Operator algebras with hyperarithmetic theory

We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\Gamma)$ for $\Gamma$ a finitely generated group with solvable word problem, $C^*(\Gamma)$ for $\Gamma$ a finitely presented group, $C^*_\lambda(\Gamma)$ for $\Gamma$ a finitely generated group with solvable word problem, $C(2^\omega)$, and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ and the universal UHF algebra $\mathcal Q$ have hyperarithmetic theories provided certain embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. C$^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. C$^*$-algebras).


I
It is a typical question in logic to ask whether or not a given mathematical structure is decidable, that is, whether or not there is an algorithm that, upon input a first-order sentence in the language of that structure, can decide whether or not the sentence is true or false. For example, theorems of Tarski establish that the real and complex fields are decidable, while the famous Gödel Incompleteness Theorem states that the ring of integers is undecidable.
When a structure is undecidable or one cannot determine its decidability, it is also common to instead prove results that say that the structure is no more complicated than another structure (in the sense of Turing reduction; see Section 2 below). It is the purpose of this paper to prove results along these lines for various familiar operator algebras. Temporarily, let us say that an operator algebra is hyperarithmetic if it is no more complicated than the ring of integers. (Technically speaking, this is a fairly low level hyperarithmetic set and the precise definitions will be given in Section 2.) In this paper, we prove that the following operator algebras are hyperarithmetic: (1) the hyperfinite II 1 factor R; (2) L(Γ ) for Γ a finitely generated group with solvable word problem; (3) C * (Γ ) for Γ a finitely presented group; (4) C * λ (Γ ) for Γ a finitely generated group with solvable word problem; (5) C(2 ω ); (6) C(P) (where P is the pseudoarc). I. Goldbring was partially supported by NSF CAREER grant DMS-1349399. 1 We also show that the following operator algebras are hyperarithmetic provided certain embedding problems have positive solutions: (7) the Cuntz algebra O 2 (provided every C * -algebra embeds into an ultrapower of O 2 ); (8) the universal UHF algebra Q (provided every stably finite C * -algebra embeds into an ultrapower of Q).
To ensure that the above results are nontrivial, we note that the main result of [3] implies that there are continuum many different theories of II 1 factors (resp. C * -algebras) whence there can be only countably many hyperarithmetic II 1 factors and C * -algebras.
Our results are proven using two main techniques. The first is to quote a result of Camrud, McNicholl, and the current author from [5], which states that if an algebra has a hyperarithmetic presentation, then it is hypearithmetic; this technique applies to examples (1)-(4) from the above list (where the results of [12] are used to show that the algebras have hyperarithmetic presentations).
Our other main technique is to prove the continuous analogue of a classical result (see [16,Chapter 7,Section 2]), which shows that if T is an arithemetic theory, then its finite forcing companion is hyperarithmetic. This technique is how we deduce the remaining examples.
It is an open problem in the model theory of operator algebras whether or not there are two distinct theories of existentially closed (e.c.) II 1 factors (resp. C * -algebras). Here, an e.c. operator algebra is the model-theoretic formulation for what it means for the operator algebra to be "algebraically closed" in the sense of field theory. In this paper, we prove the continuous analogue of a fact from classical logic due to Simpson ([16, Chapter 7, Section 4]), which, when specialized to the above context, states if there is an e.c. II 1 factor (resp. C * -algebra) that is not hyperarithmetic, then there are actually continuum many distinct theories of e.c. II 1 factors (resp. C *algebras). In the case of groups, one can show that the theory of the socalled infinitely generic groups interprets second-order arithmetic (so is not hyperarithmetic), leading to a quick proof that there are continuum many distinct theories of e.c. groups. We conjecture that the same fact should be true in the II 1 factor and C * -algebra contexts.
The appropriate logical framework for studying operator algebras is continuous logic and there have been a plethora of attempts to approach this logic with operator algebraists in mind (e.g. [10]). For this reason, and to keep the paper relatively short, we assume that the reader is somewhat familiar with continuous model theory.
However, we do not assume that the reader is familiar with the requisite computability theory. For that reason, in Section 2 we give a rapid account of the basic facts needed to follow the proofs in this paper. We also take this opportunity to treat Gödel numbering in continuous logic and to discuss some subtleties about relative computability of theories in continuous logic.
We would like to thank Bradd Hart, Timothy McNicholl, and Andreas Thom for helpful discussions around this work.
2. P 2.1. Computability theory. We begin this section with a brief discussion of the basic notions from computability theory; a good and accessible reference is [8] and a more advanced reference is [23].
