2 A little continuous logic

For the purposes of this paper, we need to state what we mean by a computable language, theory, and structure in the continuous setting. There are some choices involved, but we make the following definitions, following [Reference Camburd, Goldbring and McNicholl5]:

By a computable uniform continuity modulus we will mean a strictly increasing computable function $\mu \colon \mathbb {N} \rightarrow \mathbb {N}$ . We say that a function $f:(X,d) \rightarrow (X',d')$ between two metric spaces has uniform continuity modulus $\mu $ if: for all $k\geq 1$ and for all $a,b \in X$ , if $d(a,b) < 1/\mu (k)$ , then $d'(f(a),f(b)) \leq 1/k$ . We say that a function $g:\mathbb {N}^n \rightarrow \mathbb {R}$ is computable if there is an algorithm which, when given $k_1,\ldots ,k_n\in \mathbb N$ and $m\geq 1$ as inputs, returns a rational number $q\in \mathbb Q$ such that $|g(k_1,\ldots ,k_n) - q| < 1/m$ .

Recall that a language in continuous logic is a triple ${\mathcal L} = (\mathcal {S}, \mathcal {F}, \mathcal {R})$ consisting of a set of sorts, a set of function symbols, and a set of relation symbols. We say that such a language is computable if:

1. (1) there is a computable enumeration of all pairs $(S,d_S)$ , where $S \in \mathcal S$ and $d_S$ is the metric symbol for S;

2. (2) there is a computable enumeration of all tuples $(f,D,C,\mu )$ , where $f \in \mathcal F$ , D is the domain of f, C is the codomain of f, and $\mu $ is a computable continuity modulus which acts as the uniform continuity modulus for f; and

3. (3) there is a computable enumeration of all tuples $(R,D,B,\mu )$ , where $R \in \mathcal R$ , D is the domain of R, B is a positive integer representing the bound on R, and $\mu $ is a computable continuity modulus which acts as the uniform continuity modulus for R.

Notice that necessarily a computable language has only countably many symbols. We will have to address this when we consider theories like the theory of tracial von Neumann algebras which a priori have continuum many symbols. A similar consideration appears when we consider formulas in continuous logic, the issue being the number of connectives. On this front we will be explicit and say that the only connectives we will consider are compositions of rational polynomials in several variables together with the binary function . We will call this set of connectives restricted.

For a separable ${\mathcal L}$ -structure M, we call an $\mathcal S$ -indexed sequence of maps $p = (p_S : S \in \mathcal {S})$ , with $p_S:\mathbb {N} \rightarrow S(M)$ for every $S \in \mathcal S$ , a presentation of M if the range of $p_S$ is dense in $S(M)$ for all $S \in \mathcal {S}$ . If ${\mathcal L}$ is a computable language, then we say that such a presentation p is computable if for every ${\mathcal L}$ -atomic formula $\varphi (x_1,\ldots ,x_n)$ with variables $x_1,\ldots ,x_n$ from sorts $S_1,\ldots ,S_n$ , the map $\operatorname {val}_\varphi :\mathbb {N}^n \rightarrow \mathbb {R}$ given by

$$\begin{align*}\operatorname{val}_\varphi(k_1,\ldots,k_n) = \varphi^M(p_{S_1}(k_1),\ldots,p_{S_n}(k_n)) \end{align*}$$

is computable, uniformly in $\varphi $ .

We now fix a computable continuous language ${\mathcal L}$ . (In the next section, ${\mathcal L}$ will be the language of tracial von Neumann algebras, restricted to complex rational scalars.) We call an ${\mathcal L}$ -formula restricted if it only uses restricted connectives and an ${\mathcal L}$ -sentence is restricted if it is a restricted ${\mathcal L}$ -formula which has no free variables.

Given an ${\mathcal L}$ -structure M, the theory of M is the function $\operatorname {Th}(M)$ whose domain is the set of ${\mathcal L}$ -sentences and which is defined by $\operatorname {Th}(M)(\sigma ):=\sigma ^M$ . The universal theory of M, denoted $\operatorname {Th}_\forall (M)$ , is the restriction of $\operatorname {Th}(M)$ to the set of universal ${\mathcal L}$ -sentences.

