# Model theory and Ultrapower Embedding Problems in Operator Algebras

Isaac Goldbring  Department of Mathematics
University of California, Irvine, 340 Rowland Hall (Bldg.# 400), Irvine, CA
92697-3875<EMAIL_ADDRESS>http://www.math.uci.edu/ isaac

###### Abstract.

We survey the model theoretic approach to a variety of ultrapower embedding
problems in operator algebras.

Goldbring was partially supported by NSF grant DMS-2054477.

##  1\. Introduction

Suppose that $T$ is an $\forall\exists$-axiomatizable theory in some
countable, classical language $L$. It is often too much to ask that there
exists a countable model of $T$ into which all other countable models embed.
(This does happen, for example, when the theory $T$ admits an
$\aleph_{0}$-categorical model companion.) Nevertheless, under the mild
assumption that the theory $T$ has the _joint embedding property_ (JEP),
meaning that any two models of $T$ mutually embed into a third model of $T$,
we can infer the existence of countable models of $T$ whose ultrapower with
resepct to any nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$ embeds
all countable models of $T$; we refer to such models of $T$ as _locally
universal_ models of $T$. Indeed, if $M$ is any _existentially closed_ model
of $T$ and $N$ is any countable model of $T$, then by jointly embedding $M$
and $N$ into a (without loss of generality) countable model $P$ of $T$, we see
that $P$ embeds in $M^{\mathcal{U}}$ (this follows from the fact that $M$ is
e.c. and $M^{\mathcal{U}}$ is a countably saturated model of its theory)
whence $N$ also embeds into $M^{\mathcal{U}}$. (By using so-called _good
ultrafilters_ , one can obtain ultrapowers of $M$ which embed larger models of
$T$.) This discussion holds verbatim for continuous theories: if $T$ is an
$\forall\exists$-axiomatizable theory with JEP in a separable language, then
separable locally universal models of $T$ exist. It is clear that any model of
$T$ that contains a locally universal model of $T$ is itself locally
universal, whence countable (separable) locally universal models of $T$ are
ubiquitous.

Seemingly unaware of this abstract model-theoretic discussion, operator
algebraists have posed a number of problems which ask whether or not concrete
operator algebras are locally universal for the corresponding classes to which
they belong (which are not always elementary classes). The most famous of
these problems is the _Connes embedding problem_ , which appeared in Alain
Connes’ fundamental work [8] from 1976, in which he showed that any separable
injective II1 factor is necessarily hyperfinite (and for which he received the
Fields medal in 1982). Part of the proof of his main theorem involved showing
that a particular separable II1 factor embedded into an ultrapower of the
_hyperfinite II 1 factor_ $\mathcal{R}$. He casually remarked that such an
embedding “ought to” exist for any separable II1 factor, that is,
$\mathcal{R}$ should be a locally universal object for the elementary class of
II1 factors. (Incidentally, since $\mathcal{R}$ embeds into any II1 factor,
the CEP is equivalent to the assertion that all II1 factors have the same
universal theory.) Connes’ reason why this “ought to” be the case is not
entirely compelling: he points out that such an embedding exists for
$L(\mathbb{F}_{2})$, the group von Neumann algebra associated to the free
group on two generators, and it is for this reason that such an embedding
should exist for all separable II1 factors. While some operator algebraists
refer to Connes’ offhand remark as a “Conjecture,” most prefer to call it a
“Problem.”

Over the years, the CEP has gained significant interest due its connections
with a wide variety of areas of mathematics, including
$\mathrm{C}^{*}$-algebra theory (in connection with _Kirchberg’s QWEP
problem_), quantum information theory (in connection with _Tsirelson’s
problem_), free probability, group theory, and noncommutative real algebraic
geometry, to name a few.

Remarkably, in early 2020, a negative resolution was obtained to the CEP via
its equivalence with Tsirelson’s problem, which was itself refuted using a
remarkable theorem in quantum complexity theory known as
$\operatorname{MIP}^{*}=\operatorname{RE}$. Independent of its connection with
the CEP, this latter result is widely considered to be a landmark scientific
achievement; the reader interested in understanding the entire story behind
these connections can consult the author’s survey [16].

That being said, someone wishing to understand the proof of the negative
solution to the CEP using $\operatorname{MIP}^{*}=\operatorname{RE}$ must
tread the deep waters connecting these two seemingly distant results. In joint
work with Bradd Hart [20], we showed how, using basic ideas from continuous
model theory (most notably the Completeness Theorem for first-order continuous
logic and the theory of definable sets in continuous logic), one can obtain a
more direct proof of the failure of CEP from
$\operatorname{MIP}^{*}=\operatorname{RE}$. Moreover, the model-theoretic
approach offers more insight into this refutation and allows one to prove
extra results, such as “many counterexamples” to the CEP, that is, many
different universal theories of II1 factors, as well as a Gödelian-style
refutation stating that any effectively axiomatizable class of II1 factors
will contain a counterexample to the CEP. We present this model theoretic
approach to the negative solution to the CEP in Section 3.

In Sections 4 and 5, we consider two $\mathrm{C}^{*}$-algebraic analogs of the
CEP, the so-called _Kirchberg embedding problem_ and the _MF problem_ , which,
in some sense, can be thought of as the “infinite” and “finite”
$\mathrm{C}^{*}$-versions of the CEP. The former problem asks whether or not
the _Cuntz algebra_ $\mathcal{O}_{2}$ is locally universal for the class of
all $\mathrm{C}^{*}$-algebras. This problem has eluded model-theoretic
techniques thus far and we discuss what we know about this still-open problem;
the material presented in this section represents joint work of the author and
Thomas Sinclair [22]. The latter problem is the so-called _MF problem_ , which
asks whether or not the _universal UHF algebra_ $\mathcal{Q}$ is a locally
universal object for the elementary class of stably finite
$\mathrm{C}^{*}$-algebras. While a negative solution to the MF problem can
readily be deduced from the failure of the CEP, the model-theoretic approach
allows us to establish a similar Gödelian-style refutation of the MF problem.

A variant of the MF problem, known as the _quasidiagonality (QD) problem_ ,
asks if $\mathcal{Q}$ is a locally universal object for the (non-elementary!)
class of stably finite _nuclear_ $\mathrm{C}^{*}$-algebras. While the QD
problem is still open, a major breakthrough was obtained by Tikuisis, Winter,
and White [38], who resolved this problem in the positive (for simple such
algebras) assuming a technical assumption known as the _universal coefficient
theorem_ (UCT). The model-theoretic content of the UCT is not widely
understood at the time of the writing of this paper. Nevertheless, we discuss
some model-theoretic ideas around the QD problem representing more joint work
of the author and Sinclair [23].

In the final subsection of Section 4, we return to the ideas of the author and
Hart from [20] and show how the Gödelian-style refutation of the MF problem
extends to a larger class of algebras which, in particular, allow us to refute
a _stably projectionless_ version of the MF problem which asks if every stably
projectionless algebra embeds into an ultrapower of a very important algebra
in the classification program for nuclear $\mathrm{C}^{*}$-algebras known as
the _Jiang Su algebra_ $\mathcal{Z}$. Currently, this latter result has no
purely operator-algebraic proof.

In the final section, we consider the simpler case of (unital) abelian
$\mathrm{C}^{*}$-algebras. There, an $\aleph_{0}$-categorical model completion
exists, namely the theory of $C(2^{\mathbb{N}})$. This result is essentially
(after some category-theoretic considerations) a restatement of the existence
of an $\aleph_{0}$-categorical model completion for the classical theory of
Boolean algebras, namely the theory of atomless Boolean algebras. However, an
interesting phenomenon arises when restricting to the class of
_projectionless_ abelian $\mathrm{C}^{*}$-algebas, whose models are of the
form $C(X)$ for $X$ a _continuum_ (that is, a connected compact Hausdorff
space). In this case, a theorem of K.P. Hart [33] states that all such
algebras (except for the trivial case of $C(\text{a point})\cong\mathbb{C}$)
have the same universal theory, whence all nontrivial objects of this class
are locally universal! We provide a fairly detailed proof of this result
below. However, this theorem does not represent the end of the story for this
class of algebras, for the question of a model companion for this class is
still open. Concerning existentially closed projectionless abelian
$\mathrm{C}^{*}$-algebras, we discuss the positive solution, due to
Christopher Eagle, Alessandro Vignati, and the author [11] of a question of
Bankston, who asked if a fairly important (and generic) continuum, the so-
called _pseudoarc_ $\mathbb{P}$, is co-existentially closed (which just means
that $C(\mathbb{P})$ is an existentially closed projectionless abelian
$\mathrm{C}^{*}$-algebra).

Many of the results to be discussed below involve a particular approach to the
Henkin construction known as building models by games (as first popularized by
Hodges in his book [27]). We discuss the essential properties of this
construction in the next section.

##  2\. Building models by games

In this section, we recall the basic facts from the theory of model-theoretic
forcing needed throughout this paper. The relevant version of model-theoretic
forcing for us is the game-theoretic approach, originally presented in Hodges’
classic book [27] and adapted to the continuous setting by the author in [15].
That being said, for some of what is to follows, we need to consider a
slightly more general setting and so we take the opportunity to extend the
context here.

Throughout this section, we fix a countable (continuous) language $L$. By a
$\forall\bigvee\exists$-sentence we mean an $L_{\omega_{1},\omega}$-sentence
of the form

$\sup_{x_{1}}\cdots\sup_{x_{k_{n}}}\bigvee_{m\in\mathbb{N}}\varphi_{m}(x_{1},\ldots,x_{k_{n}}),$

where each $\varphi_{m}$ is an existential $L$-formula and the symbol
$\bigvee$ denotes a countable infimum. (We note that there are several
different approaches to $L_{\omega_{1},\omega}$ in the literature; in the
above sentences, there is no requirement on a common modulus of uniform
continuity for the formulae appearing in the countable infimum). By a
$\forall\bigvee\exists$-theory we mean a collection of
$\forall\bigvee\exists$-sentences. We say that a class $\mathcal{K}$ of
$L$-structures is $\forall\bigvee\exists$-axiomatizable if there is a
$\forall\bigvee\exists$-theory $T$ such that, for all $L$-structures $A$, we
have $A\in\mathcal{K}$ if and only if $\sigma^{A}=0$ for all $\sigma\in T$.
The need to consider such infinitary theories arises as many important classes
of $\mathrm{C}^{*}$-algebras are not first-order axiomatizable but are
$\forall\bigvee\exists$-axiomatizable, such as simple
$\mathrm{C}^{*}$-algebras and nuclear $\mathrm{C}^{*}$-algebras (see, for
example, [12], where the class is called _definable by a uniform family of
formulae_). Motivated by this, if $T$ is a
$\forall\bigvee\exists$-axiomatizable theory and $P$ is a property that may or
may not hold of models of $T$, we say that $P$ is
$\forall\bigvee\exists$-axiomatizable (relative to $T$) if the collection of
models of $T$ having property $P$ is itself
$\forall\bigvee\exists$-axiomatizable.

In the remainder of this subsection, we fix a
$\forall\bigvee\exists$-axiomatizable $L$-theory $T$. We note that this
assumption implies that the class of models of $T$ is an inductive class (that
is, is closed under direct limits), whence every model of $T$ is contained in
an e.c. model of $T$ of the same density character.

We now fix a countably infinite set $C$ of constant symbols enumerated
$(c_{n})_{n<\omega}$. A condition (relative to $T$) is a finite set $p$ of
expressions of the form $\varphi<r$, where $\varphi$ is a quantifier-free
$L(C)$-sentence, such that $T\cup p$ is satisfiable.