Computability theory studies the question of what it means for a function f : N k → N to be "computable." Naïvely speaking, such a function f should be computable if there is an algorithm 1 such that, upon input (a 1 , . . . , a k ) ∈ N k , runs and eventually halts, outputting the result f(a 1 , . . . , a k ). There are many approaches to formalizing this heuristic (e.g. Turing-machine computable functions and recursive functions) and all known formalizations can be proven to yield the same class of functions. This latter fact gives credence to the Church-Turing thesis, which states that this aformentioned class of functions is indeed the class of functions that are computable in the naïve sense described above. In the rest of this paper, we will never argue about this class of functions using any formal definition but will only argue informally in terms of some kind of algorithm or computer; this is often referred to as arguing using the Church-Turing Thesis. We call a subset of N k computable if its characteristic function is computable.
We will also consider a relative form of computability in the following sense: given a function g : N l → N, a g-algorithm is an algorithm in the usual sense that is also allowed to make "queries" to an "oracle" which has access to the function g. Thus, for example, in the course of running the algorithm, the computer is allowed to query the oracle and ask what is the value of g (15) and then use that information in the computation. A function f as above is then said to be g-computable (or that g computes f) if it is computable using a g-algorithm. Of course, if g were itself computable, then f would be computable as well. We say that a subset of N k is g-computable if its characteristic function is g-computable.
If f : N k → N and g : N l → N are functions, we say that f Turing reduces to g, denoted f ≤ T g, if f is g-computable. This is a preorder on the set of functions from finite cartesian powers of N to N. We say that f and g as above are Turing equivalent, denoted f ≡ T g, if f ≤ T g and g ≤ T f. This is indeed an equivalence relation whose equivalence classes are called Turing degrees. We typically denote Turing degrees by d. We often abuse notation and refer to a d-algorithm when we mean a g-algorithm for some g ∈ d. (The choice of representative of d is often irrelevant). Similarly, we speak of d-computable functions and sets. The preorder ≤ T induces a partial order, also denoted ≤ T , on the set of Turing degrees. We write The Turing degree consisting of precisely the computable functions is denoted by 0. It is clear that 0 is the minimum element of the poset of Turing degrees.
Given a Turing degree d, there is a unique Turing degree d ′ , called the jump of d, such that d < T d ′ and whenever d < T e, then d ′ ≤ T e. (d ′ is the Turing degree of the halting problem relativized to d. ) We can iterate the jump operation a finite number of times and we let d (n) denote the n th iterated jump of d. We will shortly discuss how to iterate the jump an infinite number of times as well.
It will become convenient for us to use an equivalent formulation of these concepts in terms of definability in first-order arithmetic. We consider the first-order language for studying the natural numbers that contains function symbols for addition and multiplication, a relation symbol for the ordering, and constant symbols for 0 and 1. One defines a hierarchy of formulae, called the arithmetic hierarchy, first by defining ∆ 0 formulae to consist of only those formulae defined using "bounded" quantifiers (such as ∀x < m and ∃x < m). Supposing by recursion that one has defined Σ m and Π m formulae, one defines Σ m+1 formulae to be those of the form ∃xϕ, where ϕ is Π m , and Π m+1 formulae to be of the form ∀xψ, where ψ is Σ m . A set A ⊆ N is called Σ m or Π m if it can be defined by a formula of that type, and it is called ∆ m if it can be defined both by a Σ m formula and a Π m formula.
Since the ∆ 0 formulae are precisely those that define computable sets, given a degree d, if we change the definition of ∆ 0 formulae above to consider those formulae which define d-computable sets, and then continue the recursive definitions as before, one arrives at the relativized hierachy of formulae denoted Σ d n , Π d n , and ∆ d n . We will often use: A set is called arithmetical if it is Σ n for some n. Thus, a consequence of Post's theorem is that a set is arithmetical if and only if it is Turing reducible to 0 (n) for some n.
As referred to in the title of this paper, we need to work with so-called hyperarithmetic sets, which is a class of sets properly extending the class of arithmetic sets. We first present this class in the spirit of the previous discussion. This requires us to move pass first-order logic and into second-order logic, where we now study natural numbers using a logic which, besides the usual first-order expressive power, also allows one to quantify over subsets of natural numbers and which includes a relation symbol for membership between natural numbers and sets of natural numbers. We typically use lowercase letters for variables ranging over numbers and uppercase letters for variables ranging over sets of natural numbers.
An arithmetical formula is one whose only quantifiers are number quantifiers. A Σ 1 1 formula is one of the form ∃X 1 · · · ∃X n ϕ, where ϕ is an arithmetic formula. A subset of N (or P(N)) is called Σ 1 1 if it is defined by a Σ 1 1 formula. Π 1 1 formulae are defined in same way using universal set quantifiers. A subset of N is called hyperarithmetic if it is both Σ 1 1 and Π 1 1 . There is a way to view hyperarithmetic sets in a manner analogous to Post's theorem. Without going into too many details, there is a countable ordinal ω CK 1 such that, for all ordinals γ < ω CK 1 , one can define the γ th jump of 0, denoted 0 (γ) . (See [22] for the details.) When γ ∈ ω, this jump operation agrees with the earlier notion denoting finitely many iterated jumps. One can then show that a subset of N is hyperarithmetic if and only if it Turing reduces to 0 (γ) for some γ < ω CK 1 . One can also relativize this procedure and speak of d (γ) for arbitrary Turing degrees d and ordinals γ < ω d 1 . Of particular interest to us is the "smallest" hyperarithmetic, non-arithmetic degree, namely 0 (ω) . One can show that the Turing degree of the set of Gödel numbers for true first-order sentences of arithmetic is precisely 0 (ω) .

