# Quantifier elimination in C∗-algebras

Christopher J. Eagle Department of Mathematics and Statistics
University of Victoria
PO BOX 1700 STN CSC
V8W 2Y2
http://www.math.uvic.ca/ eaglec/
Ilijas Farah Department of Mathematics and Statistics
York University
4700 Keele Street
Toronto, Ontario
http://www.math.yorku.ca/ifarah
Eberhard Kirchberg Humboldt Universität zu Berlin
Institut für Mathematik
Unter den Linden 6
D-10099 Berlin, Germany
and  Alessandro Vignati Department of Mathematics and Statistics
York University
4700 Keele Street
Toronto, Ontario
http://www.automorph.net/avignati
January 31, 2021
###### Abstract.

The only unital -algebras that admit elimination of quantifiers in continuous logic in the language of unital -algebras are , Cantor space and . We also prove that the theory of -algebras does not have model companion and show that the theory of is not -axiomatizable for any .

###### Key words and phrases:
-algebras, quantifier elimination, model completeness, logic of metric structures
###### 2010 Mathematics Subject Classification:
46L05, 03C10, 46M07
IF was partially supported by NSERC. AV is supported by a Susan Mann Scholarship

## Introduction

One of the key steps in using model theory in applications is to understand the definable objects in models of a particular theory. It is often the case that the objects which can be defined without the use of quantifiers have particularly natural descriptions, while definitions involving quantifiers are more difficult to analyze. Quantifier elimination, which is the property that every definable object can be defined without using quantifiers, is therefore a highly desirable feature for a theory to possess.

Quantifier elimination is a matter of the formal language used to study the structures of interest. It is easy to see that any theory can be extended to a theory with quantifier elimination in an expanded language by simply adding a new symbol for every object definable in the original one. While such an expansion yields quantifier elimination, it does so without simplifying the task of determining which objects are definable. The usefulness of quantifier elimination results therefore depends on using a natural language for the structures at hand, so that it is possible to give a useful description of the objects that can be defined in quantifier-free way. For this reason we consider -algebras as structures in the language for -algebras introduced in [FHS14]. This standard language for -algebras contains symbols for the natural operations in a -algebra; when we consider unital algebras we often add a symbol for the multiplicative identity to form the language of unital -algebras. These languages are sufficiently expressive that many natural classes of -algebras are either axiomatizable, or at least defined by the omission of certain types (many examples of this kind are given in [FHL16, Theorem 2.5.1 and Theorem 5.7.3]). Nevertheless, these languages are also sufficiently limited that quantifier-free formulas are quite simple, being continuous combinations of norms of -polynomials with complex coefficients. We identify with and the Cantor space with the product space .

###### Theorem 1.

The theories of unital -algebras that admit quantifier elimination (in the language of unital -algebras) are exactly the complete theories of , , and . The theories of and admit quantifier elimination in the language of -algebras without a symbol for a unit, and no theory of a noncommutative -algebra admits quantifier elimination in this language.

###### Proof.

The unital case is Theorem 2.9. The claims about not necessarily unital algebras are established in Proposition 2.11 and Theorem 2.10. ∎

We also prove that the theory of -algebras does not have model companion (Theorem 3.3) and give natural examples of -algebras whose theories are not -axiomatizable (Corollary 3.6).

Section 1 contains preliminaries and tests for quantifier elimination. In this section we also completely answer the question of which finite-dimensional -algebras have quantifier elimination. In Section 2 we prove our main results, implying in particular that is the only noncommutative -algebra whose theory admits elimination of quantifiers. In Section 3 we show that the theory of unital -algebras does not have a model companion, and also obtain results related to the -axiomatizablity of some classes of -algebras.

The word embedding has a model theoretical sense: an embedding is a unital injective -homomorphism. By we denote an ultrapower of associated with an ultrafilter . All ultrafilters are assumed to be nonprincipal ultrafilters on .

### Acknowledgments

We are indebted to Isaac Goldbring for suggesting Theorem 3.3, and an exchange that lead to Theorem 3.4. We would also like to thank Bradd Hart for helpful remarks and to the anonymous referee for a very detailed and useful report.