We now consider a two-player game whose players we denote by $\forall$ and
$\exists$. The players take turns playing conditions with the requirement that
each player’s turn extends the previous player’s turn. When they have finished
the game, they have constructed a countably infinite set
$\bar{p}=\bigcup_{n<\omega}p_{n}$ which is the union of all of the conditions
played during the game. We call the play of the game definitive if: for every
atomic $L(C)$-sentence $\theta$, there is a unique $r\in\mathbb{R}$ such that
$T\cup\bar{p}\models|\theta-r|=0$. In this case, $\bar{p}$ uniquely describes
a separable $L(C)$-structure $A^{+}(\bar{p})$ generated by $C$, called the
compiled structure, whose $L$-reduct is denoted $A(\bar{p})$. Since $\exists$
can always ensure that the play of the game is definitive, we assume from here
on out that all plays of all games are definitive, whence determine compiled
structures.

If $P$ is a property of $L(C)$-structures, we say that $P$ is enforceable if
$\exists$ has a strategy that ensures that the compiled structure has property
$P$. It is routine to check that a countable conjunction of enforceable
properties is once again enforceable.

For us, one of the most important facts about enforceability is that the
property “$A(\bar{p})$ is an e.c. model of $T$” is enforceable. While proven
in [15] for the case of an $\forall\exists$-theory $T$, the proof readily
extends to this more general case.

We will also need the following fact:

######  Proposition 2.1.

If $P$ is a $\forall\bigvee\exists$-axiomatizable property (relative to $T$)
and there is a locally universal model of $T$ with property $P$, then $P$ is
an enforceable property.

Once again, this was proven in the first-order context in [15], but the proof
readily extends. This construction of e.c. models of $T_{\forall}$ with
$\forall\bigvee\exists$-axiomatizable properties is sometimes called the
“Eastern form” of the omitting types theorem (see [30]).

Finally, we say that an $L$-structure $A$ is an enforceable structure (or an
enforceable model of $T$) if the property “$A(\bar{p})\cong A$” is an
enforceable property. It is clear that there can be at most one enforceable
structure (up to isomorphism), which is then necessarily an e.c. model of $T$.

##  3\. The Connes Embedding Problem

Recall from the introduction that the Connes Embedding Problem (CEP) asks
whether or not the hyperfinite II1 factor $\mathcal{R}$ is a locally universal
model of the theory of tracial von Neumann algebras. In this section, we
discuss the recent resolution of this problem in the negative. A much more
detailed version of this section appears in the author’s survey article [16].

###  3.1. Nonlocal games and their entangled strategies

The negative solution to the CEP follows from a result in quantum complexity
theory known as $\operatorname{MIP}^{*}=\operatorname{RE}$, which we now
proceed to explain. The central object at the heart of this result is the
notion of a nonlocal game.

######  Definition 3.1.

A nonlocal game with $k$ questions and $n$ answers is a pair
$\mathfrak{G}:=(\pi,D)$, where $\pi$ is a probability distribution on
$[k]\times[k]$ and

$D:[k]\times[k]\times[n]\times[n]\to\\{0,1\\}$

is a function, known as the decision predicate. We refer to the pair $(k,n)$
as the dimensions of the game (although this terminology is not standard).

Here, $[k]:=\\{1,\ldots,k\\}$ and similarly for $[n]$. One envisions the game
$\mathfrak{G}$ above being played by two players, usually named Alice and Bob.
The referee for the game randomly sends Alice and Bob a pair of questions
$(x,y)\in[k]\times[k]$ according to the distribution $\pi$. Somehow, Alice and
Bob respond with answers $a$ and $b$ respectively from $[n]$. The referee then
uses the function $D$ to decide if they won this particular play of the game,
that is, they win if and only if $D(x,y,a,b)=1$.

In the result $\operatorname{MIP}^{*}=\operatorname{RE}$, Alice and Bob use
so-called entangled strategies to decide how to play:

######  Definition 3.2.

  1. (1)

If $\mathcal{H}$ is a Hilbert space, a positive operator-valued measure (or
POVM for short) of length $n$ is a sequence $A_{1},\ldots,A_{n}$ of positive
operators on $\mathcal{H}$ such that $\sum_{i=1}^{n}A_{i}=I_{\mathcal{H}}$.

  2. (2)

An entangled strategy for $(k,n)$-dimensional games consists of the following
information:

    1. (a)

A pair of finite-dimensional Hilbert spaces $\mathcal{H}_{A}$ and
$\mathcal{H}_{B}$;

    2. (b)

For each $x\in[k]$, a POVM $A^{x}_{1},\ldots,A^{x}_{n}$ on $\mathcal{H}_{A}$;

    3. (c)

For each $x\in[k]$, a POVM $B^{x}_{1},\ldots,B^{x}_{n}$ on $\mathcal{H}_{B}$;

    4. (d)

A unit vector $\psi\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$.

The above definition represents the situation that Alice and Bob will perform
quantum experiments to determine how to answer the questions they are sent by
the referee. Alice has a quantum system with associated Hilbert space
$\mathcal{H}_{A}$ while Bob has another quantum system with associated Hilbert
space $\mathcal{H}_{B}$. They share some state $\psi$ of their composite
system $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, which is usually an entangled
state (whence the name entangled strategy), that is, is not necessarily a
simple tensor $\psi_{A}\otimes\psi_{B}$ with $\psi_{A}\in\mathcal{H}_{A}$ and
$\psi_{B}\in\mathcal{H}_{B}$. Upon receiving question $x$, Alice performs the
measurement corresponding to the POVM $A^{x}$ on her part of the state $\psi$
to determine how she will respond; Bob acts in a similar fashion.

Given an entangled strategy for $(k,n)$-dimensional games as above, we
calculate, for $x,y\in[k]$ and $a,b\in[n]$, the value

$p(a,b|x,y):=\langle(A^{x}_{a}\otimes B^{y}_{b})\psi,\psi\rangle,$

which we interpret as the probability that Alice and Bob respond with answers
$a$ and $b$ if asked questions $x$ and $y$ respectively. If we let $p$ denote
the strategy thus obtained, then for any nonlocal game $\mathfrak{G}$ with $k$
questions and $n$ answers, the entangled value of $\mathfrak{G}$ corresponding
to $p$ is defined to be

$\operatorname{val}(\mathfrak{G},p):=\sum_{(x,y)\in[k]\times[k]}\pi(x,y)\sum_{(a,b)\in[n]\times[n]}p(a,b|x,y)D(x,y,a,b),$

which measures the probability that Alice and Bob win the game $\mathfrak{G}$
if they play according to the strategy $p$. We let
$C_{q}(k,n)\subseteq[0,1]^{k^{2}n^{2}}$ denote the set of entangled strategies
for $(k,n)$-dimensional games. The optimal probability Alice and Bob have for
winning the game by using entangled strategies is thus

$\operatorname{val}^{*}(\mathfrak{G}):=\sup_{p\in
C_{q}(k,n)}\operatorname{val}(\mathfrak{G},p)$

and is called the entangled value of $\mathfrak{G}$.

We can now state the version of $\operatorname{MIP}^{*}=\operatorname{RE}$
relevant for us:

######  Theorem 3.3 (Ji et. al. [28]).

There is an “efficient” mapping $\bf M\mapsto\mathfrak{G}_{\bf M}$ from Turing
machines to nonlocal games such that:

  * •

If $\bf M$ halts on the empty tape, then
$\operatorname{val}^{*}(\mathfrak{G}_{\bf M})=1$.

  * •

If $\bf M$ does not halt on the empty tape, then
$\operatorname{val}^{*}(\mathfrak{G}_{\bf M})\leq\frac{1}{2}$.

The reader interested in the completxity-theoretic motivation (and
nomenclature) for this result can consult the aforementioned survey [16].

###  3.2. From $\operatorname{MIP}^{*}=\operatorname{RE}$ to the failure of
CEP: the traditional route

We now briefly describe how the above result was used to refute the CEP. We
begin by noting that one can effectively approximate
$\operatorname{val}^{*}(\mathfrak{G})$ from below (provided the distribution
$\pi$ can be effectively calculated). Indeed, given some dimension $d$, one
can enumerate a computable sequence of finite nets

$N^{d}_{1}\subseteq N^{d}_{2}\subseteq\cdots$

over all states and POVMs in dimension $d$ with $|N^{d}_{m}|=m^{O(d^{2})}$
such that, for any $p\in C_{q}(k,n)$ based on a $d$-dimensional strategy and
any $m$, there is $q\in N^{d}_{m}$ with
$|\operatorname{val}(\mathfrak{G},p)-\operatorname{val}(\mathfrak{G},q)|<\frac{1}{m}$.
If one sets $\operatorname{val}^{n}(\mathfrak{G},p)=\max_{d,m\leq n}\max_{p\in
N^{d}_{m}}\operatorname{val}(\mathfrak{G},p)$, then
$\operatorname{val}^{n}(\mathfrak{G},p)$ is computable and
$\operatorname{val}^{n}(\mathfrak{G},p)$ converges to
$\operatorname{val}(\mathfrak{G})$ from below as $n$ tends to $\infty$.

By the equality $\operatorname{MIP}^{*}=\operatorname{RE}$, we deduce that
there cannot exist an effective procedure, uniform over all games, for
approximating $\operatorname{val}^{*}(\mathfrak{G})$ from above. Indeed, if
this were possible, then, combining this procedure with the procedure from the
previous paragraph, one would be able to effectively approximate
$\operatorname{val}^{*}(\mathfrak{G})$ uniformly in the description of
$\mathfrak{G}$. Any estimate of $\operatorname{val}^{*}(\mathfrak{G}_{\bf M})$
to within $\frac{1}{4}$ would allow one to decide whether or not
$\operatorname{val}^{*}(\mathfrak{G}_{\bf M})$ was $1$ or at most
$\frac{1}{2}$, and thus one could effectively solve the halting problem!

The relevance of this conclusion is that it refutes a well-known problem in
quantum information theory known as Tsirelson’s problem. Tsirelson’s problem
questioned whether or not the entangled value
$\operatorname{val}^{*}(\mathfrak{G})$ of $\mathfrak{G}$ and its so-called
commuting value $\operatorname{val}^{co}(\mathfrak{G})$ defined using an a
priori larger set of strategies known as quantum commuting strategies
coincided. Here, a quantum commuting strategy for $(k,n)$-dimensional games
consists of:

  * •

A single (possibly infinite-dimensional) Hilbert space $\mathcal{H}$,

  * •

For each $x\in[k]$, two POVMs $A^{x}_{1},\ldots,A^{x}_{n}$ and
$B^{y}_{1},\ldots,B^{y}_{b}$ on $\mathcal{H}$ satisfying
$A^{x}_{a}B^{y}_{b}=B^{y}_{b}A^{x}_{a}$ for all $x,y\in[k]\times[k]$ and
$a,b\in[n]\times[n]$, and

  * •

A unit vector $\psi\in\mathcal{H}$.