One also uses the word theory in continuous logic as a synonym for a set of ${\mathcal L}$ -sentences. In this case, given an ${\mathcal L}$ -structure M, the theory of M is the set $\{\sigma \ : \ \sigma ^M=0\}$ and the universal theory of M is the intersection of the theory of M with the set of nonnegative universal ${\mathcal L}$ -sentences.

Given a set of ${\mathcal L}$ -sentences T, an ${\mathcal L}$ -structure N is a model of T if $\sigma ^N=0$ for all $\sigma \in T$ . Note that an ${\mathcal L}$ -structure N is a model of the universal theory of M if and only if N embeds into an ultrapower of M.

If T is a set of restricted ${\mathcal L}$ -sentences, then via usual Gödel coding, it makes sense to speak of T being decidable, effectively enumerable, etc. Since a theory is a set of sentences, we believe the following terminology is appropriate:

It is clear that each version of the theory of M can be recovered from the other version, whence, from the point of view of model theory, there is no harm in blurring the distinction. However, from the computability-theoretic perspective, there is a difference. Indeed, while it is clear that the decidability of the theory of M implies its computability, the converse need not be true.

Given an ${\mathcal L}$ -structure M, a nonnegative ${\mathcal L}$ -formula $\varphi (x)$ is called an almost-near formula for M if, for any $\epsilon>0$ , there is $\delta =\delta (\epsilon )>0$ so that, for any $a\in M$ , if $\varphi ^M(a)<\delta (\epsilon )$ , then there is $b\in M$ such that $\varphi ^M(b)=0$ and $d(a,b)\leq \epsilon $ . In this case, we refer to the function $\delta (\epsilon )$ as an almost-near modulus for $\varphi $ . If $\varphi $ is an almost-near formula for M, then we refer to the zeroset of $\varphi ^M$ in M, denoted $Z(\varphi ^M)$ , as the definable set corresponding to  $\varphi $ .

The utility of definable sets is that one can quantify over them in a first-order way. In order to explain explicitly how we use this fact, we note that, given an almost-near formula $\varphi (x)$ , [Reference Ben Yaacov, Berenstein, Henson and Usvyatsov2, Remark 2.12] establishes the existence of a nondecreasing, continuous function $\alpha :[0,\infty )\to [0,\infty )$ with $\alpha (0)=0$ and with the property that, for any $a\in M$ , we have $d(a,Z(\varphi ^M))\leq \alpha (\varphi ^M(a))$ ; moreover, the function $\alpha $ depends only on the modulus $\delta (\epsilon )$ for $\varphi $ . As shown in the proof of [Reference Ben Yaacov, Berenstein, Henson and Usvyatsov2, Proposition 9.19], it follows that

$$ \begin{align*}d(a,Z(\varphi^M))=(\inf_{x}(\alpha(\varphi(x))+d(a,x)))^M. \quad \quad (\dagger)\end{align*} $$

The import of $(\dagger )$ is that the formula on the right-hand side of $(\dagger )$ is an actual formula of continuous logic. We note that the proof appearing in [Reference Ben Yaacov, Berenstein, Henson and Usvyatsov2, Remark 2.12] shows that if the almost-near modulus $\delta $ is computable (when restricted to rational $\epsilon $ ), then the corresponding $\alpha $ is also a computable function. We summarize this discussion as follows:

In connection with the previous proposition, a sentence is existential (resp. universal) if it is of the form $\inf _x \varphi (x)$ (resp. $\sup _x \varphi (x)$ ), where x is a tuple of variables and $\varphi $ is quantifier-free.

We note also that if $\varphi $ is an almost-near formula for M, then it is also an almost-near formula for any ultrapower $M^{\mathcal {U}}$ of M with the same almost-near modulus and that the formula $(\dagger )$ and the conclusion of the previous proposition also hold for $M^{\mathcal {U}}$ as well.

