Connes embedding problem

E286302

mathematical conjecture open problem in operator algebras problem in functional analysis problem in quantum information theory

The Connes embedding problem is a central open question in operator algebras and quantum theory that asks whether every separable II₁ factor can be approximated in a specific way by finite-dimensional matrix algebras.

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All labels observed (6)

Label	Occurrences
Connes embedding problem canonical	1
Connes embedding property	1
Kirchberg conjecture	1
Kirchberg’s QWEP conjecture for C*-algebras	1
QWEP conjecture	1
Tsirelson problem	1

How this entity was disambiguated

This entity first appeared as the object of triple T2648173 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Connes embedding problem
Context triple: [Alain Connes, knownFor, Connes embedding problem]

A. Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
B. Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
C. Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
D. Yang–Mills existence and mass gap problem
The Yang–Mills existence and mass gap problem is a fundamental unsolved question in mathematical physics that asks for a rigorous proof that quantum Yang–Mills theory exists and exhibits a positive mass gap, and is one of the seven Millennium Prize Problems.
E. Gelfand–Kirillov dimension
The Gelfand–Kirillov dimension is an invariant in noncommutative algebra that measures the growth rate of algebras and modules, serving as an analogue of Krull dimension for noncommutative settings.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Connes embedding problem
Target entity description: The Connes embedding problem is a central open question in operator algebras and quantum theory that asks whether every separable II₁ factor can be approximated in a specific way by finite-dimensional matrix algebras.

A. Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
B. Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
C. Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
D. Yang–Mills existence and mass gap problem
The Yang–Mills existence and mass gap problem is a fundamental unsolved question in mathematical physics that asks for a rigorous proof that quantum Yang–Mills theory exists and exhibits a positive mass gap, and is one of the seven Millennium Prize Problems.
E. Gelfand–Kirillov dimension
The Gelfand–Kirillov dimension is an invariant in noncommutative algebra that measures the growth rate of algebras and modules, serving as an analogue of Krull dimension for noncommutative settings.
F. None of above. chosen

Statements (53)

Predicate	Object
instanceOf	mathematical conjecture ⓘ open problem in operator algebras ⓘ problem in functional analysis ⓘ problem in quantum information theory ⓘ
asksAbout	finite-dimensional matrix algebras ⓘ hyperfinite II₁ factor ⓘ separable II₁ factors ⓘ
coreQuestion	whether every separable II₁ factor can be approximated in moments by finite-dimensional matrix algebras ⓘ whether every separable II₁ factor embeds into an ultrapower of the hyperfinite II₁ factor ⓘ
disproofMethod	construction of non-hyperlinear II₁ factors ⓘ use of nonlocal games and quantum correlations ⓘ
disprovedBy	David Severin ⓘ Marius Junge ⓘ Mikael Tobiaschewski ⓘ Narutaka Ozawa NERFINISHED ⓘ Vern Paulsen ⓘ William Slofstra ⓘ collaborators in quantum information theory ⓘ
field	functional analysis ⓘ operator algebras ⓘ quantum information theory ⓘ quantum theory ⓘ von Neumann algebras ⓘ
hasEquivalentFormulation	Connes embedding problem self-linksurface differs ⓘ surface form: Kirchberg’s QWEP conjecture for C*-algebras embeddability of separable II₁ factors into R^ω ⓘ equality of certain sets of quantum correlations ⓘ
implication	existence of II₁ factors not approximable by matrices in the Connes sense ⓘ failure of equality between certain tensor products of C*-algebras ⓘ
importance	central problem in the theory of operator algebras for several decades ⓘ
motivatedResearchIn	classification of II₁ factors ⓘ free entropy and free probability ⓘ quantum nonlocality and nonlocal games ⓘ
namedAfter	Alain Connes ⓘ
relatedConcept	C-algebras ⓘ Connes embedding problem self-linksurface differs ⓘ surface form: Connes embedding property II₁ factor ⓘ Connes embedding problem self-linksurface differs ⓘ surface form: Kirchberg conjecture Connes embedding problem self-linksurface differs ⓘ surface form: QWEP conjecture Connes embedding problem self-linksurface differs ⓘ surface form: Tsirelson problem finite-dimensional approximation ⓘ free probability theory ⓘ hyperfinite II₁ factor ⓘ maximal tensor product ⓘ microstates ⓘ minimal tensor product ⓘ nonlocal games ⓘ quantum correlations ⓘ tensor product of C-algebras ⓘ tracial state ⓘ ultrapower of a von Neumann algebra ⓘ von Neumann algebras ⓘ surface form: von Neumann factors
status	disproved in general ⓘ
timePeriod	late 20th century ⓘ

How these facts were elicited

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Alain Connes → knownFor → Connes embedding problem ⓘ

Connes embedding problem → relatedConcept → Connes embedding problem self-linksurface differs ⓘ

this entity surface form: Connes embedding property

Connes embedding problem → relatedConcept → Connes embedding problem self-linksurface differs ⓘ

this entity surface form: Kirchberg conjecture

Connes embedding problem → relatedConcept → Connes embedding problem self-linksurface differs ⓘ

this entity surface form: Tsirelson problem

Connes embedding problem → relatedConcept → Connes embedding problem self-linksurface differs ⓘ

this entity surface form: QWEP conjecture

Connes embedding problem → hasEquivalentFormulation → Connes embedding problem self-linksurface differs ⓘ

this entity surface form: Kirchberg’s QWEP conjecture for C*-algebras

All labels observed (6)

How this entity was disambiguated Show

Statements (53)

How these facts were elicited Show

Referenced by (6)

How this entity was disambiguated

How these facts were elicited