The second condition in the previous definition ensures that the measurements
$A^{x}$ and $B^{y}$ can be done simultaneously (ensuring that the players
still cannot communicate). Using this strategy, the probability that Alice and
Bob respond with $a$ and $b$ if asked questions $x$ and $y$ is given by
$p(a,b|x,y)=\langle A^{x}_{a}B^{y}_{b}\xi,\xi\rangle$. The set of such quantum
commuting strategies for $(k,n)$-dimensional games is denoted $C_{qc}(k,n)$
and the quantum commuting value of $\mathfrak{G}$ is
$\operatorname{val}^{co}(\mathfrak{G})=\sup_{p\in
C_{qc}(k,n)}\operatorname{val}(\mathfrak{G},p)$. It follows from the results
in [13] and [14] that $\operatorname{val}^{co}(\mathfrak{G})$ can always be
uniformly effectively approximated from above. Thus, Tsirelson’s problem must
have a negative answer by the results of the previous paragraph.

It had already been obvserved by Fritz [13] and independently by Junge et. al.
[29] that a positive answer to a well-known problem in
$\mathrm{C}^{*}$-algebra theory known as Kirchberg’s QWEP problem would yield
a positive answer to Tsirelson’s problem, whence Kirchberg’s QWEP problem must
also have a negative answer. (Ozawa [36] had later shown that a positive
answer to Tsirelson’s problem would in turn imply a positive answer to
Kirchberg’s problem, but that is now a moot point.) A difficult result of
Kirchberg [32] states that his QWEP problem is actually equivalent to the CEP,
whence we finally obtain the negative solution to the CEP from
$\operatorname{MIP}^{*}=\operatorname{RE}$! For details on all of this, see
the aforementioned survey [16].

###  3.3. From $\operatorname{MIP}^{*}=\operatorname{RE}$ to the failure of
CEP: the model-theoretic route

We now explain how continuous model theory can be used to eliminate the use of
Tsirelson’s problem and Kirchberg’s QWEP problem in the derivation of the
negative solution of CEP from $\operatorname{MIP}^{*}=\operatorname{RE}$.
First, we set $C_{qa}(k,n):=\overline{C_{q}(k,n)}$; the elements of this set
are the strategies for $(k,n)$-dimensional games that can be approximated by
entangled strategies.

######  Definition 3.4.

A strategy $p\in C_{qa}(k,n)$ is said to be synchronous if, for all $x\in[k]$
and _distinct_ $a,b\in[n]$, we have $p(a,b|x,x)=0$.

In other words, a strategy is synchronous if there is no chance that Alice and
Bob respond differently if asked the same question. We let $C_{qa}^{s}(k,n)$
denote the set of synchronous elements of $C_{qa}$ and we let
$\operatorname{s-val}(\mathfrak{G}):=\sup_{p\in
C_{qa}^{s}(k,n)}\operatorname{val}(\mathfrak{G},p)$ denote its synchronous
value. It is an artifact of the proof of
$\operatorname{MIP}^{*}=\operatorname{RE}$ that one may replace
$\operatorname{val}^{*}$ with $\operatorname{s-val}^{*}$ without changing the
validity of the result. The relevance of the set $C_{qa}^{s}(k,n)$ for us is
the following operator-algebraic characterization:

######  Theorem 3.5 (Kim, Paulsen, and Schaufhauser [31]).

Given $p\in[0,1]^{k^{2}n^{2}}$, we have that $p\in C_{qa}^{s}(k,n)$ if and
only if, for every $x\in[k]$, there are projections
$e^{x}_{1},\ldots,e^{x}_{n}\in\mathcal{R}^{\mathcal{U}}$ such that
$\sum_{a=1}^{n}e^{x}_{a}=1$ and such that
$p(a,b|x,y)=\tau(e^{x}_{a}e^{y}_{b})$ (where $\tau$ denotes the unique trace
on $\mathcal{R}^{\mathcal{U}}$).

Consequently, if we set $z:=(z^{x}_{a})$ to be variables ranging over the
operator norm unit ball and set

$\varphi(z):=\sum_{(x,y)\in[k]\times[k]}\pi(x,y)\sum_{(a,b)\in[n]\times[n]}\tau(z^{x}_{a}z^{y}_{b})D(x,y,a,b),$

a quantifier-free formula in the language of tracial von Neumann algebras,
then for every $p\in C_{qa}^{s}(k,n)$, we have
$\operatorname{val}(\mathfrak{G},p)=\varphi(e)^{\mathcal{R}^{\mathcal{U}}}$,
where $e=(e^{x}_{a})$ is the family of projections guaranteed to exist by
Theorem 3.5.

For any $k$ and $n$, set $Z(k,n)$ to be the set consisting of all tuples
$e=(e^{x}_{a})$ of projections in $\mathcal{R}$ such that
$\sum_{a=1}^{n}e^{x}_{a}=1$ for all $x\in[k]$. In order to prove their
theorem, Kim, Paulsen, and Schaufhauser observed the following fact:

######  Lemma 3.6.

For any $k$ and $n$, $Z(k,n)$ is a definable subset of $\mathcal{R}$.

Of course, Kim, Paulsen, and Schaufhauser stated their result in operator-
algebraic terms, representing a beautiful confluence of operator-algebraic and
model-theoretic ideas! Consequently, we have:

######  Corollary 3.7.

For any nonlocal game $\mathfrak{G}$ with $k$ questions and $n$ answers, we
have

$\operatorname{s-val}^{*}(\mathfrak{G})=\left(\sup_{z\in
Z(k,n)}\varphi(z)\right)^{\mathcal{R}}.$

An analysis of how formulae with quantifiers over a definable set can be
equivalently expressed using ordinary formulae shows that the sentence
appearing above is effectively equivalent to a universal sentence, uniform in
the description of the game.

The upshot of all of this is that $\operatorname{MIP}^{*}=\operatorname{RE}$
implies that $\operatorname{Th}_{\forall}(\mathcal{R})$ cannot be effectively
approximated from above, that is, there is no algorithm which, upon input a
universal sentence $\sigma$, returns a computable sequence of upper bounds
converging to $\sigma^{\mathcal{R}}$. However, a few years prior to the
appearance of $\operatorname{MIP}^{*}=\operatorname{RE}$, the author and Bradd
Hart observed the following in [19]:

######  Theorem 3.8 (G. and Hart).

If CEP has a positive answer, then $\operatorname{Th}_{\forall}(\mathcal{R})$
_is_ effectively approximable from above.

The proof of the previous theorem proceeds as follows. If $\sigma$ is a
universal sentence in the language of tracial von Neumann algebras, then CEP
implies that the supremum of $\sigma^{M}$ as $M$ ranges over all II1 factors
is simply $\sigma^{\mathcal{R}}$. On the other hand, the Completeness theorem
for continuous logic [5] implies that this supremum is equivalently expressed
as the infimum of all dyadic rational numbers $r$ such that
$T_{II_{1}}\vdash\sigma\buildrel\textstyle\ .\over{\hbox{ \vrule
height=3.0pt,depth=0.0pt,width=0.0pt}{\smash{-}}}r$, where $T_{II_{1}}$ is the
theory of II1 factors. Since $T_{II_{1}}$ is an effectively enumerable theory,
the collection of such $r$’s is itself effectively enumerable, implying the
desired result.

Putting all of the pieces together, we arrive at the model-theoretic
explanation for why a negative solution to CEP follows from
$\operatorname{MIP}^{*}=\operatorname{RE}$.

###  3.4. Bells and whistles

The above analysis leads to a _Gödelian_ -style refutation of the CEP. To
state this precisely, we introduce the following notion.

######  Definition 3.9.

Let $M$ be an structure in some (continuous) language $L$. By the $M$EP we
mean the statement that there is an effectively enumerable subset
$T\subseteq\operatorname{Th}(M)$ such that all models of $T$ embed into an
ultrapower of $M$.

The analysis of the previous section shows the following:

######  Corollary 3.10.

The $\mathcal{R}$EP has a negative solution.

Since $\mathcal{R}$ embeds into every II1 factor, we could drop the
requirement in the above definition that $T$ be contained in
$\operatorname{Th}(M)$ and instead merely ask that $T$ extend the theory of
II1 factors, obtaining the following corollary:

######  Corollary 3.11.

There is no effectively enumerable theory $T$ extending the theory of II1
factors all of whose models embed into an ultrapower of $\mathcal{R}$.

In Subsection 5.3 below, we will see how to derive the failure of the $A$EP
for certain $\mathrm{C}^{*}$-algebras $A$ from the above arguments.

A consequence of Corollary 3.11 above is that one can infer the existence of
infinitely many universal theories of II1 factors, something the “standard”
refutation of CEP does not seem to imply:

######  Corollary 3.12.

There is a sequence $M_{1},M_{2},\ldots,$ of separable II1 factors, none of
which embed into an ultrapower of $\mathcal{R}$, satisfying, for all $i<j$,
that $M_{i}$ does not embed into an ultrapower of $M_{j}$. In particular,
there are infinitely many universal theories of II1 factors.

To construct the above sequence, let $M_{1}$ be any separable II1 factor that
does not embed into an ultrapower of $\mathcal{R}$ and let $\sigma_{1}$ be a
universal sentence such that $\sigma_{1}^{\mathcal{R}}=0$ but
$\sigma_{1}^{M_{1}}>0$. Let $r_{1}\in(0,\sigma_{1}^{M_{1}})$ be a rational
number and let $T_{1}:=T_{II_{1}}\cup\\{\sigma_{1}\buildrel\textstyle\
.\over{\hbox{ \vrule
height=3.0pt,depth=0.0pt,width=0.0pt}{\smash{-}}}r_{1}\\}$. Since $T_{1}$ is
effectively enumerable, there is a separable model $M_{2}$ of $T_{1}$ that
does not embed into an ultrapower of $\mathcal{R}$. Since
$\sigma_{1}^{M_{1}}>\sigma^{M_{2}}_{1}$, we have that $M_{1}$ does not embed
into an ultrapower of $M_{2}$. Since $M_{2}$ does not embed into an ultrapower
of $\mathcal{R}$, there is a universal sentence $\sigma_{2}$ such that
$\sigma_{2}^{\mathcal{R}}=0$ but $\sigma_{2}^{M_{2}}>0$. Let
$r_{2}\in(0,\sigma_{2}^{M_{2}})$ be rational and let
$T_{2}:=T_{1}\cup\\{\sigma_{2}\buildrel\textstyle\ .\over{\hbox{ \vrule
height=3.0pt,depth=0.0pt,width=0.0pt}{\smash{-}}}r_{2}\\}$. Once again,
$T_{2}$ is effectively enumerable, so there is a separable model $M_{3}$ of
$T_{2}$ that does not embed into an ultrapower of $\mathcal{R}$. It is clear
that neither $M_{1}$ nor $M_{2}$ embed into an ultrapower of $M_{3}$. One
constructs the remainder of the sequence analogously.

We believe the following question should have a positive answer:

######  Question 3.13.

Do there exist continuum many universal theories of II1 factors?

Another application of Corollary 3.11 above is the following, which also
appears not to follow from the “standard” refutation of the CEP:

######  Corollary 3.14.

There is a II1 factor without property Gamma that does not embed into an
ultrapower of $\mathcal{R}$.

To see this, simply let $\sigma$ be one of the sentences in the axiomatization
of property Gamma for which $\sigma^{L(\mathbb{F}_{2})}>0$ and let
$T:=T_{II_{1}}\cup\\{\sigma\buildrel\textstyle\ .\over{\hbox{ \vrule
height=3.0pt,depth=0.0pt,width=0.0pt}{\smash{-}}}r\\}$ for some rational
number $r\in(0,\sigma^{L(\mathbb{F}_{2})})$. By Corollary 3.11 above, there is
a model of $T$ that does not embed into an ultrapower of $\mathcal{R}$, which
is thus the desired factor. (Corollary 3.14 also follows from a very recent
result of Chifan, Drimbe, and Ioana [7], who prove that every II1 factor
embeds into a II1 factor with property (T).)