There is a proof system for continuous logic, first introduced in [Reference Ben Yaacov and Pederson3]. There, one defines the relation $T\vdash \sigma $ , where T is a restricted ${\mathcal L}$ -theory and $\sigma $ is a restricted ${\mathcal L}$ -sentence. A feature of this proof system is that, if T is effectively enumerable, then so is the set of $\sigma $ such that $T\vdash \sigma $ . The following version of the completeness theorem, first proven in [Reference Ben Yaacov and Pederson3], will play a large role in the sequel:

Fact 2.4. For any restricted ${\mathcal L}$ -theory T and any restricted ${\mathcal L}$ -sentence $\sigma $ , we have

Suppose, in the previous display, that $\sigma $ is a nonnegative universal sentence and that the common value is $0$ . If T is effectively enumerable and we begin to enumerate the theorems of T, then we may never see the fact that $T\vdash \sigma $ even though for all n. This motivates the following definition:

For some structures (such as $\mathcal R$ ), the computability of the universal theory of the structure is equivalent to it being weakly effectively enumerable:

As mentioned in [Reference Goldbring and Hart10] (and elaborated on in [Reference Goldbring and Hart11]), $\mathcal R$ has a computable presentation. The proof of the nontrivial direction of the previous proposition follows by using the computable presentation to perform a brute force lower bound approximation to the value of any positive universal sentence.

3 Proof of the main theorem

We remind the reader that if $(C_a : a\leq m)$ is a PVM on H, then the projections are automatically pairwise orthogonal, that is, $C_aC_b=0$ for all distinct $a,b\leq m$ .

Definition 3.2. A nonlocal game $\mathfrak G$ with n questions and m answers is a probability distribution $\mu $ on $n \times n$ together with a decision function

$$\begin{align*}D:n\times n \times m \times m \rightarrow \{0,1\}. \end{align*}$$

We call the nonlocal game synchronous if $D(v,v,i,j)=0$ whenever $i\not =j$ .

Definition 3.3. For each nonlocal game $\mathfrak G$ , the entangled value of $\mathfrak G$ is the quantity

$$ \begin{align*}\operatorname{val}^*(\mathfrak G)=\sup_{p\in C_{qa}(n,m)}\sum_{v,w}\mu(v,w)\sum_{i,j}D(v,w,i,j)p(i,j|v,w).\end{align*} $$

We also define the synchronous value of $\mathfrak G$ to be the quantity

$$ \begin{align*}\operatorname{s-val}^*(\mathfrak G)=\sup_{p\in C_{qa}^s(n,m)}\sum_{v,w}\mu(v,w)\sum_{i,j}D(v,w,i,j)p(i,j|v,w).\end{align*} $$

In general, $\operatorname {s-val}^*(\mathfrak G)\leq \operatorname {val}^*(\mathfrak G)$ .

The following is the main result of [Reference Ji, Natarajan, Vidick, Wright and Yuen13]:

We let $\varphi _{n,m}(x_{v,i})$ ( $i=1,\ldots ,m$ , $v=1,\ldots ,n$ ) denote the (computable!) formula

$$ \begin{align*}\max\left(\max_{v,i}\|x_{v,i}^2-x_{v,i}\|_2,\max_{v,i}\|x_{v,i}^*-x_{v,i}\|_2,\max_v\|\sum_ix_{v,i}-1\|_2\right).\end{align*} $$

We let $X^N_{n,m}$ denote the zeroset of $\varphi _{n,m}$ in N. Note that elements of $X^N_{n,m}$ are n-tuples of PVMs in N, where each PVM in the tuple consists of m orthogonal projections.

