A survey of the model theory of tracial von Neumann algebras Isaac - - PowerPoint PPT Presentation

A survey of the model theory of tracial von Neumann algebras

Isaac Goldbring

University of Illinois at Chicago

Harvard-MIT Logic Seminar November 11, 2013

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 1 / 34

Introduction to von Neumann algebras

1

Introduction to von Neumann algebras

2

Isomorphic ultrapowers

3

Connes Embedding Problem

4

Model companions

5

Computability theory

6

Future directions

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 2 / 34

Introduction to von Neumann algebras

Von Neumann algebras

Throughout, H is a complex Hilbert space and B(H) is the ∗-algebra of bounded operators on H. For X ⊆ B(H), we set X ′ := {T ∈ B(H) : TS = ST for all S ∈ X}. Observe that:

X ′ is a unital subalgebra of B(H) that is closed under ∗ if X is. X ⊆ X ′′.

Definition A von Neumann algebra is a ∗-subalgebra M of B(H) such that M = M′′. Equivalently, M is a ∗-subalgebra of B(H) that is closed in either the weak operator topology or the strong operator topology.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 3 / 34

Introduction to von Neumann algebras

Examples of vNas

Example B(H) is a von Neumann algebra. Example Suppose that (X, µ) is a finite measure space. Then L∞(X, µ) acts on the Hilbert space L2(X, µ) by left multiplication, yielding an embedding L∞(X, µ) ֒ → B(L2(X, µ)), the image of which is a von Neumann algebra. (Actually, all abelian von Neumann algebras are isomorphic to some L∞(X, µ), whence von Neumann algebra theory is sometimes dubbed “noncommutative measure theory.”)

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 4 / 34

Introduction to von Neumann algebras

Group von Neumann algebras

Example Suppose that G is a locally compact group and α : G → B(H) is a unitary group representation. Then the group von Neumann algebra of α is α(G)′′. (Understanding α(G)′′ is tantamount to understanding the invariant subspaces of α.) In the important special case that α : G → B(L2(G)) (where G is equipped with its haar measure) is given by left translations α(g)(f)(x) := f(g−1x), we call α(G)′′ the group von Neumann algebra of G and denote it by L(G).

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 5 / 34

Introduction to von Neumann algebras

Group von Neumann algebras

Example Suppose that G is a locally compact group and α : G → B(H) is a unitary group representation. Then the group von Neumann algebra of α is α(G)′′. (Understanding α(G)′′ is tantamount to understanding the invariant subspaces of α.) In the important special case that α : G → B(L2(G)) (where G is equipped with its haar measure) is given by left translations α(g)(f)(x) := f(g−1x), we call α(G)′′ the group von Neumann algebra of G and denote it by L(G).

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 5 / 34

Introduction to von Neumann algebras

R

Example Let M2 denote the set of 2 × 2 matrices with entries from C. We consider the canonical embeddings M2 ֒ → M2 ⊗ M2 ֒ → M2 ⊗ M2 ⊗ M2 ֒ → · · · and set M := ∞

n=1

• n M2.

The normalized traces on

n M2 form a cohesive family of traces,

yielding a trace tr : M → C. We can define an inner product on M by A, B := tr(B∗A). Set H to be the completion of M with respect to this inner product. M acts on H by left multiplication, whence we can view M as a ∗-subalgebra of B(H). We set R to be the von Neumann algebra generated by M. R is called the hyperfinite II1 factor.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 6 / 34

Introduction to von Neumann algebras

Tracial von Neumann algebras

Suppose that A is a von Neumann algebra. A tracial state (or just trace) on A is a linear functional τ : A → C satisfying: τ(1) = 1; τ(x∗x) ≥ 0 for all x ∈ A; τ(xy) = τ(yx) for all x, y ∈ A. A tracial von Neumann algebra is a pair (A, τ), where A is a von Neumann algebra and τ is a trace on A. In the case that τ is also faithful, meaning that τ(x∗x) = 0 ⇒ x = 0, the function x, yτ := τ(y∗x) is an inner product on A, yielding the so-called 2-norm · 2 on A. The associated metric is complete on any bounded subset of A. (A, τ) is called separable if the metric associated to the 2-norm is separable.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 7 / 34

Introduction to von Neumann algebras

II1 Factors

A von Neumann algebra A is said to be a factor if A ∩ A′ = C · 1. Fact If A is a von Neumann algebra, then A ∼ = ⊕

X Ax (a direct integral)

where each Ax is a factor. A factor is said to be of type II1 if it is infinite-dimensional and admits a trace. Fact A II1 factor admits a unique weakly continuous trace, which is automatically faithful.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 8 / 34

Introduction to von Neumann algebras

II1 Factors

A von Neumann algebra A is said to be a factor if A ∩ A′ = C · 1. Fact If A is a von Neumann algebra, then A ∼ = ⊕

X Ax (a direct integral)

where each Ax is a factor. A factor is said to be of type II1 if it is infinite-dimensional and admits a trace. Fact A II1 factor admits a unique weakly continuous trace, which is automatically faithful.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 8 / 34

Introduction to von Neumann algebras

Examples-revisited

B(H) is a factor. If dim(H) < ∞, then B(H) admits a trace, but is not a II1 factor. If dim(H) = ∞, then B(H) admits no trace. Thus, B(H) is never a II1 factor. L∞(X, µ) admits a trace f →

• X f dµ but is not a factor.

