Type II von Neumann algebra - Imported Knowlpedia Content

Definition

A von Neumann algebra $M$ is type II if it is semifinite and has no nonzero abelian projection. Here a projection $p\in M$ is abelian when the corner $pMp$ is a commutative algebra. Thus type II algebras possess enough finite projections or, equivalently in the standard formulation, a faithful normal semifinite trace, but none of their nonzero projection corners is type I and abelian. This definition applies to general von Neumann algebras, not only to factors.

The subdivisions

\mathrm{II}_1

and

\mathrm{II}_\infty

A type II factor is of type $\mathrm{II}_1$ when its identity is finite; it then has a unique normalized faithful normal trace. It is of type $\mathrm{II}_\infty$ when its identity is infinite; it has a faithful normal semifinite trace but no finite trace normalized on the identity. A general type II algebra admits a canonical central decomposition into $\mathrm{II}_1$ and $\mathrm{II}_\infty$ parts.

Distinguishing nearby conditions

Having no nonzero minimal projections does not by itself imply type II: a diffuse abelian von Neumann algebra also has no minimal projections, but it has nonzero abelian projections and is type I. The no-abelian-projection condition is what excludes this case. Type III algebras are also non-type-I, but, unlike type II algebras, they have no nonzero finite projections.

References

R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, American Mathematical Society, 1997. DOI record. Relevant: Section 6.5 on the type decomposition.
M. Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on finite, semifinite, and type II von Neumann algebras.