Given a nonlocal game $\mathfrak G$ , let $\psi _{\mathfrak {G}}(x_{v,i})$ denote the formula

$$ \begin{align*}\sum_{v,w}\mu(v,w)\sum_{i,j}D(v,w,i,j)\operatorname{\mathrm{tr}}(x_{v,i}x_{w,j}).\end{align*} $$

Theorem 3.7. For any game $\mathfrak G$ , we have

$$ \begin{align*}\operatorname{s-val}^*(\mathfrak G)=\left(\sup_{x_{v,i}\in X_{n,m}}\psi_{\mathfrak{G}}(x_{v,i})\right)^{\mathcal R}.\end{align*} $$

Proof For simplicity, set $X:=X_{n,m}$ and $x:=(x_{v,i})$ . Set $\sigma :=\sup _{x\in X}\psi _{\mathfrak {G}}(x_{v,i})$ . Note first that, since $\psi _{\mathfrak {G}}$ is 1-Lipshitz, we have . Now given $\eta>0$ , one can effectively find an existential restricted formula $\varphi _n$ such that, for all $x\in \mathcal R^{\mathcal {U}}$ , we have $|d(x,X^{\mathcal R^{\mathcal {U}}})-\varphi _n(x)^{\mathcal R^{\mathcal {U}}}|<\eta $ . It follows that

Since the latter formula in the above display is equivalent to a universal restricted formula, the computability of the universal theory of $\mathcal R$ allows us to compute it to within $\eta $ , and thus we can compute $\sigma ^{\mathcal {R}}$ to within $2\eta $ . By the previous theorem, this is equivalent to being able to compute $\operatorname {s-val}^*(\mathfrak G)$ to within $2\eta $ .

It is clear that these considerations are uniform in the description of $\mathfrak G$ .

4 A reformulation in terms of noncommutative moments

In this section, we offer a reformulation of our main theorem in terms that might be more appealing to operator algebraists.

Given positive integers n and d, we fix variables $x_1,\ldots ,x_n$ and enumerate all *-monomials in the variables $x_1,\ldots ,x_n$ of total degree at most d, $m_1,\ldots ,m_L$ . (Of course, $L=L(n,d)$ depends on both n and d.) We consider the map $\mu _{n,d}:\mathcal R_1^n \rightarrow D^L$ given by $\mu _{n,d}(\vec a)=(\tau (m_i(\vec a)) \ : \ i=1,\ldots ,L)$ . (Here, D is the complex unit disk.)

We let $X(n,d)$ denote the range of $\mu _{n,d}$ and $X(n,d,p)$ be the image of the unit ball of $M_p(\mathbb C)$ under $\mu _{n,d}$ . Notice that $\bigcup _{p\in \mathbb N} X(n,d,p)$ is dense in $X(n,d)$ .

Proof First suppose that the universal theory of $\mathcal R$ is computable. We produce a computable function F as in (2). Fix n, d, and k, and set $\epsilon :=\frac {1}{3k}$ . Computably find $s_1,\ldots ,s_t$ , an $\epsilon $ -net in $D^L$ . For each $i=1,\ldots ,t$ , ask the universal theory of $\mathcal R$ to compute intervals $(a_i,b_i)$ with $b_i-a_i<\epsilon $ and with $\left (\inf _{\vec x}|\mu _{n,d}(\vec x)-s_i|\right )^{\mathcal {R}}\in (a_i,b_i)$ . For each $i=1,\ldots ,t$ such that $b_i<2\epsilon $ , let $p_i\in \mathbb N$ be the minimal p such that when you ask the universal theory of $\mathbb M_p(\mathbb C)$ to compute intervals of shrinking radius containing $\left (\inf _{\vec x}|\mu _{n,d}(\vec x)-s_i|\right )^{M_p(\mathbb C)}$ , there is a computation that returns an interval $(c_i,d_i)$ with $d_i<2\epsilon $ . Let p be the maximum of these $p_i$ ’s. We claim that setting $F(n,d,k):=p$ is as desired. Indeed, suppose that $s\in X(n,d)$ and take $i=1,\ldots ,t$ such that $|s-s_i|<\epsilon $ . Then $\left (\inf _{\vec x}|\mu _{n,d} (\vec x)-s_i|\right )^{\mathcal {R}}<\epsilon $ , whence $b_i<2\epsilon $ . It follows that there is an interval $(c_i,d_i)$ as above with $\left (\inf _{\vec x}|\mu _{n,d}(\vec x)-s_i|\right )^{M_p(\mathbb C)}<d_i<2\epsilon $ . Let $a\in M_p(\mathbb C)$ realize the infimum. Then $|\mu _{n,d}(\vec a)-s|<3\epsilon =\frac {1}{k}$ , as desired.

