Type I von Neumann algebra - Imported Knowlpedia Content

Definition

A von Neumann algebra $M$ is type I if every nonzero central projection $z\in M$ dominates a nonzero abelian projection: there is a projection $0\neq p\leq z$ such that the corner $pMp$ is abelian. Equivalently, the central support of the family of abelian projections, meaning the supremum of their central supports, is $1$ . This definition applies to algebras with nontrivial center; it does not assert that $M$ is itself a factor. The Roman numeral I names the first Murray–von Neumann type and must not be confused with the separate notion of a type I $C^*$ -algebra.

Factor case and homogeneous pieces

A von Neumann factor is type I exactly when it is isomorphic to $B(H)$ for some Hilbert space $H$ . It is called type $\mathrm{I}_n$ when $\dim H=n<\infty$ , and type $\mathrm{I}_\infty$ when $H$ is infinite-dimensional. A general type I von Neumann algebra is assembled over its center from such homogeneous type I pieces; the dimensions of the factor fibers need not be constant.

Abelian and concrete examples

Every abelian von Neumann algebra is type I because each projection $p$ has an abelian corner $pMp$ . Matrix algebras $M_n(\mathbb C)$ and full operator algebras $B(H)$ are type I factors. By contrast, a type II factor has no nonzero abelian projection, and a type III factor has no nonzero finite projection. These contrasts concern factor types; a general von Neumann algebra may have nonzero central summands of several types.

Classification convention

The type I, II, and III decomposition is a decomposition by unique central projections whose sum is $1$ . Thus phrases such as “ $M$ has a type I part” refer to a central summand, whereas “ $M$ is type I” means that the other two central parts vanish. The abelian-projection formulation and the factor description are treated in Takesaki, Chapter V.

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory, American Mathematical Society, 1997. AMS record. Relevant: §6.5 on the type decomposition.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on types of von Neumann algebras and traces.