Math 2.0

Functional analysis von Neumann algebras Object

Hyperfinite type II1 factor

Builds on Von Neumann algebraWeak operator topologyHilbert space

Description

The hyperfinite type II1\mathrm{II}_1 factor RR is the von Neumann algebra obtained as the closure of an increasing union of matrix algebras. Embed M2n(ℂ)M_{2^n}(\C) into M2n+1(ℂ)M_{2^{n+1}}(\C) by x↦I2⊗x=(x00x)x \mapsto I_2 \otimes x = \begin{pmatrix} x & 0 \\ 0 & x \end{pmatrix}. These embeddings preserve the normalized trace τ(x)=2−ntr⁡(x)\tau(x) = 2^{-n} \operatorname{tr}(x), so τ\tau is well defined on the union A∞=⋃nM2n(ℂ)A_\infty = \bigcup_n M_{2^n}(\C). Let HH be the Hilbert space completion of A∞A_\infty for the inner product ⟨x,y⟩=τ(y∗x)\langle x, y \rangle = \tau(y^* x), and let A∞A_\infty act on HH by left multiplication. Then RR is the closure of A∞A_\infty in the weak operator topology on HH.

Remarks

The algebra RR is a factor: its center consists of scalars. The trace τ\tau extends to a faithful normal tracial state on RR, and the values of τ\tau on projections fill [0,1][0, 1], so RR has type II1\mathrm{II}_1. It is hyperfinite: it contains an increasing sequence of finite-dimensional subalgebras whose union is weakly dense. Murray and von Neumann proved that a hyperfinite II1\mathrm{II}_1 factor with separable predual is unique up to isomorphism. Connes proved that, among II1\mathrm{II}_1 factors with separable predual, RR is also the unique injective one. Other constructions give the same factor, for example the group von Neumann algebra of the group of finitely supported permutations of ℕ\N. Every type II1\mathrm{II}_1 factor contains a von Neumann subalgebra isomorphic to RR.

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Hilbert space, concept, no evidence levelHilbert spaceconceptVon Neumann algebra, concept, no evidence levelVon Neumann algebraconceptWeak operator topology, concept, no evidence levelWeak operator topologyconceptHyperfinite type II1 factor, object, no evidence levelHyperfinite type II1 …object
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  1. Hilbert space concept
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  3. Weak operator topology concept
  4. Hyperfinite type II1 factor object

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Further reading

  1. Web https://en.wikipedia.org/wiki/Hyperfinite_type_II_factor
About this page
TagNZ4NY8SK1C
URIm2://NZ4NY8SK1C
Kindobject
WritingW0 stub
MSC46L36, 46L10
Instance ofVon Neumann algebra
UsesVon Neumann algebra, Weak operator topology, Hilbert space
Changed2 November 2026

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