Type III von Neumann algebra

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 of type III.

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.