The following question is surprisingly more difficult:

######  Question 3.15.

Are there two separable II1 factors $M$ and $N$, each of which embed into an
ultrapower of $\mathcal{R}$, neither of which have property Gamma, for which
$M\not\equiv N$?

###  3.5. The existence of the enforceable II1 factor

In this subsection, we mention a model-theoretic variant of the CEP that is
still open and, in this author’s opinion, is one of the more interesting open
problems in the model theory of operator algebras.

As discussed in the article by the author and Hart in this volume,
$\mathcal{R}$ is an e.c. model of its universal theory and the CEP is thus
equivalent to the assertion that $\mathcal{R}$ is an e.c. II1 factor. The
ideas in this subsection elaborate further on this observation. We first note
the following:

######  Lemma 3.16.

Being hyperfinite is a $\forall\bigvee\exists$-axiomatizable property of II1
factors.

The proof of this lemma has not appeared explicitly in the literature but is
similar to the proof of the main results in [6].

Armed with this and Proposition 2.1 above, we arrive at the following:

######  Corollary 3.17.

$\mathcal{R}$ is the enforceable model of its universal theory.

In particular, if CEP were to hold, then $\mathcal{R}$ would be the
enforceable II1 factor. (Of course, the converse is also true, but moot at
this point.) Nevertheless, the following question is open and tantalizing:

######  Question 3.18.

Does the enforceable II1 factor exist?

How likely is it that the enforceable II1 factor exists? That is of course
difficult to say. If it did exist, then it would “rival” $\mathcal{R}$ for
being “the most important II1 factor” for it would be generic from the model-
theoretic point of view. In [17], some properties of the enforceable II1
factor were established (of course presuming its existence).

It is worth pointing out one notable case when the enforceable object does not
exist, namely for the (classical) theory of groups. (This seems implicit in
Hodges’ book [27] but is written down explicitly in the article [21] by
Kunnawalkam Elayavalli, Lodha, and the author.) However, the ingredients
involved in the proof are a blend of recursion-theoretic and combinatorial
group-theoretic tools that seem to be currently unavailable to us in the II1
factor setting.

Another interesting variant of the CEP is the following:

######  Question 3.19.

Is the property of being isomorphic to a group von Neumann algebra an
enforceable property?

##  4\. The Kirchberg Embedding Problem

In this and the next section, we consider $\mathrm{C}^{*}$-algebra versions of
the CEP. For simplicity, henceforth all $\mathrm{C}^{*}$-algebras will be
assumed to be unital. (Much of what is said below can be adapted to the not
necessarily unital situation, but this assumption simplifies the exposition.)

If one phrases the CEP as the statement that every tracial von Neumann algebra
embeds into the tracial ultrapower of an _injective_ II1 factor, then a
natural $\mathrm{C}^{*}$-algebra analog of CEP would be to ask whether or not
every $\mathrm{C}^{*}$-algebra embeds into an ultrapower of a nuclear
$\mathrm{C}^{*}$-algebra (for a $\mathrm{C}^{*}$-algebra $A$ is nuclear if and
only if its enveloping von Neumann algebra $A^{**}$ is injective). By
Kirchberg’s celebrated theorem [33], every separable nuclear
$\mathrm{C}^{*}$-algebra embeds into the Cuntz algebra $\mathcal{O}_{2}$ (see
Szabó’s article in this volume for the definition of $\mathcal{O}_{2}$),
whence it is equivalent to ask whether or not $\mathcal{O}_{2}$ is a locally
universal $\mathrm{C}^{*}$-algebra. We refer to this problem as the Kirchberg
embedding problem (KEP). (We attribute this problem to Kirchberg as we first
learned of this problem from Ilijas Farah, who in turn first learned of this
problem during a discussion with Kirchberg in 2007. The first mention of this
problem in the literature appears to be in the author’s article [22] with
Sinclair.)

At the moment of the writing of this article, the KEP remains an open problem.
In this section, we mention the connection between it and the model theory of
$\mathrm{C}^{*}$-algebras in a way that parallels the situation with the CEP.
Recall that the CEP is equivalent to the statement that $\mathcal{R}$ is an
e.c. tracial von Neumann algebra. The analogous statement for the KEP holds:

######  Proposition 4.1.

The KEP has a positive solution if and only if $\mathcal{O}_{2}$ is an e.c.
$\mathrm{C}^{*}$-algebra.

The proof is analogous to the proof of the same statement for tracial von
Neumann algebras, using the fact that any two embeddings of $\mathcal{O}_{2}$
into its ultrapower are unitarily conjugate (a consequence of the fact that
$\mathcal{O}_{2}$ is a strongly self-absorbing $\mathrm{C}^{*}$-algebra) and
that the theory of $\mathrm{C}^{*}$-algebras has the joint embedding property.

The following variation on the preceding proposition is also of interest:

######  Proposition 4.2.

$\mathcal{O}_{2}$ is the only possible separable $\mathrm{C}^{*}$-algebra that
is nuclear and e.c. Consequently, a positive solution to the KEP is equivalent
to the statement that there is a $\mathrm{C}^{*}$-algebra that is both nuclear
and e.c.

The proof of this proposition is quite interesting. Indeed, suppose that $A$
is a separable $\mathrm{C}^{*}$-algebra that is both nuclear and e.c. A
consequence of being e.c. is that $A$ is simple (see [22]). By another
fundamental result of Kirchberg [33], the fact that $A$ is simple, separable
and nuclear implies that $A\otimes\mathcal{O}_{2}\cong\mathcal{O}_{2}$.
However, a consequence of $A$ being e.c. is that $A$ is
“$\mathcal{O}_{2}$-stable,” that is, $A\otimes\mathcal{O}_{2}\cong A$; this
follows from the fact that being $\mathcal{O}_{2}$-stable is an
$\forall\exists$-axiomatizable property of $\mathrm{C}^{*}$-algebras (see
[12]) together with the fact that every $\mathrm{C}^{*}$-algebra embeds into
an $\mathcal{O}_{2}$-stable algebra (namely by tensoring the algebra with
$\mathcal{O}_{2}$ itself and using the fact, due to Cuntz, that
$\mathcal{O}_{2}\otimes\mathcal{O}_{2}\cong\mathcal{O}_{2}$). It follows that
$A\cong\mathcal{O}_{2}$, as desired. An alternative, slightly more elementary
proof, can be found in [23, Remark 21].

Another similarity with the CEP concerns enforceability. First, we will need
the following result:

######  Proposition 4.3.

Being nuclear is a $\forall\bigvee\exists$-axiomatizable property of
$\mathrm{C}^{*}$-algebras.

Two proofs for the preceding proposition are offered in [12], one “soft” and
model-theoretic, the other “concrete,” writing down specific axioms for
nuclearity. Let us sketch the former argument. Given a
$\mathrm{C}^{*}$-algebra $A$ and $k,n\in\mathbb{N}$, define the predicate
$\operatorname{nuc}_{k,n}^{A}:A_{1}^{k}\to\mathbb{R}$ by
$\operatorname{nuc}_{k,n}^{A}(\vec{a})=\inf_{\phi,\psi}\|(\psi\circ\phi)(\vec{a})-\vec{a}\|$,
where $\phi$ ranges over all ucp maps $A\to M_{n}(\mathbb{C})$ and $\psi$
ranges over all ucp maps $M_{n}(\mathbb{C})\to A$. An argument using the Beth
definability theorem shows that the predicates $\operatorname{nuc}_{k,n}$ are
actually existentially definable relative to the theory of
$\mathrm{C}^{*}$-algebras, that is, there are existential formulae (really,
uniform limits of existential formulae) $\Phi_{k,n}(\vec{x})$ such that, for
every $\mathrm{C}^{*}$-algebra $A$, every $k,n\in\mathbb{N}$, and every
$\vec{a}\in A^{k}_{1}$, we have
$\operatorname{nuc}_{k,n}^{A}(\vec{a})=\Phi_{k,n}^{A}(\vec{a})$. It remains to
note that a $\mathrm{C}^{*}$-algebra $A$ is nuclear if and only if, for every
$k\in\mathbb{N}$, we have
$\left(\sup_{\vec{x}}\bigvee_{n\in\mathbb{N}}\Phi_{k,n}(\vec{x})\right)^{A}=0$.

As with the CEP, we arrive at the following formulation of the KEP, whose
proof uses everything we have discussed thus far:

######  Theorem 4.4 (G. [15]).

The following are equivalent:

  1. (1)

The KEP has a positive solution.

  2. (2)

The property of being nuclear is enforceable.

  3. (3)

$\mathcal{O}_{2}$ is the enforceable $\mathrm{C}^{*}$-algebra.

The previous theorem yielocus an interesting local, finitary reformulation of
the KEP first identified by Sinclair and the author in [22]. First, we say a
condition (relative to the theory of $\mathrm{C}^{*}$-algebras) $p(\vec{x})$,
where $\vec{x}$ is a $k$-tuple, has good nuclear witnesses if, for every
$\epsilon>0$, there is a $\mathrm{C}^{*}$-algebra $A$, $\vec{a}\in A^{k}$, and
an $n\in\mathbb{N}$ such that $\vec{a}$ satisfies the condition $p(\vec{x})$
and for which $\operatorname{nuc}_{k,n}^{A}(\vec{a})<\epsilon$. The previous
theorem then yields the following corollary:

######  Corollary 4.5.

The KEP has a positive solution if and only if every condition has good
nuclear witnesses.

The import of the previous corollary is a (seemingly) significant weakening of
the demand that every condition be satisfied in a nuclear
$\mathrm{C}^{*}$-algebra (equivalently, satisfied in $\mathcal{O}_{2}$), for
one only asks that the witness admit a good ucp factorization through a matrix
algebra, and, moreover, the witness and the dimension of the matrix algebra
can vary as the level of approximation varies.

##  5\. The MF problem and the quasidiagonality problem

###  5.1. A negative solution to the MF problem

Recall that the CEP is also equivalent to the statement that every separable
tracial von Neumann algebra embeds into
$\prod_{\mathcal{U}}M_{n}(\mathbb{C})$, a _tracial_ ultraproduct of matrix
algebras with respect to a nonprincipal ultrafilter on $\mathbb{N}$. It is
thus natural to formulate a $\mathrm{C}^{*}$-algebra version of CEP by asking
that every separable $\mathrm{C}^{*}$-algebra embed into a _$\mathrm{C}^{*}$
-algebra_ ultraproduct $\prod_{\mathcal{U}}M_{n}(\mathbb{C})$ of matrix
algebras with respect to a nonprincipal ultrafilter on $\mathbb{N}$. Using the
same notation for both tracial von Neumann algebra ultraproducts and
$\mathrm{C}^{*}$-algebra ultraproducts is potentially dangerous (and some
authors even use different notations for the two ultraproducts); to prevent
confusion, in the remainder of this section, unless explicitly stated
otherwise, all ultraproducts will be $\mathrm{C}^{*}$-algebra ultraproducts.

There is an immediate obstruction to the statement “every separable
$\mathrm{C}^{*}$-algebra embeds into $\prod_{\mathcal{U}}M_{n}(\mathbb{C})$”
from being true, namely the $\mathrm{C}^{*}$-algebra
$\prod_{\mathcal{U}}M_{n}(\mathbb{C})$ is stably finite, as is any subalgebra.
Thus, we may modify the problem as follows:

######  Definition 5.1.

The MF problem is the problem of whether or not every separable stably finite
$\mathrm{C}^{*}$-algebra embeds into $\prod_{\mathcal{U}}M_{n}(\mathbb{C})$.