Research presented in the Appendix was supported by the Fields undergraduate summer research program in July and August 2014. This research gave the initial impetus to study that resulted in the present paper.

## 1. Quantifier elimination

In this section we recall the model-theoretic framework for studying -algebras, as well as tests for quantifier elimination. The reader interested in a more complete discussion of the model theory of -algebras can consult [FHS14] or [FHL16]. For more on quantifier elimination in metric structures in general, see [BYBHU08, Section 13].

###### Definition 1.1.

The formulas for -algebras are recursively defined as follows. In each case, denotes a finite tuple of variables (which will later be interpreted as elements of a -algebra).

1. If is a -polynomial with complex coefficients, then is a formula.

2. If are formulas and is continuous, then is a formula.

3. If is a formula and , then and are formulas.

We think of and as replacements for the first-order quantifiers and , respectively. A formula constructed using only clauses (1) and (2) of the definition is therefore said to be quantifier-free.

The definition above is slightly different from the one in [FHS14]. In particular, we have replaced their domains of quantification by requiring that our suprema and infima range over closed -balls of finite radius, but the difference is clearly cosmetic.

If is a formula, is a -algebra, and is a tuple of elements of of the same length as the tuple , there is a natural way to evaluate in with replaced by ; the result is a real number denoted .

###### Definition 1.2.

Let be a -algebra, and be a tuple of elements from . The type of in , denoted , is defined to be the set of all formulas such that . Similarly, the quantifier-free type of , denoted , is the set of all quantifier-free formulas such that . If the algebra is clear from the context we omit it from the notation.

A formula without free variables is a sentence. A theory is a set of sentences, and a -algebra is a model of (written ) if every sentence in takes the value when interpreted in . The theory of , , is the set of all sentences which take value when interpreted in . If then we say that and are elementarily equivalent and write .

A formula is weakly stable if for every there exists such that for every -algebra and every , implies that the distance from to the zero-set of in is . In the language of logic of metric structures, the zero-sets of weakly stable formulas are precisely the definable sets (as defined in [BYBHU08, Definition 9.16]). See [CCF14, Lemma 2.1] and [FHL16, Lemma 3.2.4] for details. It is shown in [BYBHU08, Theorem 9.17] (see also [FHL16, Theorem 3.2.2] for the treatment of sets definable in a not necessarily complete theory) that every formula involving quantification over a definable set is equivalent to a standard formula. We will use without mention the fact that in a unital -algebra the sets of unitaries, self-adjoints, positive elements, and projections are definable (see [FHL16, Example 3.2.6]).

###### Definition 1.3.

A theory has quantifier elimination if for every formula and every there is a quantifier-free formula such that whenever and , being the length of , is a tuple of elements of norm less than or equal to , we have

 ∣∣φA(¯a)−ψAϵ(¯a)∣∣≤ϵ.

By a standard abuse of language, we say that a -algebra has quantifier elimination if does.

Determining whether or not a theory has quantifier elimination directly from the definition is often difficult. Fortunately, there are several tests for quantifier elimination that are more useful in practice. We will make use of two such tests. The first test, which we will use most often, is the following well-known strengthening of [BYBHU08, Proposition 13.6]. We include a proof for the convenience of the reader. The density character of a metric structure is the minimal cardinality of a dense subset of its underlying metric space, and in particular it is if and only if the metric structure is separable.

###### Proposition 1.4.

Let be a language of metric structures and let be an -theory. Then the following are equivalent:

1. has quantifier elimination;

2. if and are models of of density character then every embedding of a finitely generated substructure of into can be extended to an embedding of into an elementary extension of .

###### Proof.

In [BYBHU08, Proposition 13.6] (see also [HI02, pp. 84-91]) it was proved that the following version of (2) is equivalent to (1):

1. if and are models of of density then every embedding of a substructure of into can be extended to an embedding of into an elementary extension of .

Since (3) clearly implies (2), it only remains to prove that (2) implies (3).