Now suppose that F is as in (2). We show that the universal theory of $\mathcal R$ is computable. Towards this end, fix a restricted universal sentence

$$\begin{align*}\sigma = \sup_{\vec x} f(\tau(m_1),\ldots,\tau(m_\ell)), \end{align*}$$

where $\vec x = x_1,\ldots ,x_n$ and $m_1,\ldots ,m_\ell $ are *-monomials in $\vec x$ of total degree at most d. Fix also rational $\epsilon>0$ . We show how to compute the value of $\sigma ^{\mathcal {R}}$ to within $\epsilon $ . Since f is a restricted connective, it has a computable modulus of continuity $\delta $ . Consequently, we can find $k\in \mathbb N$ computably so that $\frac {1}{k} \leq \delta (\epsilon )$ . Set $p = F(n,d,2k)$ . Computably construct a sequence $\vec a_1,\ldots ,\vec a_t\in (M_p(\mathbb {C})_1)^n$ that is a $\frac {1}{2k}$ cover of $(M_p(\mathbb C)_1)^n$ (with respect to the $\ell ^1$ metric corresponding to the 2-norm). Consequently, $\mu _{n,d}(\vec a_1),\ldots ,\mu _{n,d}(\vec a_t)$ is a $\frac {1}{2k}$ -cover of $X(n,d,p)$ . Set

$$\begin{align*}r:=\max_{i=1,\ldots,t}f(\tau(m_1(\vec a_i)),\ldots,\tau(m_l(\vec a_i))). \end{align*}$$

By assumption, $X(n,d,p)$ is $\frac {1}{2k}$ -dense in $X(n,d)$ . It follows that $r\leq \sigma ^{\mathcal {R}}\leq r+\epsilon $ , as desired.

5 A general perspective on Embedding Problems

As stated above, a structure N embeds into an ultrapower of another structure M (in the same language) if and only if N is a model of the universal theory of M. All of the embedding problems in operator algebras attempt to find a small subset of the universal theory of some canonical object so that modeling that small subset suffices to conclude that one models the entire universal theory. For example, the Connes Embedding Problem asks whether or not modeling the theory of tracial von Neumann algebras (which is a subset of the universal theory of $\mathcal R$ ) is enough to know that one models the entire universal theory of $\mathcal R$ . Similarly, the MF Problem asks whether or not modeling the theory of stably finite C*-algebras (which, again, is part of the universal theory of $\mathcal Q$ ) is enough to know that one models the entire universal theory of $\mathcal Q$ .

Now that we know that the Connes Embedding Problem is false, it is natural to ask whether or not one can “reasonably” enlarge the theory of tracial von Neumann algebras in such a way that then modeling that enlarged theory does indeed imply that you model the entire universal theory of $\mathcal R$ . We show that, under one interpretation of “reasonable,” this is impossible. We first offer the following general definition:

Note that this definition allows the possibility that the extra information being allowed need not be universal information, but rather can have arbitrary quantifier-complexity. On the other hand, the restriction that T be effectively enumerable is somewhat severe (although natural from the logical point of view).

We have the following general statement:

Proof Suppose that there is an effectively enumerable subset T of the theory of M such that whenever $N\models T$ , then N embeds into an ultrapower of M. It follows that, for any universal sentence $\sigma $ , we have, using the Completeness Theorem, that

The result now follows.

Recalling that weak effective enumerability is equivalent to computability for the universal theory of $\mathcal R$ , we now have the following strengthening of the fact that CEP has a negative solution:

Recall that a tracial von Neumann algebra $\mathcal S$ is called locally universal if every tracial von Neumann algebra embeds into an ultrapower of $\mathcal S$ . As shown in [Reference Farah, Hart and Sherman7], there is a locally universal tracial von Neumann algebra and it is clear that they all have the same universal theory. Since the theory of tracial von Neumann algebras is recursively axiomatizable, it follows that if $\mathcal S$ is a locally universal tracial von Neumann algebra, then the $\mathcal S$ EP has a positive solution. The same remark can be made for locally universal C*-algebras.

