# About the Connes embedding conjecture

@article{Ozawa2013AboutTC, title={About the Connes embedding conjecture}, author={Narutaka Ozawa}, journal={Japanese Journal of Mathematics}, year={2013}, volume={8}, pages={147-183} }

In his celebrated paper in 1976, A. Connes casually remarked that any finite von Neumann algebra ought to be embedded into an ultraproduct of matrix algebras, which is now known as the Connes embedding conjecture or problem. This conjecture became one of the central open problems in the field of operator algebras since E. Kirchberg’s seminal work in 1993 that proves it is equivalent to a variety of other seemingly totally unrelated but important conjectures in the field. Since then, many more…

## 109 Citations

Addendum to “Connesʼ embedding conjecture and sums of hermitian squares” [Adv. Math. 217 (4) (2008) 1816–1837]

- Mathematics
- 2014

Synchronous correlation matrices and Connes' embedding conjecture

- Mathematics
- 2015

In a recent paper, the concept of synchronous quantum correlation matrices was introduced and these were shown to correspond to traces on certain C*-algebras. In particular, synchronous correlation…

The Connes Embedding Problem: A guided tour

- MathematicsBulletin of the American Mathematical Society
- 2022

This work introduces all of the background material necessary to understand the proof of the negative solution of the Connes Embedding Problem and outlines two such proofs, one following the “traditional” route that goes via Kirchberg’s QWEP problem inC-algebra theory and Tsirelson's problem in quantum information theory and a second that uses basic ideas from logic.

A negative resolution to Connes’ Embedding Problem and Tsirelson’s problem

- Mathematics
- 2021

Connes’ Embedding Problem is a deep question on approximability of certain tracial von Neumann algebras by finite-dimensional matrix algebras. We survey the connections between operator algebras,…

Grothendieck's Theorem, past and present UNCUT updated and still expanding...

- Mathematics
- 2015

Probably the most famous of Grothendieck’s contributions to Banach space theory is the result that he himself described as \the fundamental theorem in the metric theory of tensor products". That is…

Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture

- Mathematics
- 2015

This monograph presents some cornerstone results in the study of sofic and hyperlinear groups and the closely related Connes' embedding conjecture. These notions, as well as the proofs of many…

Remark on negative solution to the Tsirelson conjecture about quantum correlations

- Mathematics
- 2021

Recently it was proved [arXiv:2001.04383] that a closure of the set of spatial quantum correlations is strictly less than the set of commuting quantum correlations. Thus, the famous Tsirelson…

R We Living in the Matrix

- Mathematics
- 2019

Introduction In the early years of the twentieth century, the foundations for quantum mechanics were laid out by Dirac, Heisenberg, Bohr, Schrödinger, and others. In his work on the foundations of…

Unbounded expectations to some von Neumann algebras

- Mathematics
- 2020

For any injective von Neumann algebra R and any discrete, countable group G, which acts by *-automorphisms on R, we construct an idempotent mapping of an ultra-weakly dense subspace of B(H) onto the…

On Noncommutative Joinings

- Mathematics
- 2018

This paper extends the classical theory of joinings of measurable dynamical systems to the noncommutative setting from several interconnected points of view. Among these is a particularly fruitful…

## References

SHOWING 1-10 OF 38 REFERENCES

Real Closed Separation Theorems and Applications to Group Algebras

- Mathematics
- 2011

In this paper we prove a strong Hahn-Banach theorem: separation of disjoint convex sets by linear forms is possible without any further conditions, if the target field $\R$ is replaced by a more…

Grothendieck's Theorem, past and present

- Mathematics
- 2011

Probably the most famous of Grothendieck's contributions to Banach space theory is the result that he himself described as "the fundamental theorem in the metric theory of tensor products". That is…

Connes' embedding problem and Tsirelson's problem

- Mathematics
- 2011

We show that Tsirelson's problem concerning the set of quantum correlations and Connes' embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchberg's QWEP…

Operator system quotients of matrix algebras and their tensor products

- Mathematics
- 2011

An operator system modulo the kernel of a completely positive linear map of the operator system gives rise to an operator system quotient. In this paper, operator system quotients and quotient maps…

Algebraic reformulation of connes embedding problem and the free group algebra

- Mathematics
- 2011

We give a modification of the I. Klep and M. Schweighofer algebraic reformulation of Connes’ embedding problem by considering *-algebra of the countably generated free group. This allows one to…

A positivstellensatz for non-commutative polynomials

- Mathematics
- 2004

A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the…

Sums of squares on real algebraic surfaces

- Mathematics
- 2006

AbstractConsider real polynomials g1, . . . , gr in n variables, and assume that the subset K = {g1≥0, . . . , gr≥0} of ℝn is compact. We show that a polynomial f has a representation
in which the…

Tsirelson's problem and asymptotically commuting unitary matrices

- Mathematics
- 2013

In this paper, we consider quantum correlations of bipartite systems having a slight interaction, and reinterpret Tsirelson's problem (and hence Kirchberg's and Connes's conjectures) in terms of…

TSIRELSON'S PROBLEM AND KIRCHBERG'S CONJECTURE

- Mathematics
- 2012

Tsirelson's problem asks whether the set of nonlocal quantum correlations with a tensor product structure for the Hilbert space coincides with the one where only commutativity between observables…