Type II∞ factor - Imported Knowlpedia Content

Definition

A type $\mathrm{II}_\infty$ factor is a von Neumann factor $M$ that is type II and whose identity projection $1_M$ is infinite. Equivalently, it is a semifinite, properly infinite factor with no nonzero abelian projection. Semifiniteness supplies nonzero finite projections below every nonzero projection, while the infinite identity excludes the finite type $\mathrm{II}_1$ case. A faithful normal semifinite trace on $M$ therefore takes the value $+\infty$ at $1_M$ ; unlike the normalized trace on a $\mathrm{II}_1$ factor, it is determined only up to a positive scalar.

Projection and trace structure

Every nonzero finite projection $p\in M$ has a type $\mathrm{II}_1$ corner $pMp$ after its trace is normalized. Conversely, the identity can be decomposed into mutually orthogonal finite projections; countable decomposability permits a countable such decomposition. These facts explain how finite corners coexist with a properly infinite ambient algebra Kadison–Ringrose, §6.5.

The defining distinction from a type III factor is precisely this supply of finite projections. Both classes have infinite identity and no minimal projections, but a type III factor has no nonzero finite projection at all.

Examples and non-examples

If $N$ is a type $\mathrm{II}_1$ factor and $H$ is an infinite-dimensional Hilbert space, then

N\mathbin{\overline{\otimes}}B(H)

is type $\mathrm{II}_\infty$ . Its tensor-product trace is semifinite and infinite on the identity.

The algebra $B(H)$ itself is not type $\mathrm{II}_\infty$ : it is a type $\mathrm{I}_\infty$ factor because it contains minimal projections. A type $\mathrm{II}_1$ factor is also a near miss, failing only the requirement that its identity be infinite.

References

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 type decomposition and type $\mathrm{II}_\infty$ factors.
M. Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on finite and semifinite von Neumann algebras.