2.2.
Gödel numbering in continuous logic. Throughout this paper, L denotes a continuous first-order language. As in the classical case, it makes sense to speak of whether or not L is computable. Roughly speaking, L is computable if there is an enumeration of the symbols for which one can computably tell what kind of symbol a given number represents, its arity, and, new to the continuous case, in case of function and predicate symbols, one can also uniformly computably calculate moduli of uniform continuity of the symbols. A precise definition can be found in [11]. Suffice it to say, the languages that occur in the operator algebra context are all computable. We thus make the convention that, in the rest of this paper, L denotes a computable continuous language.
The set of restricted L-formulae is the smallest collection of L-formulae containing the atomic formulae and closed under the unary connectives x → 0, x → 1, and x → x 2 , the binary connective (x, y) → x − . y := max(x − y, 0), and the quantifiers sup x and inf x . We let Form L and Sent L denote the set of restricted L-formulae and restricted L-sentences respectively.
Since L is computable, one can assign Gödel numbers to restricted Lformulae and we let ϕ denote the Gödel number of ϕ. We let Form L and Sent L denote the sets of Gödel numbers of restricted L-formulae and restricted L-sentences respectively. If k ∈ Form L , we let ϕ k denote the formula that it codes. Every restricted L-formula is equivalent to one in prenex normal form and we use the usual notation ∀ n and ∃ n here as in the classical case. Also as in the classical case, for each n, the set of Gödel numbers of ∀ n and ∃ n L rest -sentences are computable; we refer to these sets as ∀ n -Sent L and ∃ n -Sent L respectively. We let ϕ Sent L (x) be an arithmetical formula defining Sent L and similarly for ϕ ∀-Sent L (x), ϕ ∃-Sent L (x), ϕ qf L (x), the latter of which defines the codes of quantifier-free L-formulae. We also let Φ Sent L (X) := ∀p(p ∈ X ↔ ϕ Sent L (p)), another arithmetic formula. If Φ Sent L (X) holds, we set T X := {σ ∈ Sent L : σ ∈ X}.
The following lemma is obvious: There are recursive functions f, g : Occasionally we will need to consider a set C := {c 1 , c 2 , . . .} of new constant symbols and we set L(C) := L ∪ C. In this case, we can also assume that L(C) is computable and that one can computably tell whether or not an element of Form L(C) actually belongs to Form L or not.