Let , , , and an embedding be as in (3). We may assume that and is the identity map. Consider the expansion of obtained by adding a constant for every . Define an -theory to be the union of the elementary diagram of , , and the atomic diagram of , , (see [FHL16, 2.3(a)]). Then has a model if and only if there exists an elementary extension of and an embedding of into that extends ([FHL16, Theorem 2.3.4 and Theorem 2.3.5]). It therefore suffices to prove that is consistent. Fix a finite and let be the substructure of that contains all such that appears in . Let (, respectively), be an elementary submodel of (, respectively) of density character that contain all such that appears in . Then (2) implies that is consistent. Since was arbitrary, the compactness theorem implies that is consistent and (3) holds. ∎

If we assume that and are separable, we note that the elementary extension of required for the statement (2) can be found inside a countably saturated model (see, for example, the proof of [HI02, Proposition 13.17]). Specializing to -algebras we can therefore state a more appealing weakening of the assertion (2) of Proposition 1.4:

1. [label=()]

2. whenever is a finitely-generated -algebra, and are embeddings then there is an embedding that makes the diagram commute.

If the language of interest has a specified symbol for the unit then all algebras in play are assumed to be unital, as well as the embeddings. To avoid redundancy of notation, we will refer to the unital or the nonunital version of 1. We emphasize that the ‘nonunital version of 1’ is applied to unital -algebras if we are considering the language without a specified symbol for the unit. In Section 2 we usually take 1 as an hypothesis, meaning that the results proved hold for algebras satisfying the nonunital version of 1 and for unital algebras satisfying the unital version of 1.

The second quantifier elimination test we will use applies to a more restricted class of theories. Recall that a theory is -categorical if it has a unique separable model (up to isomorphism). The following result is well-known, but does not appear to have been explicitly stated in the literature, so we provide a brief proof. This test was used in [EV15, Theorem 5.26] to show that has quantifier elimination.

###### Proposition 1.5.

Let be an -categorical theory. The following are equivalent:

1. has quantifier elimination

2. the separable model of is near ultrahomogeneous, in the sense that if is separable and have the same quantifier-free type, then for each there is an automorphism of such that the distance between and is less than .

Since and have the same quantifier-free type if and only if the map extends to an isomorphism between the metric structures generated by and , (2) is equivalent to the following relative of Proposition 1.4 (2).

1. If is a finitely-generated substructure of and is a tuple generating it then for every and every embedding there exists an automorphism of such that the distance between and is less than .

###### Proof.

The direction (1) implies (2) is [BYBHU08, Corollary 12.11], together with the fact that in a theory with quantifier elimination two tuples with the same quantifier-free type have the same type.

For (2) implies (1), it suffices to show that if we assume (2) then in every model of any two tuples with the same quantifier-free type have the same type. So suppose and we have tuples from such that . By the continuous logic version of the Ryll-Nardzewski theorem ([BYBHU08, Theorem 12.10]) both and are isolated, and hence there are and from such that and (see [BYBHU08, Theorem 12.6]). In particular, , so it follows from (2) that , and hence . ∎

The two quantifier elimination tests above apply to any theory of metric structures. We now record some general consequences of quantifier elimination more specifically for -algebras. We will apply these results in the subsequent sections to show no noncommutative -algebra other than admits elimination of quantifiers. The first of these results, Lemma 1.6, is straightforward but very useful as it gives an analytic description of a quantifier-free type of a tuple of commuting normal elements. The joint spectrum of commuting normal elements , , is the set of all such that generates a proper ideal.

###### Lemma 1.6.

In any -algebra, for two finite tuples of commuting normal elements and the following conditions are equivalent:

1. and have the same quantifier-free type

2. the -algebras generated by and are isomorphic via an isomorphism that sends to .

Consequently, if a -algebra has quantifier elimination, then two finite tuples of commuting normal elements in have the same type if and only if they have the same joint spectrum.

###### Proof.

Let and be as in the hypothesis. Statements (1) and (3) are obviously equivalent. By [EV15, Proposition 5.25] the joint spectrum is quantifier-free definable from , and hence if then .

