Hyperfinite II₁ factor - Imported Knowlpedia Content

Definition

The hyperfinite type $\mathrm{II}_1$ factor is a type $\mathrm{II}_1$ factor $M$ with separable predual for which there are finite-dimensional unital $*$ -subalgebras

A_1\subseteq A_2\subseteq\cdots\subseteq M

whose union is strong-operator dense in $M$ . Such a factor is also called approximately finite-dimensional. Murray and von Neumann proved that any two factors satisfying these conditions are isomorphic, so the definition determines a single isomorphism class, conventionally denoted $R$ Murray–von Neumann, uniqueness theorem for approximately finite factors. Hyperfiniteness is an approximation property; $R$ itself is infinite-dimensional.

Standard construction

Equip the infinite tensor product

\bigotimes_{n=1}^{\infty}M_2(\mathbb C)

with its product tracial state and take the weak closure in the associated GNS representation. The finite tensor factors form an increasing sequence of matrix algebras with strongly dense union, and the resulting von Neumann algebra is $R$ . Replacing $M_2(\mathbb C)$ by many other nontrivial sequences of matrix algebras produces the same factor up to isomorphism.

Injectivity and uniqueness

For factors with separable predual, hyperfiniteness is equivalent to injectivity. Connes proved this implication in the course of classifying injective factors; in type $\mathrm{II}_1$ , it identifies every injective factor with $R$ Connes, main classification theorem in the type II₁ case. This equivalence is a theorem, not part of the defining approximation property.

Examples and non-examples

Every matrix algebra is finite-dimensional and hence trivially hyperfinite, but it is type I rather than type $\mathrm{II}_1$ , so it is not $R$ . This near-miss shows why both the factor type and the approximation condition are present in the core definition.

References

Francis J. Murray and John von Neumann, “On Rings of Operators IV,” Annals of Mathematics 44 (1943), 716–808. DOI record. Relevant: the uniqueness theorem for approximately finite type II₁ factors.
Alain Connes, “Classification of Injective Factors. Cases II₁, II∞, IIIλ, λ ≠ 1,” Annals of Mathematics 104 (1976), 73–115. DOI record. Relevant: the main classification theorem in the type II₁ case.