Theories.
By an L-theory, we simply mean a subset of Sent L . If T is an L-theory, we set T : Given an L-structure M, we define the theory of M to be the function As it stands, the theory of M is not actually a theory in the above sense of the word. However, if one considers the zeroset of Th(M), that is, the set {k : ϕ M k = 0}, then one gets a theory in the above sense (after identifying k with ϕ k ). Under this identification, the complete theories are precisely those of the form Th(M) for some structure M.
This paper is all about relative computability of theories. Given the above two paragraphs, there is a subtlety that needs to be explained. First, a definition: algorithm such that, for any k ∈ Sent L and any rational ǫ > 0, returns an interval I with rational endpoints such that |I| < ǫ and ϕ M k ∈ I. This translation is not perfect as far as relative computability goes: First suppose that Th(M) is d-computable and consider j : N 2 → N which, upon input (k, n), if k = σ , uses d to calculate an interval I n with |I n | < 1 n and with σ M ∈ I n , and then returns 1 if 0 ∈ I n and otherwise returns 0. (If k is not the Gödel number of a sentence, set j(k, n) = 0.) Note that j is a d-computable function. Let h(k) be the least n such that j(k, n) = 0 when such n exists and be undefined otherwise.
(This uses a different reformulation of Σ d 1 sets, namely the domains of d-computable partial functions.) Conversely, suppose that Th(M) is Π d 1 . Fix σ ∈ Sent L and rational ǫ > 0. Using d, one can enumerate Sent L \ Th(M) . Whenever one sees σ − . r in this enumeration, one knows that σ M > r. Likewise, whenever one sees s − . σ , one knows that σ M < s. Thus, after some amount of time, one will arrive at such a pair (r, s) with s − r < ǫ and the interval (r, s) contains ϕ M .

Remark 2.5. The above proof can easily be modified to show that if
There is a proof system for continuous logic presented in [2]. There one defines the relation T ⊢ σ for T an L-theory and σ a restricted L-sentence.
We will also need the following form of the Completeness Theorem for continuous logic; here D denotes the dyadic rational numbers. Fact 2.6 (Pavelka-Style Completeness [2]). For any σ ∈ Sent L , one has The backwards direction is immediate. For the forwards direction, if T is consistent, then by the Completeness Theorem, T is satisfiable (that is, has a model), whence sup{(1 − . sup x d(x, x)) M : M | = T } = 1 and thus T ⊢ (1 − . sup x d(x, x)) − . 1 2 . We set Φ Theorem (X, p) := Φ Sent (X) ∧ ϕ Sent L (p) ∧ T X ⊢ ϕ p , an arithmetic formula. We also set Con(X) to be the statement Thus, Con(X) is an arithmetical formula and, by the previous lemma, Con( T ) is true if and only if T is consistent.

Presentations of operator algebras.
In this subsection, we present the basic definitions from computable structure theory for metric structures [11] (building upon the work in [4], [6], [19], [20], [21]) in the context of operator algebras. Throughout this subsection, M denotes either a separable C * -algebra or a separable tracial von Neumann algebra whose unit ball is denoted by M 1 . Given x, y ∈ M 1 , a rounded combination of x and y is an element of the form λx + µy, where λ, µ ∈ C satisfy |λ| + |µ| ≤ 1. The rounded combination will be called rational if λ and µ belong to Q(i).  The following theorem appears in [5]:

3.2.
Theories of group operator algebras. We now apply the previous corollary to group operator algebras. Fix a finitely generated group Γ , say generated by the finite set X. We let F X denote the free group on the alphabet X and we fix an effective enumeration (g n ) of the group algebra Q(i)F X and let π : Q(i)F X → Q(i)Γ denote the canonical surjection.
We first consider the case of the universal group C * -algebra C * (Γ ). Following [12], we let · u denote the operator norm on B(H u ), where Γ ֒→ U(H u ) is the universal representation of Γ . Let B := {g n : π(g n ) u ≤ 1} and let this be enumerated as g n k . Since the operators π(g n k ) are dense in the unit ball of C * (Γ )), we have a presentation of C * (Γ ) which we refer to as the standard presentation. (This technically depends on the choice of finite generating set and the effective enumeration, but that will not affect what is to follow.) We will need the following, which is [12, Corollary 2.2] 5 : Fact 3.6. Suppose that Γ is finitely presented. Then there is an algorithm which, upon input n, computes a decreasing sequence of rational numbers that converges to π(g n ) u .

