Type III factor - Imported Knowlpedia Content

Definition

A type III factor is a von Neumann factor $M$ that is a type III von Neumann algebra. Thus $M$ has no nonzero finite projection: every nonzero projection $p\in M$ is infinite. The factor condition separately requires that the center of $M$ consist only of scalars. Neither condition follows from the other. This is the factorial case of the Murray–von Neumann type III class; the finer labels $\mathrm{III}_0$ , $\mathrm{III}_\lambda$ , and $\mathrm{III}_1$ require modular invariants and are not part of this definition.

Projection and corner structure

Every nonzero corner $pMp$ of a type III factor is again a type III factor. Indeed, its center remains scalar and any finite projection in the corner would be finite in $M$ . Nonzero projections have central support $1$ , and the absence of finite projections makes them properly infinite. These projection properties distinguish type III factors from both finite factors and semifinite infinite factors.

Traces and modular structure

A type III factor admits no nonzero normal semifinite trace: semifiniteness would produce nonzero finite projections. Normal states can nevertheless exist, and faithful normal states exist under standard countable decomposability hypotheses. Their modular automorphism groups carry structure that cannot generally be removed by choosing a trace. The resulting modular invariants underlie Connes's finer classification; see Takesaki, Chapter XII.

Examples and scope

Type III factors arise from infinite tensor products, nonsingular group actions, and local algebras in algebraic quantum field theory. Merely being infinite-dimensional is not enough: $B(\ell^2)$ is an infinite-dimensional type I factor, and the hyperfinite $\mathrm{II}_1$ factor is infinite-dimensional but finite.

References

M. Takesaki, Theory of Operator Algebras III, Springer, 2003. DOI record. Relevant: Chapter XII on type III factors and their classification.
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, American Mathematical Society, 1997. DOI record. Relevant: §6.5 on the Murray–von Neumann types.