The terminology MF comes from the fact that a separable
$\mathrm{C}^{*}$-algebra is called matricially finite (or MF) if it embeds
into $\prod_{\mathcal{U}}M_{n}(\mathbb{C})$. Consequently, the MF problem asks
if the notions of stably finite and MF coincide for separable
$\mathrm{C}^{*}$-algebras.

As in the case of the CEP, the MF problem can be reformulated in terms of
ultrapowers of a single object. Indeed, the MF problem is equivalent to the
problem of whether every separable stably finite $\mathrm{C}^{*}$-algebra
embeds into a nonprincipal ultrapower $\mathcal{Q}^{\mathcal{U}}$ of the
universal UHF algebra $\mathcal{Q}$ (see [12, Lemma 4.4.1]).

An immediate consequence of the negative solution of the CEP is that the MF
problem also has a negative solution:

######  Corollary 5.2.

The MF problem has a negative solution.

To prove the previous corollary, suppose that $M$ is a II1 factor that does
not embed into $\mathcal{R}^{\mathcal{U}}$ (here we mean the tracial von
Neumann algebra ultrapower). We claim then that $M$ does not embed (as a
C*-algebra) into a nonprincipal C*-ultrapower of $\mathcal{Q}$; by considering
a separable elementary subalgebra of $M$ (in the language of
$\mathrm{C}^{*}$-algebras), we obtain the desired counterexample to the MF
problem. Suppose, towards a contradiction, that
$i:M\hookrightarrow\mathcal{Q}^{\mathcal{U}}$ is an embedding. Recall that
$\mathcal{Q}$ has a unique trace $\tau$ and the von Neumann algebra generated
by $\mathcal{Q}$ with respect to the GNS represenation corresponding to $\tau$
is $\mathcal{R}$. Let
$\pi:\mathcal{Q}^{\mathcal{U}}\to\mathcal{R}^{\mathcal{U}}$ denote the
composition of the quotient map
$\mathcal{Q}^{\mathcal{U}}\to\mathcal{Q}^{\mathcal{U}}/I$, where

$I=\\{(x_{n})^{\bullet}\in\mathcal{Q}^{\mathcal{U}}\ :\
\lim_{\mathcal{U}}\|x_{n}\|_{2}=0\\}$

is the trace ideal, with the natural inclusion
$\mathcal{Q}^{\mathcal{U}}/I\hookrightarrow\mathcal{R}^{\mathcal{U}}$ obtained
by viewing operator norm bounded balls in $\mathcal{Q}$ as
$\|\cdot\|_{2}$-dense subsets of the corresponding balls in $\mathcal{R}$.
Since $M$ has a unique trace, which is faithful, we get that the composition
$\pi\circ i:M\to\mathcal{R}^{\mathcal{U}}$ is a trace-preserving
*-homomorphism, a contradiction.

The following question seems wide open:

######  Question 5.3.

Does $\mathcal{R}$ embed, as a $\mathrm{C}^{*}$-algebra, into
$\mathcal{Q}^{\mathcal{U}}$?

The negative solution to the MF problem also has a Gödelian-style refutation,
that is, the $\mathcal{Q}$EP has a negative solution as well; we postpone the
discussion of this fact until Subsection 5.3 below.

Unlike most of the embedding problems discussed in this paper, it is not even
clear that there _ought_ to be a locally universal object for the class of
stably finite $\mathrm{C}^{*}$-algebras as the following question appears to
be open:

######  Question 5.4.

Does the class of stably finite $\mathrm{C}^{*}$-algebras have the JEP?

A natural guess would be that the minimal tensor product of two stably finite
$\mathrm{C}^{*}$-algebras would once again be stably finite. However, the
validity of this statement is far from clear. In fact, the question of whether
or not the minimal tensor product of two _simple_ stably finite
$\mathrm{C}^{*}$-algebras is once again stably finite is equivalent to a well-
known open problem, namely whether or not every stably finite
$\mathrm{C}^{*}$-algebra admits a trace (see [23]). It is worth mentioning
that the class of $\mathrm{C}^{*}$-algebras admitting a trace does have JEP
(the tensor product trace on the minimal tensor product witnesses this) and
consequently any e.c. object for this class (which exists since the class is
inductive) is locally universal.

###  5.2. The quasidiagonality problem

Connes’ original motivation for considering the question of which tracial von
Neumann algebras embed into ultrapowers of $\mathcal{R}$ came from his
striking result proving that injective II1 factors were hyperfinite, thus
completing the classification of injective II1 factors [8]. A crucial
ingredient in his proof was that injective factors did indeed admit embeddings
into ultrapowers of $\mathcal{R}$.

In $\mathrm{C}^{*}$-algebra theory, the analogous problem would be trying to
classify simple, nuclear $\mathrm{C}^{*}$-algebras, where simple is the analog
of being a factor and (as already mentioned) nuclear is the analog of being
injective. In trying to mimic Connes’ approach in classifying simple, nuclear
$\mathrm{C}^{*}$-algebras, it thus becomes natural to try to prove that they
admit embeddings into $\mathcal{Q}^{\mathcal{U}}$. As stated in the previous
subsection, an immediate obstruction to proving such a result is that the
$\mathrm{C}^{*}$-algebra in question must be stably finite. Thus, one is
naturally led to:

######  Definition 5.5 (Quasidiagonality problem-simple version).

Does every simple, stably finite, nuclear $\mathrm{C}^{*}$-algebra embed into
$\mathcal{Q}^{\mathcal{U}}$?

A word about the nomenclature in the previous definition is in order. A
$\mathrm{C}^{*}$-algebra $A$ is said to be quasidiagonal if there is an
embedding $A\hookrightarrow\prod_{\mathcal{U}}M_{n}(\mathbb{C})$ that admits a
ucp lift $A\to\prod_{n\in\mathbb{N}}M_{n}(\mathbb{C})$. Thus, quasidiagonal
$\mathrm{C}^{*}$-algebras form a special subclass of the class of MF-algebras.
However, by the Choi-Effros lifting theorem, if a _nuclear_
$\mathrm{C}^{*}$-algebra is MF, then the aforementioned ucp lift automatically
exists, whence there is no difference in the two notions. Halmos defined what
it meant for a set of bounded operators to be quasidiagonal and then a
$\mathrm{C}^{*}$-algebra is called quasidiagonal if it admits a concrete
representation for which the operators in the image of the representation form
a quasidiagonal set. Voiculescu then proved that this definition of
quasidiagonal $\mathrm{C}^{*}$-algebra agrees with the one given at the
beginning of this paragraph (see [39]).

As with the MF problem, it is not evident that the class of simple, stably
finite nuclear $\mathrm{C}^{*}$-algebras should have a locally universal
object for this class is not known to have JEP.

Amazingly enough, this modified version of the MF problem has almost been
shown to be true, modulo one technical assumption:

######  Theorem 5.6 (Tikuisis, Winter, White [38]).

Every simple, stably finite, nuclear $\mathrm{C}^{*}$-algebra _satisfying the
UCT_ is quasidiagonal.

Here, the UCT is short for the Universal Coefficient Theorem. Assuming that a
$\mathrm{C}^{*}$-algebra satisfies the UCT is a technical K-theoretic
assumption on the algebra. (See [37] for more information on the UCT.) One of
the major open questions in $\mathrm{C}^{*}$-algebra theory is:

######  Question 5.7 (UCT problem).

Do all separable nuclear $\mathrm{C}^{*}$-algebras satisfy the UCT?

Of all of the adjectives appearing in the statement of Theorem 5.6, all but
the UCT have been shown to have model-theoretic meaning: being stably finite
and MF are universally axiomatizable properties whilst being simple and being
nuclear are $\forall\bigvee\exists$-axiomatizable properties. (It turns out
that quasidiagonality in general is also
$\forall\bigvee\exists$-axiomatizable; see [12, Section 5.13].)

It is interesting to ask:

######  Question 5.8.

Is satisfying the UCT a $\forall\bigvee\exists$-axiomatizable property of
separable nuclear $\mathrm{C}^{*}$-algebras?.

The previous question notwithstanding, Barlak and Szabó [4] proved the
following interesting fact:

######  Theorem 5.9.

If $A$ is an e.c. subalgebra of $B$ and $B$ is a nuclear
$\mathrm{C}^{*}$-algebra satisfying the UCT, then so does $A$.

This theorem allows one to deduce the truth of the simple version of the
quasidiagonality problem from a weakening of the UCT problem:

######  Theorem 5.10 (G. and Sinclair [23]).

Suppose that every simple, stably finite, nuclear $\mathrm{C}^{*}$-algebra
embeds into a simple, stably finite, nuclear $\mathrm{C}^{*}$-algebra
satisfying the UCT. Then the simple version of the quasidiagonality problem is
true.

Indeed, suppose that $A$ is an e.c. simple, stably finite, nuclear
$\mathrm{C}^{*}$-algebra (which exists since this class is
$\forall\bigvee\exists$-axiomatizable). By the assumption of the theorem and
Theorem 5.9 above, it follows that $A$ itself satisfies the UCT, whence $A$ is
quasidiagonal by Theorem 5.6 above. Now if $B$ is any separable, simple,
stably finite, nuclear $\mathrm{C}^{*}$-algebra, then $A\otimes B$ is also
stably finite; this uses the quasidiagonality of $A$ (see [23, Lemma 4]).
Since $A$ is e.c., we have that $A\otimes B$, and thus $B$, embeds into an
ultrapower of $A$, whence $A$ is locally universal for the class of simple,
stably finite, nuclear $\mathrm{C}^{*}$-algebras. Since $A$ is quasidiagonal,
the result follows.

One can remove the simplicty assumption in the quasidiagonality problem,
arriving at:

######  Definition 5.11 (Quasidiagonality problem-general version).

Does every stably finite, nuclear $\mathrm{C}^{*}$-algebra embed into
$\mathcal{Q}^{\mathcal{U}}$?

The quasidiagonality problem has several model-theoretic equivalents:

######  Theorem 5.12.

The following are equivalent:

  1. (1)

The quasidiagonality problem has a positive solution.

  2. (2)

Being UHF is an enforceable property of stably finite, nuclear
$\mathrm{C}^{*}$-algebras.

  3. (3)

$\mathcal{Q}$ is the enforceable stably finite, nuclear
$\mathrm{C}^{*}$-algebra.

  4. (4)

$\mathcal{Q}$ is an e.c. stably finite, nuclear $\mathrm{C}^{*}$-algebra.

The implication (1) implies (2) in the previous proposition follows from
Proposition 2.1 and the fact that being UHF is
$\forall\bigvee\exists$-axiomatizable (see [6]). The implication (3) implies
(4) follows from the fact that being e.c. is enforceable while the implication
(4) implies (1) proceeds along the lines of Theorem 5.10 above. Finally, to
see (2) implies (3), one first notes that the property of being
$\mathcal{Q}$-stable, that is, that $A\otimes\mathcal{Q}\cong A$, is also
enforceable. Indeed, this property is $\forall\exists$-axiomatizable (for the
same reason as in the case of $\mathcal{O}_{2}$) and is thus true of any e.c.
object in this class as any object in this class is a subalgebra of a
$\mathcal{Q}$-stable object in the class (by tensoring with $\mathcal{Q}$). It
remains to note that $\mathcal{Q}$ is the only $\mathcal{Q}$-stable UHF
algebra.

