Type II₁ factor - Imported Knowlpedia Content

Definition

A type $\mathrm{II}_1$ factor is a von Neumann factor $M$ that is type II and whose identity is a finite projection. Equivalently, it is a type II finite von Neumann algebra with trivial center. Thus $M$ has no nonzero abelian projection, but all its projections are finite. It possesses a unique normalized faithful normal trace $\tau$ , characterized by $\tau(1_M)=1$ and $\tau(xy)=\tau(yx)$ . Infinite-dimensionality is automatic: a finite-dimensional factor is a matrix algebra and hence type I.

Continuous dimension

The trace classifies projections up to Murray–von Neumann equivalence:

p\sim q\quad\Longleftrightarrow\quad \tau(p)=\tau(q).

Moreover every value in $[0,1]$ occurs as $\tau(p)$ for some projection $p$ . This continuous range of projection dimensions explains the older name “finite continuous factor.” These statements are part of the basic dimension theory of finite factors Kadison–Ringrose, §6.5.

Examples and contrasts

The hyperfinite type $\mathrm{II}_1$ factor is obtained as the weak closure, in its tracial representation, of an increasing union of matrix algebras. Group von Neumann algebras of many infinite conjugacy class discrete groups give further examples.

The matrix factor $M_n(\mathbb C)$ is finite but type I, not type $\mathrm{II}_1$ , because it has minimal abelian projections. A type $\mathrm{II}_\infty$ factor has no nonzero abelian projections but its identity is infinite.

Conventions and scope

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 finite continuous factors and dimension theory.
M. Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on finite factors and traces.