A character of a -algebra is a unital -homomorphism into . By the Gelfand–Naimark theorem every unital abelian -algebra is naturally isomorphic to where is the space of its characters with respect to the weak-topology. As the joint spectrum of is equal to the set of all where ranges over all characters of , characters of are in one-to-one correspondence with the elements of and . This proves that (2) implies (3). ∎

By the Weyl-von Neumann theorem (see e.g., [Dav96, Corollary II.4.2]) it is true that if and are self-adjoint elements of the Calkin algebra such that , then . This is not true, however, for normal elements. If is the unilateral shift in then its image under the quotient map is a unitary with full spectrum and (because of the Fredholm index obstruction) it satisfies for all unitaries . As pointed out in the introduction to [PW07], this failure of quantifier elimination is one of the reasons why it was difficult to construct an outer automorphism of the Calkin algebra.

### 1.1. Finite-dimensional C∗-algebras

To conclude this section we treat the case of finite-dimensional -algebras. We will need the fact if is a metric structure each of whose domains of quantification is compact, then the diagonal embedding of into its ultrapower is surjective. This is because every ultrafilter limit converges in a compact metric space. In particular, if is a finite-dimensional -algebra then the diagonal embedding of into its ultrapower is surjective, and so and are isomorphic. The Keisler-Shelah theorem (see [BYBHU08, Theorem 5.7]) asserts that two structure are elementarily equivalent if and only if they have isomorphic ultrapowers (for some ultrafilter on a sufficiently large—and possibly uncountable—index set). It follows from these facts, together with the fact that compactness of domains of quantification is preserved by elementary equivalence, that a finite-dimensional -algebra is the unique model of its theory. This is the continuous logic analogue of the well-known fact in discrete model theory that any finite structure is the unique model of its theory. For more on this, see [BYBHU08, §5], in particular the remark preceding Proposition 5.3.

We say that a projection is minimal if there is no proper subprojection and abelian if is abelian. We are interested in a strengthening of these two properties and we say that is scalar if . The set of scalar projections in a -algebra is definable. In fact a projection is scalar if and only if , where

 φ(p)=sup∥a∥≤1infλ∈C,|λ|≤1∥pap−λp∥.

Although the above expression quantifies over the complex unit disc, it is possible to interpret the expression as a formula in our formal language, so we may treat as a formula; see [FHL16, Remark 3.4.3] for details. It is not difficult to see that is -valued on projections, and therefore both the set of projections with and the set of projections with are definable, since the set of projections is definable (see [FHL16, Example 3.2.6]). In fact, if , we have that contains a -dimensional vector space. In this case there is an element with such that whenever .

In particular, if is scalar then is scalar also when seen in . Note that if  is finite-dimensional, scalar and minimal projections coincide.

###### Theorem 1.7.

For a finite-dimensional -algebra the following are equivalent.

1. Every commutative subalgebra of is isomorphic to or to .

2. is isomorphic to one of , or .

3. has quantifier elimination in the language of unital -algebras.

4. satisfies the unital version of 1.

###### Proof.

The equivalence of (1) if and only if (2) is an easy consequence of the fact that every finite-dimensional -algebra is isomorphic to a direct sum of full matrix algebras. Clauses (3) and (4) are equivalent for finite-dimensional algebras by Proposition 1.4 and since a finite-dimensional -algebra is the only model of its theory.

We prove that (3) implies (1). If (1) fails, then there are two projections and in which are both minimal, are orthogonal, and are such that . If has quantifier elimination then every nontrivial projection has the same type as , and in particular, is minimal. This contradicts the fact that is a nontrivial nonminimal projection.

We now prove that has quantifier elimination, using Proposition 1.4. Every ultrapower of is isomorphic to . If and are isomorphic unital subalgebras of , then by the equivalence of (2) and (1) and easy computation the isomorphism of and is implemented by a unitary in . Therefore the isomorphism extends to an automorphism of , and this completes the proof.

We omit the proofs that and have quantifier elimination, which are similar but easier (see also Lemma 1.6). ∎

###### Proposition 1.8.

For a finite-dimensional -algebra the following are equivalent.