The following question is the natural stably finite analog of Theorem 4.2
above; see [23] for partial progress towards its resolution:

######  Question 5.13.

Suppose that $A$ is an e.c. stably finite $\mathrm{C}^{*}$-algebra that is
also nuclear. Must we have $A\cong\mathcal{Q}$?

At the moment, it is unclear if the general version of the quasidiagonality
problem could be deduced from the simple version, even assuming a positive
solution to the UCT problem. However, using model-theoretic forcing again, one
can prove the following result:

######  Theorem 5.14 (G. and Sinclair [23]).

Suppose the following hold:

  1. (1)

Every stably finite nuclear $\mathrm{C}^{*}$-algebra embeds into a stably
finite, nuclear $\mathrm{C}^{*}$-algebra satisfying the UCT.

  2. (2)

There is a simple, stably finite, nuclear $\mathrm{C}^{*}$-algebra that is
locally universal for the class of stably finite nuclear
$\mathrm{C}^{*}$-algebras.

Then the general version of the quasidiagonality problem holds.

Note that the second item in the hypotheses of the previous theorem is indeed
a weakening of the statement of the quasidiagonality problem as $\mathcal{Q}$
itself is simple. The proof of the preceding theorem proceeds similarly as in
the proof of Theorem 5.10 above. Indeed, the second condition, Proposition
2.1, and the fact that being simple is $\forall\bigvee\exists$-axiomatizable
allows one to construct an e.c. stably finite nuclear $\mathrm{C}^{*}$-algebra
$A$ that is simple. Moreover, the first condition and Theorem 5.9 above allows
one to conclude that $A$ satisfies the UCT. Thus, Theorem 5.6 above allows one
to conclude that $A$ is quasidiagonal. It follows that the quasidiagonality
problem has a positive answer just as in the conclusion of the proof of
Theorem 5.10.

By the negative solution to the MF problem, there is a universal sentence
$\sigma$ in the language of $\mathrm{C}^{*}$-algebras for which
$\sigma^{\mathcal{Q}}=0$ and yet $\sigma^{A}=r>0$ for some stably finite
$\mathrm{C}^{*}$-algebra $A$. Let $T$ be the theory of stably finite
$\mathrm{C}^{*}$-algebras together with the existential condition
$r\buildrel\textstyle\ .\over{\hbox{ \vrule
height=3.0pt,depth=0.0pt,width=0.0pt}{\smash{-}}}\sigma=0$. We adapt the
terminology from Section 4 above and say that a condition $p(\vec{x})$ (with
$\vec{x}=(x_{1},\ldots,x_{k})$) relative to the theory $T$ has good nuclear
witnesses if, for every $\epsilon>0$, there is $A\models T$ and $\vec{a}\in A$
satisfying $p$ such that $\operatorname{nuc}_{k,n}^{A}(\vec{a})<\epsilon$.

######  Theorem 5.15.

Using the terminology in the previous paragraph, suppose that every condition
has good nuclear witnesses. Then the quasidiagonality problem has a negative
solution.

Indeed, the assumption that every condition has good nuclear witnesses allows
one to construct a model of $T$ that is nuclear; being a model of $T$, the
algebra is also stably finite but not embeddable in an ultrapower of
$\mathcal{Q}$ (that is, not quasidiagonal). Considering the contrapositive of
this theorem, if the quasidiagonality problem has a positive solution, then
whenever one has a counterexample to the MF problem as in the paragraph above,
then there must be some condition $p(\vec{x})$ relative to the associated
theory $T$ which does not have good nuclear witnesses, meaning that there is
some some $\epsilon>0$ such that, in every model $A$ of $T$, every $k$-tuple
$\vec{a}$ from $A$ satisfying $p$ must satisfy
$\operatorname{nuc}_{k,n}^{A}(\vec{a})\geq\epsilon$ for all $n\in\mathbb{N}$.

###  5.3. Monotracial $\mathrm{C}^{*}$-algebras and the Jiang-Su Embedding
Problem

One can adapt the techniques used to show that the $\mathcal{R}$EP fails to
show that the $A$EP fails for a large class of $\mathrm{C}^{*}$-algebras. We
will be concerned with monotracial $\mathrm{C}^{*}$-algebras, that is,
$\mathrm{C}^{*}$-algebras which admit a unique tracial state. For example, the
universal UHF algebra $\mathcal{Q}$ is monotracial. Suppose that $A$ is a
monotracial $\mathrm{C}^{*}$-algebra whose unique trace is $\tau_{A}$. Let $N$
denote the von Neumann algebra generated by $A$ via the GNS representation of
$A$ associated to $\tau_{A}$. Then $N$ is a tracial von Neumann algebra when
equipped with the extension $\tau_{N}$ of the original trace $\tau_{A}$ to
$N$. Since $A$ has a unique trace, it follows that $\tau_{N}$ is the unique
trace on $N$, whence $N$ is in fact a tracial factor. If we further assume
that $A$ is infinite-dimensional, then we can conclude that $N$ is a II1
factor.

Now suppose that $\sigma$ is a universal sentence in the language of tracial
von Neumann algebras. We can also view $\sigma$ as a sentence in the language
of tracial $\mathrm{C}^{*}$-algebras (one that simply does not refer to the
operator norm in any way) and can thus compare the values
$\sigma^{(A,\tau_{A})}$ and $\sigma^{(N,\tau_{N})}$. Since $A$ is
$\|\cdot\|_{2}$-dense in any operator norm bounded subset of $N$, it is clear
that $\sigma^{(N,\tau_{N})}\leq\sigma^{(A,\tau_{A})}$. On the other hand, if
$A$ is further assumed to be simple, then $A$ embeds into $N$ and thus,
$\sigma^{(A,\tau_{A})}=\sigma^{(N,\tau_{N})}$.

Summarizing thus far: if $A$ is a unital, simple, infinite-dimensional,
monotracial $\mathrm{C}^{*}$-algebra whose weak closure in the GNS
representation we denote by $N$, then for any sentence $\sigma$ in the
language of tracial von Neumann algebras, we have that
$\sigma^{(A,\tau_{A})}=\sigma^{(N,\tau_{N})}$. If we further assume that $N$
embeds in an ultrapower of $\mathcal{R}$, then this common value equals
$\sigma^{(\mathcal{R},\tau_{R})}$.

It is thus tempting to try to conclude that the $A$EP must fail for any
$\mathrm{C}^{*}$-algebra $A$ satisfying the conditions appearing in the
previous paragraph. Indeed, if, towards a contradiction, there was an
effectively enumerable subset $T\subseteq\operatorname{Th}(A)$, all of whose
models embed into an ultrapower of $A$, then by adding to these axioms the
(effectively enumerable) axioms for tracial $\mathrm{C}^{*}$-algebras, one
might hope that by running proofs from this new theory $T^{\prime}$, one might
be able to obtain effective upper bounds for
$\operatorname{Th}_{\forall}(\mathcal{R})$ and thus contradict
$\operatorname{MIP}^{*}=\operatorname{RE}$ as before. The issue with this is
that if $(M,\tau)\models T^{\prime}$, one is only guaranteed that $M$ embeds
into $A^{\mathcal{U}}$ as a $\mathrm{C}^{*}$-algebra, that is, the embedding
need not preserve the trace $\tau$ on $M$. Consequently, we would not know
that $\sup\\{\sigma^{(M,\tau)}\ :\ (M,\tau)\models T^{\prime}\\}$ coincides
with $\sigma^{(A,\tau_{A})}=\sigma^{(\mathcal{R},\tau_{R})}$ and thus our
usual Completeness Theorem argument need not go through. However, if $M$ were
itself monotracial, then the above embedding would be guaranteed to be trace-
preserving and the above argument would work. Unfortunately, being monotracial
is not an axiomatizable property of $\mathrm{C}^{*}$-algebras. That being
said, there is a an axiomatizable property of $\mathrm{C}^{*}$-algebras known
as the uniform Dixmier property which implies being monotracial. (To be fair,
the uniform Dixmier property itself is not axiomatizable. Instead, there are
quantitative versions known as the $(m,\gamma)$-uniform Dixmier property for
some parameters $m\in\mathbb{N}$ and $\gamma\in(0,1)$, which are each
axiomatizable; having the uniform Dixmier property means having the
$(m,\gamma)$-Dixmier property for some choice of parameters $m$ and $\gamma$.
See [1] for details.)

In summary, we have:

######  Theorem 5.16 (G. and Hart [20]).

Suppose that $A$ is a unital, infinite-dimensional, simple,
$\mathrm{C}^{*}$-algebra $A$ with the uniform Dixmier property whose
associated GNS von Neumann algebra $N$ embeds in an ultrapower of
$\mathcal{R}$. Then the $A$EP has a negative solution.

There are many examples of $\mathrm{C}^{*}$-algebras satisfying the hypotheses
appearing in the previous theorem. In particular, it follows from [25] and [1,
Corollary 3.11] that $\mathcal{Q}$ satisfies all of the above hypotheses,
leading to the aforementioned strengthed refutation of the MF problem:

######  Corollary 5.17.

The $\mathcal{Q}$EP has a negative solution.

We can use Theorem 5.16 to prove a new, purely operator algebra-theoretic
result, which refutes another natural ultarpower embedding problem.

One of the most important algebras in modern $\mathrm{C}^{*}$-algebra
classification theory is the Jiang-Su algebra $\mathcal{Z}$. (See Vignati’s
article in this volume for more information on $\mathcal{Z}$.) It follows from
the works in [25] and [1, Remark 3.18 and Corollary 3.22] that $\mathcal{Z}$
also satisfies hypotheses of Theorem 5.16, whence we have:

######  Corollary 5.18.

The $\mathcal{Z}$EP has a negative solution.

One of the defining features of $\mathcal{Z}$ is that it is stably
projectionless, meaning that for any $n\in\mathbb{N}$ and any projection $p\in
M_{n}(\mathcal{Z})$, there is a projection $q\in M_{n}(\mathbb{C})$ unitarily
conjugate to $p$. Being stably projectionless is axiomatizable by the
following (effective) list of axioms, one for each $n\in\mathbb{N}$:

$\sup_{p\in M_{n}(A)}\inf_{q\in M_{n}(\mathbb{C})}\inf_{u\in
U(M_{n}(A))}d(upu^{*},q)=0.$

A couple of words are in order about this axiomatization. First, it is know
that the matrix amplifications $M_{n}(A)$ belong to the imaginary sorts of the
theory of $\mathrm{C}^{*}$-algebras, whence the first and third quantifiers
are not problematic. Similarly, being (locally) compact, adding the matrix
algebras $M_{n}(\mathbb{C})$ to the theory of $\mathrm{C}^{*}$-algebras is
also harmless. Finally, the above axioms only seem to say that every
projection in $M_{n}(A)$ is approximately unitarily equivalent to a projection
in $M_{n}(\mathbb{C})$; however, it is well-known that two projections that
are sufficiently close are actually unitarily conjugate, whence the axioms do
indeed express that an algebra is stably projectionless.

Combining Corollary 5.18 with the discussion in the previous paragraph yields
the following stably projectionless analog of the negative solution to the MF
problem, which currently has no purely operator-algebraic proof:

######  Corollary 5.19 (G. and Hart [20]).

There is a stably projectionless $\mathrm{C}^{*}$-algebra that does not embed
into an ultrapower of $\mathcal{Z}$.