As stated in the introduction, Corollary 5.3 allows us to provide “many” counterexamples to CEP:

6 An application to C*-algebras

Recall that the MF problem, first posed by Blackadar and Kirchberg, asks whether or not every stably finite $\mathrm {C}^*$ -algebra embeds into an ultrapower of the universal UHF algebra $\mathcal Q$ . The following consequence of the failure of CEP was pointed out to us by Thomas Sinclair and Aaron Tikuisis:

In this section, we improve upon this result by showing that the $\mathcal Q$ EP has a negative solution. This result will follow from a more general result applying to a wider class of C*-algebras.

Definition 6.3. Given $m\in \mathbb N$ and $0<\gamma <1$ , we say that a unital $\mathrm {C}^*$ -algebra A has the $(m,\gamma )$ -uniform Dixmier property if, for all self-adjoint $a\in A$ , there are unitaries $u_1,\ldots ,u_m\in U(A)$ and $z\in Z(A)$ such that

$$ \begin{align*}\left\|\sum_{i=1}^m \frac{1}{m}u_iau_i^*-z\right\|\leq \gamma\|a\|.\end{align*} $$

We say that A has the uniform Dixmier property if it has the $(m,\gamma )$ -Dixmier property for some m and $\gamma $ .

Clearly if A has the $(m,\gamma )$ -Dixmier property, then it has the $(m,\gamma ')$ -Dixmier property for any $0<\gamma <\gamma '<1$ , whence we may always assume that $\gamma $ is dyadic rational.

Given m and $\gamma $ , let $\theta _{m,\gamma }$ denote the following sentence in the language of $\mathrm {C}^*$ -algebras:

Here, the supremum is over self-adjoint contractions, the first infimum is over contractions, and the second infimum is over the unit disk in $\mathbb C$ . If A is a simple unital $\mathrm {C}^*$ -algebra with the $(m,\gamma )$ -uniform Dixmier property, then $\theta _{m,\gamma }^A=0$ . Conversely, if $\theta _{\gamma ,m}^B=0$ and B admits a trace, then this trace on B is unique (see [Reference Archbold, Robert and Tikuisis1, Proposition 1.4 and Theorem 3.2]).

Given a tracial C*-algebra $(A,\tau _A)$ , one lets $N_{(A,\tau _A)}$ denote the weak closure of A in the GNS representation corresponding to $\tau _A$ . It is known that $N_{(A,\tau _A)}$ is isomorphic to the algebra obtained from taking the $\|\cdot \|_{2,\tau _A}$ -completion of each bounded ball of A.

We are now ready to prove our main theorem of this section:

Proof Suppose, towards a contradiction, that the AEP has a positive solution as witnessed by the theory $T_0$ . Fix m and $\gamma $ with $\gamma $ a dyadic rational such that A has the $(m,\gamma )$ -uniform Dixmier property. Let ${\mathcal L}$ be the language of tracial $\mathrm {C}^*$ -algebras and let T be the union of the following three ${\mathcal L}$ -theories:

• • $T_0$ ;

• • the ${\mathcal L}$ -theory of tracial $\mathrm {C}^*$ -algebras;

• • the single condition $\theta _{m,\gamma }=0$ .