1. is isomorphic to .

2. has quantifier elimination in the language of -algebras without a symbol for a unit.

3. satisfies the nonunital version of 1.

###### Proof.

(1) follows from the fact that the only finitely-generated substructure of is itself and that every embedding of into itself is necessarily unital. That (2) is equivalent to (3) is given by the fact that every finite-dimensional -algebra is the only model of its theory, therefore we are left to prove that (2) implies (1). For this, note that if is finite-dimensional and not isomorphic to then it has a nontrivial projection. It therefore admits both unital and nonunital embeddings of , violating Proposition 1.4 (2). ∎

If we expand the language of -algebras to include a trace, then every matrix algebra (considered with its canonical trace) has quantifier elimination. This follows from [BYBHU08, Proposition 13.6] and the fact that unital embeddings of matrix algebras are trace preserving. Making this change to the language does not affect questions of definability, since the trace is already definable in matrix algebras (see [FHL16, Lemma 3.5.3 and Theorem 3.5.5]).

## 2. Noncommutative C∗-algebras

Our goal is to prove that no noncommutative infinite-dimensional -algebra admits quantifier elimination. The proof proceeds by showing that if is noncommutative and infinite-dimensional and it admits quantifier elimination, then is purely infinite and simple (Proposition 2.6).

###### Lemma 2.1.

Assume is a -algebra with no scalar projections. Then it contains a positive contraction of full spectrum.

###### Proof.

Passing to a subalgebra, we may assume is separable. Let be a locally compact metric space such that is isomorphic to a masa of . By the continuous functional calculus we need to find whose range is a nontrivial interval. Since has no scalar projections, has no isolated points and is therefore uncountable.

Let us first consider the case when has an uncountable connected component . Choose a point and small enough to have and that is relatively compact. Define by if , if , and elsewhere. Then defined by is in and its range is equal to .

If there is no such then every connected component of consists of a single point and therefore is zero-dimensional. Being locally compact and with no isolated points, has a clopen subset homeomorphic to the Cantor set. Since the Cantor set maps continuously onto , we can find as required. ∎

###### Proposition 2.2.

If is noncommutative, infinite-dimensional, and satisfies 1 then does not have a scalar projection.

###### Proof.

We first observe that if satisfies the hypotheses of the theorem and has one scalar projection, then in fact every projection in is scalar. Let be a scalar projection, and let be a projection that is not scalar. Consider the embedding given by , and the embedding given by . Since the set of scalar projections is definable, remains a scalar projection in , so . There can be no embedding of into as in the conclusion of 1, because such an embedding would embed into , which is impossible as

Now fix a scalar projection . Since is infinite-dimensional, so is (since is a multiplier of this is a subalgebra of even if is nonunital). If is a nonzero projection in then is a non-scalar projection in , contradicting the above.

It will therefore suffice to prove that has a nonzero projection. Suppose otherwise. Since being projectionless is axiomatizable (by the argument of [FHL16, 3.6(a)]), is projectionless. Clearly is isomorphic to .

By Lemma 2.1 there is a positive element with . Let . Then since and therefore is isomorphic to and in turn . Let send to and send to . Let the embedding extending whose existence is assured by 1. Then is a projection , a contradiction. ∎

###### Definition 2.3 ([Cun78]).

For positive elements and in a -algebra we write , and say that is Cuntz-subequivalent to , if there is a sequence such that

 limn∥znbz∗n−a∥=0.

We write if . Note that is an equivalence relation.

###### Lemma 2.4.

For every in every -algebra one has . Moreover, for every there exists with such that .

###### Proof.

For let be defined by if and if . Let . Clearly . Also we have the following computation, which takes place in the commutative algebra :

 z∗nxx∗zn=fn(x∗x)(x∗x)2fn(x∗x)=gn(x∗x),

where if and if . Since we have that , as required. ∎

For let us temporarily write if

1. and are positive,

2. for every there is such that and , and

3. for every there is such that and .

Note that and implies , and that implies is Cuntz-equivalent to (i.e. and ) for all . Also, for all , the relation is encoded in .