Theorem 3.7. Suppose that Γ is a finitely presented group. Then the standard presentation of
Proof. Since the sequence (g n k ) is closed under all the symbols, it suffices to show that one can compute π(g n k ) u uniformly in k from 0 ′ . Given k, by the previous fact, one can use 0 ′ to determine n k and then one can use the previous fact and 0 ′ again to compute π(g n k ) u .
We now turn to the reduced group C * -algebra C * λ (Γ ). This time, we let · λ denote the operator norm on B(ℓ 2 (Γ )), where Γ ֒→ U(ℓ 2 (Γ )) is the leftregular representation. We now set B := {g n : π(g n ) λ ≤ 1} and once again let this be enumerated as g n k . Since the operators π(g n k ) are dense in the unit ball of C * λ (Γ ) (), we have a presentation of C * λ (Γ ) which we refer to as the standard presentation.
We let τ denote the canonical trace on CΓ . We will need the following fact [12, Lemma 3.1]: Fact 3.8. For any a ∈ CΓ , we have a λ = sup{τ((a * a) n ) 1/2n : n ∈ N}. Theorem 3.9. Suppose that Γ is a finitely generated group with word problem solvable from the Turing degree d. Then the standard presentation of C * λ (Γ ) is d ′computable. Consequently, Th(C * λ (Γ )) ≤ T d (ω) . In particular, if Γ has solvable (or even arithmetic) word problem, then Th(C * λ (Γ )) ≤ T 0 (ω) . Proof. By the assumption on the solvability of the word problem, the function n → τ(π(g n )) : N → Q can be calculated using d. Consequently, using the previous fact, we once again see that n k can be computed uniformly in k from d ′ and that π(g n k ) λ can be computed from d ′ .
We finally mention the corresponding fact for L(Γ ). The only wrinkle here is that we need to use · λ to determine that elements belong to the operator norm unit ball but then use τ to compute the 2-norm · 2,τ . Nevertheless, the above arguments show: Theorem 3.10. Suppose that Γ is a finitely generated group with word problem solvable from the Turing degree d. Then the standard presentation 6 (ω) . In particular, if Γ has solvable (even arithmetic) word problem, then Th(L(Γ )) ≤ T 0 (ω) . Remark 3.11. When Γ is an amenable, ICC group (whence L(Γ ) ∼ = R) with solvable word problem, the previous theorem implies that R has a 0 ′ -computable presentation. However, in [14], we claimed that R has a computable presentation. Since no proof was offered there, we say a few words here.
Indeed, one can effectively enumerate n M n (Q(i)) as (A n ). Since, given a matrix A, one has A 2 = A * A and the latter equals the largest eigenvalue of A * A, using familiar algorithms from numerical analysis, one can compute, given n, a sequence p m,n of rational upper bounds for A n . By letting B m,n := 1 pm,n A n , one obtains an effectively enumerated dense sequence from the unit ball of R. It is then routine to effectively calculate the 2-norm · 2 of B m,n using the normalized trace on matrix algebras.
It is not clear to us whether some fundamental C * -algebras such as the Cuntz algebra O 2 or the universal UHF algebra Q have computable (or even hyperarithmetic) presentations. Thus, in order to deduce that these algebras have hyperarithmetic theories, we need a new technique, which is the content of the next section.