##  6\. Abelian $\mathrm{C}^{*}$-algebras

In this section, we move on from the more difficult embedding problems in the
previous sections and instead consider the case of abelian
$\mathrm{C}^{*}$-algebras. Not surprisingly, the ensuing discussion becomes
topological in nature.

As in the previous two sections, all $\mathrm{C}^{*}$-algebras in this section
are assumed to be unital.

###  6.1. Preliminaries on ultracoproducts of compact Hausdorff spaces

Some of the model theory of abelian $\mathrm{C}^{*}$-algebras to be discussed
below follows immediately from classical model-theoretic facts about Boolean
algebras together with a categorical understanding of the relevant
ultraproduct constructions. The ideas presented here are an elaboration of
those in [10, Section 5].

To begin, let $\operatorname{ZDComp}$ and $\operatorname{Bool}$ denote the
categories of zero-dimensional compact Hausdorff spaces and Boolean algebras
respectively. Consider the Stone functor
$\operatorname{ZDComp}\to\operatorname{Bool}$ given by sending the zero-
dimensional compact Hausdorff space $X$ to the Boolean algebra $Cl(X)$ of
clopen subsets of $X$. This functor is contravariant and is a duality of
categories whose inverse is given by the functor taking a Boolean algebra
$\mathbb{B}$ to its spectrum, that is, the set of ultrafilters on
$\mathbb{B}$, or, equivalently, the set of Boolean algebra homomorphisms
$\mathbb{B}\to\\{0,1\\}$.

Letting $\operatorname{Comp}$ and $\operatorname{AbC*}$ denote the categories
of compact Hausdorff spaces and unital abelian $\mathrm{C}^{*}$-algebras, then
we may also consider the Gelfand functor
$\operatorname{Comp}\to\operatorname{AbC*}$ given by sending the compact
Hausdorff space $X$ to the unital abelian $\mathrm{C}^{*}$-algebra $C(X)$ of
complex-valued continuous functions on $X$. Like the Stone functor, the
Gelfand functor is contravariant and is a duality of categories, this time the
inverse given by the functor taking a unital abelian $\mathrm{C}^{*}$-algebra
$A$ to its spectrum $\Sigma(A)$ consisting of all unital $*$-homomorphisms
$A\to\mathbb{C}$.

In a sense, the Gelfand functor is an “extension” of the Stone functor. More
precisely, recall first that a unital $\mathrm{C}^{*}$-algebra is called real
rank zero if the set of invertible self-adjoint elements is dense in the set
of self-adjoint elements. A unital abelian $\mathrm{C}^{*}$-algebra $C(X)$ is
real rank zero if and only if $X$ is zero-dimensional. Let
$\operatorname{RRZ}$ denotes the category of real-rank zero unital abelian
$\mathrm{C}^{*}$-algebras. Then the covariant functor
$\operatorname{RRZ}\to\operatorname{Bool}$ given by composing the inverse of
the Gelfand functor (restricted to $\operatorname{RRZ}$) with the Stone
functor is an equivalence of categories. In this case, for
$X\in\operatorname{ZDComp}$, $C(X)$ is the closed linear span of its space of
projections $P(X)$, which in turn is naturally isomorphic to $Cl(X)$. We refer
to this equivalence of categories as the “forgetful functor” as it forgets the
$\mathrm{C}^{*}$-algebra structure and only remembers the Boolean algebra
structure on the set of projections.

Next recall that one can present the $\mathrm{C}^{*}$-algebra ultraproduct
construction in purely categorical language. Indeed, suppose that
$(A_{i})_{i\in I}$ is a family of $\mathrm{C}^{*}$-algebras and $\mathcal{U}$
is an ultrafilter on $I$. For each $J\in\mathcal{U}$, let $A_{J}:=\prod_{j\in
J}A_{j}$ denote the direct product and note that the family $(A_{J},\pi_{JK})$
forms a directed family, where $\pi_{JK}:A_{K}\to A_{J}$ is the canonical
projection map when $J,K\in\mathcal{U}$ are such that $J\subseteq K$. There is
then a natural isomorphism $\prod_{\mathcal{U}}A_{i}\cong\varinjlim A_{J}$.
This fact is actually completely general and holds for ultraproducts of
$L$-structures for any (classical or continuous) language $L$. In particular,
the same observation holds verbatim for Boolean algebras and their
ultraproducts. (See [18, Chapter 6, Section 10] for more details.)

Now suppose that $(X_{i})_{i\in I}$ is a family of compact Hausdorff spaces
and $\mathcal{U}$ is an ultrafilter on $I$. Since
$\prod_{\mathcal{U}}C(X_{i})$ is once again a unital abelian
$\mathrm{C}^{*}$-algebra, it makes sense to consider the compact Hausdorff
space $\Sigma(\prod_{\mathcal{U}}C(X_{i}))$. One can give a purely topological
description of $\Sigma(\prod_{\mathcal{U}}C(X_{i}))$. Towards this end, for
$J\in\mathcal{U}$, set $X_{J}:=\coprod_{i\in J}X_{i}=\beta(\oplus_{i\in
J}X_{i})$, the Stone-Cech compactification of the direct sum of the $X_{i}$’s,
which is the coproduct construction for the category $\operatorname{Comp}$.
Since the Gelfand functor is a duality of categories, it follows that
$\prod_{i\in J}C(X_{i})\cong C(X_{J})$. Applying the inverse of the Gelfand
functor to the isomorphism $\prod_{\mathcal{U}}C(X_{i})\cong\varinjlim
C(X_{J})$ yields the isomorphism
$\Sigma(\prod_{\mathcal{U}}C(X_{i}))\cong\varprojlim X_{J}$. The compact
Hausdorff space $\varprojlim X_{J}$ is called the ultracoproduct of the family
$(X_{i})_{i\in I}$ with respect to $\mathcal{U}$, denoted
$\coprod_{\mathcal{U}}X_{i}$. If each $X_{i}=X$, we speak of the ultracopower
of $X$ with respect to $\mathcal{U}$, denoted $X_{\mathcal{U}}$.

Suppose now that each $X_{i}$ in the previous paragraph is also assumed to be
zero-dimensional. One can then apply the forgetful functor to the isomorphisms

$C\left(\coprod_{\mathcal{U}}X_{i}\right)\cong\prod_{\mathcal{U}}C(X_{i})\cong\varinjlim
C(X_{J})$

to get the isomorphisms

$Cl\left(\coprod_{\mathcal{U}}X_{i}\right)\cong\prod_{\mathcal{U}}Cl(X_{i})\cong\varinjlim
Cl(X_{J}).$

Arguing in a similar fashion, one sees that the forgetful functor sends the
diagonal embedding $C(X)\hookrightarrow C(X)^{\mathcal{U}}$ to the
corresponding diagonal embedding $Cl(X)\hookrightarrow Cl(X)^{\mathcal{U}}$.

###  6.2. The model companion of $T_{\operatorname{AbC*}}$

The discussion in the previous subsection, together with classical facts about
the model theory of Boolean algebras, will allow us to immediately deduce the
existence of an $\aleph_{0}$-categorical model companion for the theory
$T_{\operatorname{AbC*}}$ of unital abelian $\mathrm{C}^{*}$-algebras. We
first observe:

######  Proposition 6.1.

Suppose that $X$ and $Y$ are zero-dimensional compact Hausdorff spaces. Then
$C(X)\equiv C(Y)$ if and only if $Cl(X)\equiv Cl(Y)$.

The shortest proof of the previous proposition appeals to the Keisler-Shelah
theorem. Indeed, if $\mathcal{U}$ and $\mathcal{V}$ are ultrafilters, then by
the results discussed at the end of the previous subsection,
$C(X)^{\mathcal{U}}\cong C(Y)^{\mathcal{V}}$ if and only if
$X_{\mathcal{U}}\cong Y_{\mathcal{V}}$ (homeomorphic) if and only if
$Cl(X)^{\mathcal{U}}\cong Cl(Y)^{\mathcal{V}}$.

######  Proposition 6.2.

Suppose that $X$ is a compact Hausdorff space. Then $C(X)$ is an e.c. model of
$T_{\operatorname{AbC*}}$ if and only if (i) $X$ is zero-dimensional, and (ii)
$Cl(X)$ is an e.c. Boolean algebra.

For the forward direction, to prove item (i), we note that being real rank $0$
is $\forall\exists$-axiomatizable (see [12, Section 3.6.2]) and every model of
$T_{\operatorname{AbC*}}$ embeds in a real rank zero model of
$T_{\operatorname{AbC*}}$ (since every separable compact metric space is a
continuous image of Cantor space). Item (ii) follows from our analysis in the
previous subsection: if $Cl(X)\subseteq Cl(Y)$, then $C(X)\subseteq C(Y)$ and
thus there is an embedding $C(Y)\hookrightarrow C(X)^{\mathcal{U}}$ that
restricts to the diagonal embedding on $C(X)$. Applying the forgetful functor
shows that $Cl(X)$ is e.c. in $Cl(Y)$. The backwards direction is proven in a
similar manner.

We remind the reader that the theory of Boolean algebras has an
$\aleph_{0}$-categorical model completion, namely the theory of atomless
Boolean algebras. In particular, $Cl(2^{\mathbb{N}})$ is the unique countable
model of this model completion.

With everything in place, we can now conclude:

######  Theorem 6.3.

$\operatorname{Th}(C(2^{\mathbb{N}}))$ is $\aleph_{0}$-categorical and is the
model completion of $T_{\operatorname{AbC*}}$.

The $\aleph_{0}$-categoricity of $\operatorname{Th}(C(2^{\mathbb{N}}))$
follows from Proposition 6.1 above, the fact that being real-rank zero is
elementary, and the $\aleph_{0}$-categoricity of
$\operatorname{Th}(Cl(2^{\mathbb{N}}))$. Propositions 6.1 and 6.2, together
with the fact that $\operatorname{Th}(Cl(2^{\mathbb{N}}))$ is the model
companion of the theory of Boolean algebras, shows that
$\operatorname{Th}(C(2^{\mathbb{N}}))$ axiomatizes the e.c. models of
$T_{\operatorname{AbC*}}$. To see that the model companion is in fact a model
completion, we simply use the fact that $T_{\operatorname{AbC*}}$ has the
amalgamation property, which follows from the fiber product construction for
compact spaces: if $C(X)$ is embedded in $C(Y)$ and $C(Z)$, then there are
surjections $\pi_{Y}:Y\to X$ and $\pi_{Z}:Z\to X$ that induce these
embeddings. The corresponding fiber product is the space
$W=Y\times_{X}Z:=\\{(y,z)\in Y\times Z\ :\ \pi_{Y}(y)=\pi_{Z}(z)\\}$. The
natural projection mappings $\theta_{Y}:W\to Y$ and $\theta_{Z}:W\to Z$
satisfy $\pi_{Y}\circ\theta_{Y}=\pi_{Z}\circ\theta_{Z}$. It follows that the
embeddings of $C(Y)$ and $C(Z)$ into $C(W)$ corresponding to $\theta_{Y}$ and
$\theta_{Z}$ yield the desired amalgamation.

