Connes Embedding Problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A tracial von Neumann algebra $(M,\tau)$ is a weak-operator closed unital $*$-algebra of bounded operators together with a faithful normal tracial state $\tau$ ($\tau(1)=1$ and $\tau(xy)=\tau(yx)$). Let $R$ denote the hyperfinite type $\mathrm{II}_1$ factor with its normalized trace $\tau_R$: the infinite-dimensional factor obtained as the weak closure of an increasing union of finite-dimensional matrix algebras. A factor has center $\mathbb C1$. For an index set $I$ and ultrafilter $\mathcal U$ on $I$, set \[ R^{\mathcal U}=\ell^\infty(I,R)/\mathcal I_{\mathcal U},\qquad \mathcal I_{\mathcal U}=\{(x_i):\lim_{i\to\mathcal U}\|x_i\|_2=0\}, \qquad\|x\|_2=\tau_R(x^*x)^{1/2}, \] where $\ell^\infty(I,R)$ consists of operator-norm bounded families. Its trace is $\tau_{\mathcal U}([(x_i)])=\lim_{i\to\mathcal U}\tau_R(x_i)$. An embedding is a unital injective $*$-homomorphism preserving the specified trace.

Statement recorded in the supplied source.

Does every tracial von Neumann algebra $(M,\tau)$ admit a trace-preserving embedding $M\hookrightarrow R^{\mathcal U}$ into some tracial ultrapower of $R$? The index set $I$ may depend on $M$. Equivalently, for every separable tracial von Neumann algebra $M$ and any nonprincipal ultrafilter $\mathcal U$ on $\mathbb N$, does such an embedding into $R^{\mathcal U}$ exist? Here separability means that the predual $M_*$ is separable. [S2]

Short English statement

Can every finite tracial von Neumann algebra be represented, without changing its trace, inside an ultrapower of the hyperfinite type II-one factor?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1104 (ALG-004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Isaac Goldbring, The Connes Embedding Problem: A Guided Tour, arXiv:2109.12682v1 (2021), §3.6, pp.20--22, official statement and its larger-index-set variation. The abstract and §§6--7 explain that the classical universal assertion has a negative answer as a consequence of MIP$^*=$RE. https://arxiv.org/abs/2109.12682.