Model-theoretic forcing: a recap.
In this section, we summarize the relevant background on building models by games in the continuous setting from [13].
We fix an L-theory T . A pre-condition is a finite set p of expressions of the form ϕ < r, where ϕ ∈ Sent L(C) is quantifier-free and r is a positive rational number. A pre-condition p is a condition (for T ) if T ∪ p is satisfiable.
We consider a two-person game involving the players ∀ and ∃. Players ∀ and ∃ take turns playing conditions extending the previous players move. Thus, ∀ starts by playing the condition p 0 , whence ∃ follows up by playing the condition p 1 ⊇ p 0 , and then ∀ follows that play with some condition p 2 ⊇ p 1 , etc... After ω many steps, the two players have together played a chain p 0 ⊆ p 1 ⊆ p 2 ⊆ · · · of conditions whose union we will denote byp.
We call the above play definitive if, for every atomic L(C)-sentence ϕ, there is a unique r ∈ [0, 1] such that T ∪p | = ϕ = r. In this case,p describes an L(C)-prestructure A + 0 (p) whose completion will be denoted by A + (p) and will be referred to as the compiled structure. The reduct of A + (p) to L will be denoted by A(p). Ifp is clear from context, we will denote A + (p) and A(p) simply by A + and A respectively.
Note that, regardless of player ∀'s moves, player ∃ can always ensure that the play of the game is definitive. Definition 4.1. Let P be a property of L(C)-structures. The game G(P) is the game whose moves are as above and such that Player ∃ wins G(P) if and only ifp is definitive and A + (p) has property P. We say that P is enforceable if Player ∃ has a winning strategy in G(P). We say that the condition p forces P, denoted p P, if, for any position (p 0 , . . . , p k ) of the game G(P), if p ⊆ p k , then the position is winning for ∃ in G(P). We write P when ∅ p.   Proof. First suppose that F p (ϕ) ≤ r. Then for every n > 1, p ϕ < r + 1 n , whence, by the Conjunction Lemma, p ϕ ≤ r.
Conversely, suppose p ϕ ≤ r. Then for every s > r, we have p ϕ < s, so F p (ϕ) ≤ s.
We will require the following fact about the function F p , whose proof can be found in [

4.2.
Finitely generic theories are hyperarithmetic. Until further notice, T is an L-theory that is Σ d 1 for some Turing degree d. We let Pre-Cond T denote the set of numbers e ∈ N such that e codes a tuple (e 1 , . . . , e n ), where each e i codes a pair (k, r) with k the code of a quantifier-free restricted L(C)sentence and r ∈ D >0 . We let Cond T denote those e ∈ Pre-Cond T such that e codes a condition for T . If e ∈ Pre-Cond T , we let p e denote the precondition it codes.
Note that p $ is an L(C)-theory for which A | = T ∪ p if and only if there is θ ∈ p $ such that A | = T ∪ θ.
The following lemma is clear: There is a computable set C ⊆ N 2 such that (e, c) ∈ C if and only if e ∈ Pre-Cond T , c ∈ Sent L(C) , and ϕ c ∈ p $ . Proposition 4.8. Suppose that p is a condition for T and ψ(x) ∈ Sent L(C) is quantifier-free. Then Proof. First suppose that θ ∈ p $ is such that T ∪ {θ} ⊢ (sup x ψ(x)) − . r. Then by the Completeness Theorem, there is an L(C)-structure A such that A | = T ∪ {θ} and (sup x ψ(x)) A > r. Take constants c from C such that ψ(c) A > r. Then q := p ∪ {1 − . ψ(c) < 1 − r} is a condition and q ψ(c) > r. Consequently, p sup x ψ(x) ≤ r.
Conversely, suppose that T ∪ {θ} ⊢ (sup x ψ(x)) − . r for all θ ∈ p $ . Suppose that player I plays p. Then player II can play so that the compiled structure A is such that A | = T ∀ ∪ p. Take B | = T such that A ⊆ B and take θ ∈ p $ such that B | = θ. By assumption and the Completeness Theorem again, (sup x ψ(x)) B ≤ r, whence (sup x ψ(x)) A ≤ r. It follows that p (sup x ψ(x)) ≤ r, as desired. Theorem 4.9. For each n ≥ 0, the set {(p, k, r) : e ∈ Cond T , k ∈ ∀ 2n+1 -Sent L(C) , and F pe (ϕ k ) ≤ r} is Π d 2n+2 . Proof. First suppose that n = 0. By the previous lemma, F pe (ϕ k ) ≤ r if and only if: for every c ∈ N, either (e, c) / ∈ C or else Φ Theorem ( T ∪{c}, ϕ k − . r ). This condition is Π d 2 . Now suppose n ≥ 1 and ϕ = sup x inf y ψ(x, y), where ψ ∈ ∀ 2n−1 -Sent L(C) . Further suppose that p is a condition for T . Then the following are equivalent: • F p (ϕ) ≤ r.
• (∀c ∈ C)(∀q ⊇ p)(∀m ∈ N)(∃q ′ ⊇ q)(∃d ∈ C)F q ′ (ψ(c, d)) ≤ r + 1 m . By induction, the "matrix" in the last bullet is Π 2n in the parameters; since the Gödel code for ψ is computable from the Gödel code for ϕ, we see that the entire condition is Π d 2n+2 , as desired. Recall that T has the joint embedding property (JEP) if any two models of T can be jointly embedded into a third model of T . Before moving on, we need to recall:  Proof. Fix k ∈ ∀ 2n+1 -Sent L and ǫ > 0. Given positive r ∈ D, using d (2n+2) , one can determine whether or not F ∅ (ϕ k ) ≤ r, that is, whether or not ϕ k ≤ r. If the answer is "yes", then one concludes that ϕ T f k ≤ r. If the answer is "no", then since we know that ϕ k = ϕ T f k , then we have that ϕ T f k > r. Thus, one runs this algorithm over all possible rational r > 0 and waits until one sees that ϕ T f k belongs to an interval of radius ǫ. 4.3. Examples. We now give some applications of the previous corollary: Example 4.13. Let T be the theory of C * -algebras. Then T is Σ 1 and has the JEP. It follows that T f ≤ T 0 (ω) . If the so-called Kirchberg embedding problem (KEP) has a positive solution, that is, every C * -algebra embeds into an ultrapower of the Cuntz algebra O 2 , then it was shown in [13] Alternatively, if one lets T = Th ∀ (O 2 ), the universal theory of O 2 , then T has the JEP and (ω) . As in [14], one can show that KEP implies that Th ∀ (O 2 ) is computable, whence we arrive at the same conclusion as in the previous paragraph. Example 4.14. Let T be the theory of stably finite C * -algebras. Then T is Σ 1 7 and has the JEP. Consequently, if the so-called MF problem has a positive solution, that is, every stably finite C * -algebra embeds into an ultrapower of the universal UHF algebra Q, then it was also shown in [13] that T f = Th(Q) and thus Th(Q) ≤ T 0 (ω) . Alternatively, as in the previous example, if Th ∀ (Q) is Σ d n for some n, then Th(Q) ≤ T d (ω) .