Note that T is recursively axiomatizable. Clearly $(A,\tau _A)\models T$ . Now suppose that $(M,\tau _M)\models T$ . Since $M\models T_0$ , there is an embedding $M\hookrightarrow A^{\mathcal {U}}$ . Since $ \theta _{m,\gamma }^M=0$ , $\tau _M$ is the unique trace on M, whence this embedding is trace preserving, that is, we have an embedding $(M,\tau _M)\hookrightarrow (A,\tau _A)^{\mathcal {U}}$ . Consequently, for any universal ${\mathcal L}$ -sentence $\sigma $ , we have that

where the second equality follows from the Completeness Theorem. Thus, by running proofs from T, we obtain approximations from above to the value of $\sigma ^{(A,\tau _A)}$ . If $\sigma $ is a sentence in the language of tracial von Neumann algebras, then $\sigma $ can be construed in the language of tracial $\mathrm {C}^*$ -algebras. In this case, set $N:=N_{(A,\tau _A)}$ . Since A has a unique trace, N is a II $_1$ factor; call its trace $\tau _N$ . Note that since A is simple, $\tau _A$ is faithful and so A embeds into N. It follows that we have $\sigma ^{(A,\tau _A)}=\sigma ^{(N,\tau _N)}$ . Since N is a II $_1$ factor that embeds into an ultrapower of $\mathcal R$ , we have that $\sigma ^{(N,\tau _N)}=\sigma ^{(\mathcal R,\tau _{\mathcal {R}})}$ . We thus have that the universal theory of $\mathcal R$ is weakly effectively enumerable, which is a contradiction.

We remind the reader of a theorem of Haagerup and Zsidó [Reference Haagerup and Zsidó12], namely that a simple unital $\mathrm {C}^*$ -algebra has the Dixmier property if and only if it is monotracial. In particular, $\mathcal Q$ and $\mathcal Z$ have the Dixmier property. Relevant for our discussion is the following:

7 Tsirelson’s problem and Kirchberg’s QWEP Conjecture revisited

As discussed in the introduction, the result MIP*=RE implies a negative solution to Tsirelson’s problem. This conclusion is achieved by applying a semidefinite programming argument coupled with a noncommutative Positivstellensatz result. In this section, we show how to replace this latter argument with a simple argument using the Completeness Theorem and a result from [Reference Paulsen, Severini, Stahlke, Todorov and Winter18], whose proof is essentially just an application of the Cauchy–Schwarz inequality.

We first remind the reader of the definitions relevant to state Tsirelson’s problem.

Definition 7.1. The set $C_{qc}(n,m)$ of quantum commuting correlations consists of the correlations of the form $p(a,b|x,y)= \langle A^x_a B^y_b\xi ,\xi \rangle $ for ${x,y \leq n}$ and $a,b \leq m$ , where H is a separable Hilbert space, $\xi \in H$ is a unit vector, and for every $x, y \leq n$ , $(A^x_a: a\leq m)$ and $(B^y_b:b\leq m)$ are PVMs on H for which $A^x_a B^y_b=B^y_bA^x_a$ . As before, if $p(a,b|x,x)=0$ whenever $a\not =b$ , we call the quantum correlation synchronous and we let $C_{qc}^s(n,m)$ denote the set of synchronous quantum commuting correlations. If $\mathfrak G$ is a nonlocal game, we set

$$ \begin{align*}\operatorname{s-val}^{co}(\mathfrak G)=\sup_{p\in C_{qc}^s(n,m)}\sum_{v,w}\mu(v,w)\sum_{i,j}D(v,w,i,j)p(i,j|v,w).\end{align*} $$

Tsirelson’s problem asks whether or not $C_{qa}(n,m)=C_{qc}(n,m)$ for all n and m. Our derivation of a negative solution to Tsirelson’s problem from MIP*=RE also establishes a complexity-theoretic fact, which we now state. (To be fair, this result is also derivable from the aforementioned semidefinite programming/Positivstellensatz argument.)

The terminology $\operatorname {MIP}^{co}_0$ already exists in the literature and denotes the set of languages for which there exists an efficient mapping $z\mapsto \mathfrak G_z$ as above such that $z\in L$ if and only if $\operatorname {val}^{co}(\mathfrak G_z)=1$ , where $\operatorname {val}^{co}$ of a game is defined as $\operatorname {s-val}^{co}$ except one takes the supremum over $C_{qc}(n,m)$ instead of just over $C_{qc}^s(n,m)$ . Our aim is to prove the following:

In other words, if $L\in \operatorname {MIP}^{co,s}_{0,r}$ , then there is an algorithm which enumerates the complement of ${\mathcal L}$ .