Note that, by Lemma 2.4, .

###### Lemma 2.5.

If is noncommutative, infinite-dimensional, and satisfies 1, then there exist such that and are orthogonal positive contractions with full spectrum. Moreover, for any two orthogonal positive contractions with full spectrum and in we have .

###### Proof.

Since is noncommutative, by [Bla06, II.6.4.14], there is such that and . Then and are orthogonal positive elements of norm . Since the spectra of and both contain , they are equal (for all , and may only differ at , see e.g., [Bla06, II.1.4.2]).

Let us prove that we may assume . If then by continuous functional calculus we can find a nonzero projection . Since, by Proposition 2.2, has no scalar projections, the algebra is infinite-dimensional and by Lemma 2.1 we can find positive such that . Let . Note that and , hence . Also, if we let then and hence . Therefore by replacing with and re-evaluating and we may assume .

If and are positive orthogonal elements with we have that , so ; let . Let be the inclusion map, let be the embedding that sends to and to , and let be the embedding whose existence is guaranteed by 1. Let witness that . Then in , as witnessed by and . Since the diagonal embedding of into is elementary, we have , and in particular

 (inf∥y∥≤n(∥c−ydy∗∥))AU<1/n and% (inf∥y∥≤n(∥d−ycy∗∥))AU<1/n

hence

 (inf∥y∥≤n(∥c−ydy∗∥))A<1/n and (inf∥y∥≤n(∥d−ycy∗∥))A<1/n.

A -algebra is purely infinite and simple if it has dimension greater than 1 and for every two nonzero positive elements and we have (see [Rør02, §4.1]).

###### Proposition 2.6.

Let be noncommutative and infinite-dimensional. If  satisfies 1 then is purely infinite and simple.

###### Proof.

Since being purely infinite and simple is elementary (see [FHL16, Theorem 2.5.1] or [GS15]) it suffices to prove that is purely infinite and simple. We will use the fact that for , we have if and only if (see [FHL16, Lemma 8.1.3]).

Before doing so we will need two preliminary claims.

###### Claim 2.6.1.

Suppose that are positive contractions with full spectrum, and . Then .

###### Proof.

Choose elements , and in such that and , each with full spectrum. Such elements can be found in and Lemma 2.5 implies that contains a copy of . Then again by Lemma 2.5 we have that . Let be witnessing that and be witnessing that . Since the spectra of and are both and we have , hence . Let be the inclusion map and be the embedding sending to and to . Then by 1 there is an embedding making the diagram commute. It is easy to see that , witness that and and witness that , so . ∎

###### Claim 2.6.2.

If is unital then every positive contraction with full spectrum satisfies .

###### Proof of Claim 2.6.2.

Choose positive contractions in each with full spectrum and such that , and let . Then and , hence by Claim 2.6.1. Also . In fact, we will only need to know that and . Since , by [Cun78, Proposition 1.1] we have that . But and . In particular, is a positive contraction with full spectrum such that . The same argument used in Claim 2.6.1 and in Lemma 2.5 shows that for every as required for the claim. ∎

Now given positive contractions we show that . By Lemma 2.1 for every there exists a positive contraction in the hereditary111Recall that a subalgebra is hereditary if implies for all positive elements of . subalgebra such that . By countable saturation of there is a positive contraction with full spectrum such that , hence . It therefore suffices to prove , and by replacing with we may assume .

If is unital then by Claim 2.6.2 we have and (since ) follows.

It remains to consider the case when is nonunital. We prove that there exists a positive contraction such that and . Since and is a positive contraction the type of a positive contraction such that is consistent and such contraction exists in . For the same reason we have that there is with . Let . This is an hereditary -algebra of , and since , then as required.

