Type III von Neumann algebra - Imported Knowlpedia Content

Definition

A von Neumann algebra $M$ is type III if it has no nonzero finite projection. Equivalently, every nonzero projection of $M$ is infinite. This condition applies to the whole algebra, not only to its center. A general von Neumann algebra has a unique largest type III central summand, while saying that $M$ is type III means that this summand has central support $1$ . If $M$ is also a factor, it is called a type III factor.

Consequences

A nonzero type III von Neumann algebra admits no nonzero normal semifinite trace, because any positive finite-trace spectral cut would yield a nonzero finite projection. In particular, it has no tracial state. This does not mean that it has no normal states or no faithful normal semifinite weights: normality, semifiniteness of a weight, and the tracial identity are separate conditions. The absence of finite projections is the defining type III feature.

Factor subtypes

The modular-spectrum refinement divides type III factors into $\mathrm{III}_0$ , $\mathrm{III}_\lambda$ for $0<\lambda<1$ , and $\mathrm{III}_1$ , using modular-theoretic invariants. These labels refine type III; they do not arise by assigning a “dimension” to projections as in types I and II. The subtype definitions apply through these invariants, while deeper structure and uniqueness theorems may require hypotheses such as a separable predual; such hypotheses must accompany the theorem that uses them.

Classification convention

The Roman numeral III refers to Murray–von Neumann type. An algebra with both a semifinite central summand and a type III central summand is not itself type III under the convention used here; only its latter central summand is. The projection criterion and the central type decomposition are given in Kadison–Ringrose, §6.5.

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 type III algebras and the central type decomposition.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on factor types and traces.