Math 2.0

Functional analysis von Neumann algebras Problem

Connes embedding problem

Builds on Hyperfinite type II1 factorVon Neumann algebraUltrafilter

Problem

Let RR be the hyperfinite type II1 factor with its faithful normal tracial state τ\tau, and let ω\omega be a free ultrafilter on ℕ\N. Let ℓ∞(R)\ell^\infty(R) be the algebra of norm-bounded sequences in RR, and let IωI_\omega be the ideal of sequences (xn)(x_n) with lim⁡n→ωτ(xn∗xn)=0\lim_{n \to \omega} \tau(x_n^* x_n) = 0. The ultrapower Rω=ℓ∞(R)/IωR^\omega = \ell^\infty(R) / I_\omega is a type II1\mathrm{II}_1 factor with trace τω((xn))=lim⁡n→ωτ(xn)\tau_\omega((x_n)) = \lim_{n \to \omega} \tau(x_n). The problem asks: does every type II1\mathrm{II}_1 factor MM with separable predual, with trace τM\tau_M, admit an injective unital ∗*-homomorphism π ⁣:M→Rω\pi \colon M \to R^\omega with τω∘π=τM\tau_\omega \circ \pi = \tau_M?

Remarks

Connes raised the question in his 1976 paper on the classification of injective factors. Over the following decades it was shown to be equivalent to several problems in other areas. Among them are Kirchberg's QWEP conjecture on C*-algebras and Tsirelson's problem in quantum information theory. In 2020, Ji, Natarajan, Vidick, Wright and Yuen announced a proof of the complexity result MIP∗=RE\mathrm{MIP}^* = \mathrm{RE}. It implies a negative answer to Tsirelson's problem and hence to the Connes embedding problem. So there exist II1\mathrm{II}_1 factors with separable predual that do not embed into RωR^\omega, but no explicit example is known. Many classes of factors are known to embed, for example the group von Neumann algebras of amenable and of free groups.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Hilbert space, concept, no evidence levelHilbert spaceconceptUltrafilter, concept, no evidence levelUltrafilterconceptVon 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 …objectConnes embedding problem, problem, no evidence levelConnes embedding prob…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Hilbert space concept
  2. Ultrafilter concept
  3. Von Neumann algebra concept
  4. Weak operator topology concept
  5. Hyperfinite type II1 factor object
  6. Connes embedding problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Connes_embedding_problem
About this page
TagYYP6QNRERY
URIm2://YYP6QNRERY
Kindproblem
WritingW0 stub
MSC46L10, 46L36
StatusThis copy of the site has no build data. Run m2 status YYP6QNRERY.
AboutHyperfinite type II1 factor
UsesHyperfinite type II1 factor, Von Neumann algebra, Ultrafilter
Same as
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.