Since it suffices to prove . If this follows by Claim 2.6.1. Otherwise, since is nonunital by countable saturation we can find a positive nonzero such that . By Lemma 2.1 we may assume . Then and we can apply Claim 2.6.1 to and complete the proof. ∎

### 2.1. O2 and quantifier elimination

Any -algebra generated by isometries with orthogonal ranges with sum 1 is isomorphic to the Cuntz algebra ([Cun77]). Hence is the universal algebra defined by the relations and . This algebra plays a pivotal role in Elliott’s classification program (see [Rør02, Chapter 5]). Notably, has some properties implied by quantifier elimination; for example, every unital embedding of into itself, or into any other model of its theory, is elementary (see e.g., [GS15] or [FHRTar, Proposition 2.15]). Nevertheless, we show below that does not have quantifier elimination.

Our main goal in this section is to prove the following Theorem whose proof extends ideas used in the proof that stably finite exact -algebras are not necessarily embeddable into a stably finite nuclear -algebra (see the discussion preceding [Bro06, Corollary 4.2.3]).

###### Theorem 2.7.

If is a separable, infinite-dimensional, noncommutative -algebra then does not satisfy 1. In particular, it does not have quantifier elimination.

###### Lemma 2.8.

Let be an infinite-dimensional noncommutative -algebra that satisfies 1. Then embeds in .

###### Proof.

We write if and are Murray–von Neumann equivalent projections and note that in if and only if in . Using 1 we see that if and are commuting pairs of projections with and , then . By Proposition 2.6 we know that is purely infinite and simple. By [Rør02, Proposition 4.1.1 (iii)] every projection in is properly infinite, meaning that there are partial isometries and satisfying but and are orthogonal and such that . Therefore contains orthogonal Murray–von Neumann equivalent projections and and by the above . By transitivity , and if is such that and then and are generators of a unital copy of in . To prove the second assertion note that, if is unital can be chosen to be . ∎

In the proof of Theorem 2.7 we will make use of the reduced group -algebra of the free group on two generators , constructed from the left regular representation of on . For more information on the construction of the reduced -algebra of a group we refer to [Bla06, II.10.2.5]. The algebra is exact (see [Kir93, p. 453, 1., 1-3], or [BO08, Proposition 5.1.8]) and therefore embeds into (see [KP00]).

###### Proof of Theorem 2.7.

By Lemma 2.8, embeds into . If is unital, such an embedding can be chosen to be unital. Let be a nonprincipal ultrafilter on . Each embeds into , and hence also embeds into . We therefore have an embedding of the ultraproduct inside , denoted by . As before, if is unital, so is . Let . By [HT05, Theorem B], is MF, that is, embeds into (see, for example, [Bla06, V.4.3.6]), and we can fix a unital embedding . Let . is a -homomorphism from into .

We claim that cannot be extended to an embedding of into (and in particular it cannot be extended to an embedding of into ), as 1 would require. Otherwise, by the nuclearity of and the Choi–Effros lifting theorem ([CE76]) there exists a completely positive contraction such that , where is the quotient map.

Since each is an injective von Neumann algebra, by [Bla06, Proposition IV.2.1.4] there are coordinatewise conditional expectations . Let be the conditional expectation induced by the expectations . Then a completely positive contraction, hence

 θ∘ψ′:F→∏Mn(C)

is a completely positive contractive lifting for . We have therefore constructed an embedding of into with a completely positive contractive lifting. -algebras with this property are said to be quasidiagonal (see [Bro04]). However, by a result of Rosenberg ([BO08, Corollary 7.1.18]) quasidiagonality of implies amenability of the nonamenable group . This contradiction concludes the proof. ∎

###### Theorem 2.9.

The only -algebras with quantifier elimination in the language of unital -algebras are , , and .

###### Proof.

That the given list includes all finite-dimensional examples is Theorem 1.7, and that the list includes all noncommutative examples is Theorem 2.7. Every separable -algebra elementarily equivalent to is isomorphic to for a compact metrizable 0-dimensional space without isolated points, and therefore isomorphic to . In [EV15, Theorem 5.26], it was proved that (and therefore ) admits quantifier elimination. In the appendix (written with D.C. Amador, B. Hart, J. Kawach, and S. Kim), we show that if there is a commutative example not on our list then it is of the form for an indecomposable continuum . Finally, in [EGV16, Corollary 3.4] it was proved that no such commutative example exists. ∎

We now focus on the nonunital case. The noncommutative case follows almost directly from Theorem 2.7:

###### Theorem 2.10.

Every noncommutative -algebra fails the nonunital variant of 1. Therefore no noncommutative -algebra admits elimination of quantifiers in the language of -algebras without a symbol for a unit.