###  6.3. The projectionless case

While the model theory of the entire class of unital abelian
$\mathrm{C}^{*}$-algebras is fairly mundane (in the sense that Theorem 6.3
above is basically a classical result in disguise), the situation when one
considers the subclass of _projectionless_ unital abelian
$\mathrm{C}^{*}$-algebras is far more interesting. Note that $C(X)$ is
projectionless if and only if $X$ is a _connected_ compact Hausdorff space,
otherwise known as a continuum. The collection of projectionless unital
abelian $\mathrm{C}^{*}$-algebras does indeed form an elementary class, which
follows either from the observation that the ultracoproduct of a family of
continua is once again a continuum (see [24]) or from writing down concrete
axioms in the language of $\mathrm{C}^{*}$-algebras (see [11]). We let
$T_{cont}$ denote the theory of projectionless unital abelian
$\mathrm{C}^{*}$-algebras and note that this theory is universally
axiomatizable.

Unlike the case of arbitrary abelian $\mathrm{C}^{*}$-algebras, surprisingly
all (nondegenerate) projectionless abelian $\mathrm{C}^{*}$-algebras have the
same universal theory:

######  Theorem 6.4 (K.P. Hart [33]).

If $X$ and $Y$ are continua with $X$ nondegenerate (that is, $X$ is not a
point), then $C(X)$ embeds in an ultrapower of $C(Y)$. Consequently, all
nondegenerate models of $T_{cont}$ have the same universal theory.

We outline the proof here, including extra details not present in the
published version communicated to us directly by Hart; we thank him for his
permission to include this discussion here.

By Downward Löweinheim-Skolem, we may assume that $C(X)$ and $C(Y)$ are
separable, that is, that $X$ and $Y$ are metric continua. We fix countable
bases $\mathcal{B}$ and $\mathcal{C}$ for their lattices of closed sets and
enumerate $\mathcal{C}=(C_{n})_{n\in\omega}$. We note that
$\mathcal{B}^{\mathcal{U}}$ (the lattice ultrapower) is a base of closed sets
for the ultracopower $X_{\mathcal{U}}$ of $X$. Thus, by [9], in order to
construct a surjection $X_{\mathcal{U}}\to Y$ (and thus an embedding
$C(Y)\hookrightarrow C(X)^{\mathcal{U}}$), it suffices to find a map
$\phi:\mathcal{C}\to\mathcal{B}^{\mathcal{U}}$ satisfying:

  1. (1)

For all $F\in\mathcal{C}$, $\phi(F)=\emptyset$ if and only if $F=\emptyset$;

  2. (2)

For all $F,G\in\mathcal{C}$, if $F\cup G=Y$, then
$\phi(F)\cup\phi(G)=(X,X,X,\ldots)_{\mathcal{U}}$; and

  3. (3)

For all $F_{1},\ldots,F_{n}\in\mathcal{C}$, if
$\bigcap_{i=1}^{n}F_{i}=\emptyset$, then
$\bigcap_{i=1}^{n}\phi(F_{i})=\emptyset$.

Towards this end, we fix a surjection $f:X\to[0,1]$ (this is where the
nondegeneracy of $X$ is used) and identify $Y$ with a closed subspace of the
Hilbert cube $Q:=[0,1]^{\mathbb{N}}$. We set
$\kappa:\mathcal{P}(Y)\to\mathcal{P}(Q)$ to be the function

$\kappa(F):=\\{q\in Q\ :\ d(q,F)\leq d(q,Y\setminus F)\\}.$

In [34], it was shown that this map $\kappa$ has the following properties for
all closed $F,G\subseteq Y$:

  * (a)

$\kappa(F)\cap Y=F$;

  * (b)

$\kappa(F\cup G)=\kappa(F)\cup\kappa(G)$; and

  * (c)

$\kappa(Y)=Q$ and $\kappa(\emptyset)=\emptyset$.

Moreover, for any $F\subseteq Y$, $\kappa(F)$ is a closed subset of $Q$.

Set $E:=\\{e\subseteq\omega\ :\ \bigcap_{i\in e}C_{i}=\emptyset\\}$ and
$E_{n}:=E\cap\mathcal{P}(n)$. By (a), we have that $Y\cap\bigcap_{i\in
e}\kappa(C_{i})=\emptyset$ for all $e\in E$. For each $n\in\omega$, take
$\epsilon_{n}>0$ such that $\epsilon_{n}<\min\\{d(Y,\bigcap_{i\in
e}\kappa(C_{i}))\ :\ e\in E_{n}\\}$. Without loss of generality, we may assume
that $\lim_{n\to\infty}\epsilon_{n}=0$. Since $Y$ is compact, there is a
finite open cover $\mathbf{U}_{n}$ of $U$ by basic open sets contained in the
$\epsilon_{n}$-fattening $Y_{\epsilon_{n}}:=\\{z\in Q\ :\
d(z,Y)\leq\epsilon_{n}\\}$ of $Y$. Again, since $Y$ is compact, we may fix a
finite $\epsilon_{n}$-dense set $T_{n}\subseteq Y$ with the property that
$T_{n}\cap C_{i}\not=\emptyset$ for all $i\leq n$ for which
$C_{i}\not=\emptyset$. Since $Y$ is connected, any two points in
$\bigcup\mathbf{U}_{n}$ are connected by a piecewise linear path contained in
$\bigcup\mathbf{U}_{n}$. Consequently, we may define a continuous map
$g_{n}:[0,1]\to\bigcup\mathbf{U}_{n}\subseteq Q$ containing $T_{n}$ in its
range. In particular, this map satisfies:

  * (i)

$d(g_{n}(t),Y)<\epsilon_{n}$ for all $t\in[0,1]$.

  * (ii)

For each $y\in Y$, there is $t\in[0,1]$ such that
$d(g_{n}(t),y)<\epsilon_{n}$.

  * (iii)

$g_{n}^{-1}(C_{i})\not=\emptyset$ for all $i\leq n$ for which
$C_{i}\not=\emptyset$.

For $i<n<\omega$, set $D_{i}^{n}:=f^{-1}(g_{n}^{-1}(\kappa(C_{i})))$, a closed
subset of $X$. By items (b) and (c), whenever $C_{i}\cup C_{j}=Y$, we have
that $\kappa(C_{i})\cup\kappa(C_{j})=Q$, whence $D_{i}^{n}\cup D_{j}^{n}=X$.
Moreover, by (i), whenever $e\in E_{n}$, we have $g_{n}^{-1}(\bigcap_{i\in
e}\kappa(C_{i}))=\emptyset$, whence $\bigcap_{i\in e}D^{n}_{i}=\emptyset$.
Since $\mathcal{B}$ is a lattice base for $X$, there are elements
$B_{i}^{n}\in\mathcal{B}$ containing $D^{n}_{i}$ such that $\bigcap_{i\in
e}B_{i}^{n}=\emptyset$ for all $e\in E_{n}$.

We are finally ready to define the map
$\phi:\mathcal{C}\to\mathcal{B}^{\mathcal{U}}$ by setting
$\phi(C_{i}):=(B_{i}^{n})_{\mathcal{U}}$. We verify that $\phi$ has the
desired properties. First, if $C_{i}=\emptyset$, then
$\kappa(C_{i})=\emptyset$ by (c), whence $D_{i}^{n}=\emptyset$ and thus
$B_{i}^{n}=\emptyset$ (set $e=\\{i\\}$) for all $n>i$, whence
$\phi(C_{i})=\emptyset$. On the other hand, if $C_{i}\not=\emptyset$, then by
(iii), for all $n>i$, we have $g_{n}^{-1}(C_{i})\not=\emptyset$, whence
$D_{i}^{n}\not=\emptyset$ (since $f$ is surjective) and thus
$B_{i}^{n}\not=\emptyset$, as desired.

Since $D_{i}^{n}\subseteq B_{i}^{n}$ for all $i<n<\omega$, it follows from our
observations above that whenever $C_{i}\cup C_{j}=Y$, we have $B_{i}^{n}\cup
B_{j}^{n}=X$ for all $i<n<\omega$, whence
$\phi(C_{i})\cup\phi(C_{j})=(B_{i}^{n})_{\mathcal{U}}\cup(B_{j}^{n})_{\mathcal{U}}=(B_{i}^{n}\cup
B_{j}^{n})_{\mathcal{U}}=(X,X,X,\ldots)_{\mathcal{U}}$, establishing (b).

Finally, suppose that $e\in E_{n}$. Then $\bigcap_{i\in
e}B^{n}_{i}=\emptyset$, whence

$\bigcap_{i\in e}\phi(C_{i})=\bigcap_{i\in
e}(B_{i}^{n})_{\mathcal{U}}=\left(\bigcap_{i\in
e}B_{i}^{n}\right)_{\mathcal{U}}=\emptyset,$

establishing (c). This finishes the proof of Theorem 6.4.

In a series of papers (see, for example, [2]), Bankston studied the model
theory of continua in a fairly semantic way by dualizing most notions from
classical model theory, occasionally resorting to syntactic techniques by
working with lattices bases for the closed sets (which has the disadvantage of
not being canonical). In particular, Bankston introduced the notion of a co-
existentially closed (co-e.c.) continuum, which, when formulated in our
context, is simply a continuum $X$ for which $C(X)$ is an e.c. model of
$T_{cont}$. Bankston established many properties of co-e.c. continua,
including the fact that they are always hereditarily indecomposable (see [2,
Theorem 4.1]). A continuum is indecomposable if it cannot be written as the
proper union of two subcontinua and a continuum is hereditarily indecomposable
if all subcontinua are indecomposable. In [3], Bankston showed that the class
of hereditarily indecomposable continua is co-elementary, which, after
applying the Gelfand functor, implies that the class of models of $T_{cont}$
of the form $C(X)$ for $X$ a hereditarily indecomposable continuum is
elementary. Moreover, since the inverse limit of hereditarily indecomposable
continua is again hereditarily indecomposable, it follows that the
aforementioned class of models of $T_{cont}$ is inductive and thus
$\forall\exists$-axiomatizable.

One of Bankston’s main questions was whether or not the pseudo-arc
$\mathbb{P}$ is a co-e.c. closed continuum (see [2, Remark 4.2(i)]). Recall
that $\mathbb{P}$ is the unique metric continuum that is hereditarily
indecomposable and chainable, which is a condition that ensures that the
continuum is “arclike” in an appropriate sense. The pseudo-arc is “generic” in
the descriptive set-theoretic sense of the word [35] and thus it is also
natural to ask if it is generic from the model-theoretic perspective as well.

In joint work with Eagle and Vignati [11], we were able to answer Bankston’s
question affirmatively:

######  Theorem 6.5.

$C(\mathbb{P})$ is an e.c. model of $T_{cont}$.

Our main contribution was to show that chainability is a
$\forall\bigvee\exists$ property of models of $T_{cont}$ (or rather their
image under the inverse Gelfand functor). By Theorem 6.4, $C(\mathbb{P})$ is a
locally universal model of $T_{cont}$, whence one can enforce being chainable
by Proposition 2.1. Since being e.c. is also enforceable, it follows that
being e.c. and chainable is enforceable. Since e.c. implies hereditariliy
indecomposable, by the fact that $\mathbb{P}$ is the unique metrizable
hereditarily indecomposable, chainable continuum, we see that $C(\mathbb{P})$
is actually the enforceable model of $T_{cont}$.

One of the main open questions in the model theory of abelian
$\mathrm{C}^{*}$-algebras is the following:

######  Question 6.6.

Does $T_{cont}$ have a model companion?

As pointed out in [11], if this model companion exists, then it will _not_ be
a model completion for $T_{cont}$ does not have the amalgamation property. Of
course, if the model companion exists, then it must in fact be
$Th(C(\mathbb{P}))$.

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