Example 4.15.
If T is the theory of commutative C * -algebras, then T is Σ 1 and has the JEP. In this case, T f equals the model companion of T , namely Th(C(2 ω )), and thus we conclude Th(C(2 ω )) ≤ T 0 (ω) .
Since Th(C(2 ω )) is arguably the nicest theory of an infinite-dimensional operator algebra, it is not outlandish to ponder the following: Question 4.16. Is Th(C(2 ω )) computable? Example 4.17. If T is the theory of projectionless commutative C * -algebras, then T is Σ 1 8 and has the JEP. In [13], it was shown that T f = Th(C(P)), where P is the so-called pseudoarc. It follows that Th(C(P)) ≤ T 0 (ω) .
Although this paper is about operator algebras, we would be remiss if we did not mention the following examples: Example 4.18. Suppose that T is the theory of operator spaces (resp. operator systems). Then T is Σ 1 9 and has the JEP. In [13], it was shown that T f = Th(NG) (resp. T f = Th(GS)), where NG is the so-called noncommutative Gurarij space (resp. GS is the Gurarij operator system). Consequently, Th(NG), Th(GS) ≤ T 0 (ω) .

Musings on oracle computability of presentations of compiled structures.
In the previous subsection, we saw that having hyperarithmetic theory is an enforceable property. However, what about adding parameters?
First, we recall that a model A of T is enforceable if the property that the reduct of the compiled structure be isomorphic to A is an enforceable property.

Proposition 4.19.
Suppose that T has JEP. Then the following are equivalent: (1) There is an oracle d such that it is enforceable that the elementary diagram (with respect to some presentation) of the compiled structure is dcomputable.
(2) There is an oracle d such that it is enforceable that the compiled structure has a d-computable presentation. (3) There is an enforceable model of T .
Proof. The implications (1) implies (2) and (3) implies (1) are obvious. The implication (2) implies (3) follows from the so-called Dichotomy Theorem (see [13]) and the fact that only countably many models of T can have a d-computable presentation for any given oracle d.
Remark 4.20. At first glance, it seems that, given an oracle d, one can use a diagonalization argument to prevent the compiled structure from having a d-computable presentation. However, that strategy only prevents the "canonical" presentation of the compiled structure (that is, the one given in terms of the constants from C) from being d-computable.

