_M.

Equivalently, $0$ and $1_M$ are the only central projections in $M$ . This condition means that $M$ cannot be decomposed as a nontrivial direct sum along central projections. It does not mean that $M$ has no proper norm-closed ideals as a $C^*$ -algebra; for example, $B(H)$ on an infinite-dimensional Hilbert space is a factor but contains the compact operators as a proper norm-closed ideal.

Closed ideals and central decomposition

Ultraweakly closed two-sided ideals in a von Neumann algebra have the form $Mz$ for central projections $z$ . Hence a factor has no nonzero proper ultraweakly closed two-sided ideals. General von Neumann algebras can often be analyzed as direct integrals of factors, with their centers supplying the measurable parameter algebra.

Type classification

Factors divide into types $I_n$ , $I_\infty$ , $II_1$ , $II_\infty$ , and $III$ , according to the comparison and finiteness behavior of their projections. Type I factors are precisely the algebras $B(H)$ . Type II factors have finite projections but no nonzero abelian projections, while type III factors have no nonzero finite projections.

Representations and factorial states

If a representation $\pi$ generates the von Neumann algebra $\pi(A)''$ , the representation is called factorial when this bicommutant is a factor. A state is factorial when its GNS representation is factorial. Irreducibility is stronger: it forces $\pi(A)'=\mathbb C1$ , whereas factoriality only forces the center of $\pi(A)''$ to be scalar.

References

F. J. Murray and J. von Neumann, “On Rings of Operators,” Annals of Mathematics 37 (1936), 116–229. JSTOR record. Relevant: the introduction of factors and the projection-based type classification.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter V on factors and their types.