###### Proof.

Assume satisfies the nonunital version of 1. By Theorem 2.7 is finite-dimensional. By Theorem 1.7 we have , but has projections of ranks 1 and 2, and therefore clearly fails the nonunital version of 1. ∎

In case of nonunital abelian -algebras, we have the following.

###### Proposition 2.11.

The theories of and admit quantifier elimination in the language of -algebras without a symbol for a unit.

###### Proof.

Since has neither a proper subalgebra nor a nontrivial self-embedding, it has quantifier elimination by Proposition 1.4.

Since is generated by its projections and all countable atomless Boolean algebras are isomorphic, is (up to isomorphism) the only separable model of its theory. Let . Since for a locally compact Hausdorff space the unitization of is isomorphic to (where is the one-point compactification of ), every separable model of the theory of is isomorphic to .

The above shows that the theory of is -categorical, so by Proposition 1.5 the proof will be complete if we show that is near ultrahomogeneous. By Lemma 1.6 it suffices to show that if are finite tuples of contractions with , and , there is an automorphism of such that . This is an immediate consequence of zero-dimensionality of and homogeneity of the algebra of its clopen subsets, but we provide details for the reader’s convenience.

Fix with and . Since the joint spectrum is defined from the unitization, and have the same joint spectrum as elements of . Suppose for a moment that is finite. With denoting the cardinality of we can find nonzero projections , for , and for and such that for all . We arrange that  (the point removed from ) belongs to the clopen set corresponding to and therefore for all (because ). Since , there are nonzero projections , for , such that for all . By the choice of , belongs to the clopen set corresponding to . Any automorphism of sending the clopen set correspond to to the clopen set corresponding to for all is dual to an automorphism of that sends to . We need to ensure that sends to itself. To do so, choose so that, in addition to the above, it sends the ultrafilter of projections with in its range to itself. If is dual to then its restriction to is as required.

Now consider the case when is not necessarily finite. Fix and let be a finite -dense set. Since is zero-dimensional there exists satisfying within distance of . Similarly there exists satisfying within distance of . If is an automorphism of sending to , then is within of . This completes the proof. ∎

It seems likely that and are the only, up to isomorphism222By homogeneity of the Cantor space, we have that and are homeomorphic whenever , hence for every such we have ., -algebras that admit quantifier elimination in the language of -algebras without a symbol for a unit. By Theorem 2.10 any counterexample would have to be abelian. Some of the results of [EGV16] may be relevant.

## 3. Model completeness and model companions

A theory is said to be model complete if every embedding between models of the theory is elementary, in the sense of preserving the values of all formulas. It is easy to see that quantifier elimination implies model completeness, while the converse is false. For example, the theory of every finite-dimensional -algebra is model complete. Model completeness is a useful tool in applications of model theory to algebra; for example, the fact that the (discrete) theory of algebraically closed fields is model complete is the key ingredient in a model-theoretic proof of Hilbert’s Nullstellensatz (see [Mar02, Theorem 3.2.11]).

We shall consider two weakenings of model completeness. A sentence is (or universal) if it is of the form

 sup¯xφ(¯x)

where is quantifer-free. Similaly, a sentence is if it is of the form

 sup¯xinf¯yφ(¯x,¯y)

where is quantifier-free. A theory is universally axiomatizable (respectively, -axiomatizable) if it has a set of universal () axioms.

If a theory is model complete then it is preserved by taking inductive limits of its models; indeed, since every embedding between models of such a theory is elementary, an inductive limit of models of forms an elementary chain, and each model in the chain is elementarily embedded in the limit model (see [BYBHU08, Proposition 7.2]), from which it immediately follows that the limit model is again a model of . By a standard preservation theorem the set of models of is closed under taking inductive limits if and only if
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