Remark 4.21.
In the case of II 1 factors, C * -algebras, and stably finite C * -algebras, it is unknown if there is an enforceable model. However, in the case of II 1 factors, we conjecture that having an enforceable model is equivalent to the truth of the Connes Embedding Problem (CEP). (In [13], it was shown that CEP is equivalent to the hyperfinite II 1 factor itself being enforceable.) Recently, a purported proof of the failure of CEP has appeared [18]. If this proof is correct and the aforementioned conjecture holds, then this would show that, given any oracle d, it is not enforceable that the compiled II 1 factor have a d-computable presentation.

D
Fix an L-theory T . We recall the following definition: There is a lot to say about e.c. operator algebras (see [9] and [15] for example). In this section, we are concerned with the complexity of Th(M) for M an e.c. model of T . Our main result is the following, which is a continuous analog of a theorem of Simpson: Our approach follows [16,Chapter 7,Section 4]. For the rest of this section, we assume that T is arithmetic.
The following definitions are taken from [2]: We say that T ′ is: In the proof of the next lemma, we ask the reader to recall the definitions of the computable functions f and g from Lemma 2.2.
Since T is arithmetical, it follows that Ψ 4 (X) is arithmetical, and thus is an arithmetical formula. It is clear that this Φ 0 is as desired.
As in classical logic, if T ′ is a maximal consistent Henkin theory, then there is a canonical model M T ′ | = T ′ whose underlying universe is an appropriate quotient of the closed L-terms. We refer the reader to [2] for more details. If Φ 0 (S) holds, say S = T ′ for some maximal consistent Henkin theory T ′ , then we may write M S instead of M T ′ . Theorem 5.5. There is an arithmetic predicate Φ 1 (X) such that: (1) If Φ 1 (S) holds, then Φ 0 (S) holds and the L-reduct of M S is an e.c. model of T ∀ .
Since T is arithmetic, so is T ∀ , and thus so is Consist(X, p). Let Ψ 5 (X) be ∀p((Consist(X, p) ∧ ϕ ∃-Sent L(C) (p)) → p ∈ X). At present, it is not known whether or not there are even two distinct theories of e.c. C * -algebras or two distinct theories of e.c. tracial von Neumann algebras. Although we will not go into the definition here, there is a special kind of e.c. model called a finitely generic model (see [13] for the precise definition). If M is finitely generic, then T f = Th(M), whence is hyperarithmetic if T is hyperarithmetic. Clearly, if we can find an e.c. C *algebra (or tracial von Neumann algebra) with non-hypearithmetic theory, then its theory must differ from that of the finitely generic algebra. The previous corollary says that in fact one would obtain continuum many different theories of e.c. algebras.
In the case of groups, full second-order arithmetic Turing reduces to the theory of so-called infinitely generic groups (see, e.g. [17, Section 5.3, Exercise 14]), giving one way to prove that there are continuum many theories of e.c. groups.

Conjecture 5.8. Second-order arithmetic Turing reduces to both the theory of infinitely generic C * -algebras and the theory of infinitely generic tracial von Neumann algebras.
If one restricts to so-called embeddable tracial von Neumann algebras, that is, those that embed into an ultrapower of R, then T f = Th(R). If the previous conjecture held for this class as well, then this would show that Th(R) is not the theory of infinitely generic embeddable tracial von Neumann algebras, something that was erroneously claimed in [9].
Tangentially related to this discussion, we recall that first-order arithmetic Turing reduces to the theory of any e.c. group [17,Section 3.3,Exercise 8]. This leads us to: Conjecture 5.9. First-order arithmetic Turing reduces to the theory of any (embeddable) e.c. II 1 factor.
If the previous conjecture is true, then since R is an e.c. embeddable factor (see [9]), one would have that Th(R) is Turing-equivalent to first-order arithmetic, whence the upper bound from Section 3 would be sharp.
In [9, Question 2.5], it was asked whether or not Th(R) is ∀∃-axiomatizable. By Theorem 3.4 above, if Th(R) were ∀ n -axiomatizable, then Th(R) would be arithmetic. Consequently, if the previous conjecture were true, one would have that Th(R) does not admit any quantifier simplification.