The following is [Reference Paulsen, Severini, Stahlke, Todorov and Winter18, Corollary 5.6].

Recall the formula $\psi _{\mathfrak {G}}(x_{v,i})$ from Section 3. Let $\theta _{\mathfrak G,r}$ be the sentence

Let T be the theory of tracial C*-algebras as in the previous section. The following is immediate from Fact 7.4:

We will also need the following immediate consequence of the Completeness Theorem. In the displayed expression, the symbol “ $1$ ” refers to the ${\mathcal L}$ -sentence $1$ , while the symbol “ $\frac {1}{2}$ ” refers to the “truth value” $\frac {1}{2}$ .

Lemma 7.6. Let U be a continuous theory. Then U is satisfiable if and only if

We can now prove Theorem 7.3. Let ${\mathcal L}$ belong to $\operatorname {MIP}^{co,s}_{0,r}$ . Given a string z, let $\mathfrak G_z$ be the corresponding game. By Proposition 7.5 and Lemma 7.6, $z\notin L$ if and only if ; since this latter condition is recursively enumerable, the proof of Theorem 7.3 is complete.

One can now deduce the failure of Tsirelson’s problem from MIP*=RE as follows. Suppose, towards a contradiction, that $C_{qa}^s(n,m)=C_{qc}^s(n,m)$ for every n and m. Let $\mathcal M\mapsto \mathfrak G_{\mathcal {M}}$ be the efficient mapping from Turing machines to nonlocal games described in Theorem 3.4. Given a Turing machine $\mathcal M$ , one simultaneously starts computing lower bounds on $\operatorname {s-val}^*(\mathfrak G_{\mathcal {M}})$ while running proofs from $T\cup \{\theta _{\mathfrak G_{\mathcal M,1}}=0\}$ . Either the first computation eventually yields the fact that $\operatorname {s-val}^*(\mathfrak G_{\mathcal M})>\frac {1}{2}$ , in which case $\mathcal M$ halts, or else the second computation eventually yields the fact that , in which case $\operatorname {s-val}^*(\mathfrak G_{\mathcal {M}})<1$ , and $\mathcal M$ does not halt. In this way, we can decide the halting problem, a contradiction. Note that we derived the a priori stronger statement that $C_{qa}^s(n,m)\not =C_{qc}^s(n,m)$ for some n and m.

Although somewhat implicit in [Reference Kim, Paulsen and Schafhauser15], we can now quickly derive a negative solution to Kirchberg’s QWEP conjecture. Indeed, based on the previous paragraph, it suffices to show that if the QWEP conjecture had a positive solution, then $C_{qa}^s(n,m)=C_{qc}^s(n,m)$ for all n and m. Towards this end, fix $p\in C_{qc}^s(n,m)$ and take a tracial C*-algebra $(A,\tau )$ generated by projections $p_{v,i}$ as in Fact 7.4. Recall from [Reference Kim, Paulsen and Schafhauser15] that $C^*(\mathbb F(n,m))$ is the universal C*-algebra generated by projections as in Fact 7.4. Letting $e_{v,i}$ denote the corresponding projections in $C^*(\mathbb F(n,m))$ , we fix a surjective *-homomorphism $\pi :C^*(\mathbb F(n,m))\to A$ sending $e_{v,i}$ to $p_{v,i}$ . Let $\tau '$ be the trace on $C^*(\mathbb F(n,m))$ defined by $\tau '(a):=\tau (\pi (a))$ . Since $C^*(\mathbb F(n,m))$ has the local lifting property, if the QWEP conjecture were true, it would also have the weak expectation property, whence $\tau '$ would be an amenable trace. (See [Reference Brown and Ozawa4] for all the of the terms and facts described in the previous sentence.) By the equivalence of (1) and (3) in [Reference Kim, Paulsen and Schafhauser15, Theorem 3.6], it follows that $p\in C_{qa}^s(n,m)$ , as desired.