If G is a countable group that is ICC, namely all conjugacy classes (other than {1}) are infinite, then L(G) is a II1 factor; the trace is given by T → Tδe, δe. In particular, if n ≥ 2, then L(Fn) is a II1 factor. R is a II1 factor; the trace tr :

n

• n M2 → C extends uniquely to

the completion. Moreover, R embeds into any II1 factor.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 9 / 34

Introduction to von Neumann algebras

Continuous model theory

There is a natural language of continuous logic in which to discuss tracial von Neumann algebras. Theorem (Farah-Hart-Sherman) The class of tracial von Neumann algebras is universally axiomatizable. The class of II1 factors is ∀∃-axiomatizable.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 10 / 34

Isomorphic ultrapowers

1

Introduction to von Neumann algebras

2

Isomorphic ultrapowers

3

Connes Embedding Problem

4

Model companions

5

Computability theory

6

Future directions

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 11 / 34

Isomorphic ultrapowers

Ultrapowers of von Neumann algebras

Suppose that (A, τ) is a tracial von Neumann algebra and U is a nonprincipal ultrafilter on N. We set ℓ∞(A) := {(an) ∈ AN : an is bounded}. Unfortunately, if we quotient this out by the ideal {(an) ∈ AN : lim

U an = 0},

the resulting quotient is usually never a von Neumann algebra. Rather, we have to quotient out by the smaller ideal {(an) ∈ AN : lim

U an2 = 0},

yielding the tracial ultrapower AU of A. (This is the continuous logic ultrapower, so the result is once again a von Neumann algebra.)

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 12 / 34

Isomorphic ultrapowers

Property (Γ)

For a while, Murray and von Neumann could not figure out whether R and L(F2) were isomorphic or not. They finally figured out a property that distinguished them. Say that a II1 factor M has property (Γ) if, for any finite F ⊆ M and any ǫ > 0, there is a trace 0 unitary u such that ux − xu2 < ǫ for all x ∈ F. R has (Γ) (easy) while L(F2) does not (M-vN), so R ∼ = L(F2). Equivalently, M has property (Γ) if and only if M′ ∩ MU = C (M has nontrivial relative commutant). Note that (Γ) is axiomatizable by the sentences σn := sup

• x

infy

• yy∗ − 12 + |tr(y)| +
• [xi, y]2
• .

Therefore, R and L(F2) are not elementarily equivalent!

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 13 / 34

Isomorphic ultrapowers

Property (Γ)

For a while, Murray and von Neumann could not figure out whether R and L(F2) were isomorphic or not. They finally figured out a property that distinguished them. Say that a II1 factor M has property (Γ) if, for any finite F ⊆ M and any ǫ > 0, there is a trace 0 unitary u such that ux − xu2 < ǫ for all x ∈ F. R has (Γ) (easy) while L(F2) does not (M-vN), so R ∼ = L(F2). Equivalently, M has property (Γ) if and only if M′ ∩ MU = C (M has nontrivial relative commutant). Note that (Γ) is axiomatizable by the sentences σn := sup

• x

infy

• yy∗ − 12 + |tr(y)| +
• [xi, y]2
• .

Therefore, R and L(F2) are not elementarily equivalent!

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 13 / 34

Isomorphic ultrapowers

Property (Γ)

For a while, Murray and von Neumann could not figure out whether R and L(F2) were isomorphic or not. They finally figured out a property that distinguished them. Say that a II1 factor M has property (Γ) if, for any finite F ⊆ M and any ǫ > 0, there is a trace 0 unitary u such that ux − xu2 < ǫ for all x ∈ F. R has (Γ) (easy) while L(F2) does not (M-vN), so R ∼ = L(F2). Equivalently, M has property (Γ) if and only if M′ ∩ MU = C (M has nontrivial relative commutant). Note that (Γ) is axiomatizable by the sentences σn := sup

• x

infy

• yy∗ − 12 + |tr(y)| +
• [xi, y]2
• .

Therefore, R and L(F2) are not elementarily equivalent!

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 13 / 34

Isomorphic ultrapowers

Property (Γ)

For a while, Murray and von Neumann could not figure out whether R and L(F2) were isomorphic or not. They finally figured out a property that distinguished them. Say that a II1 factor M has property (Γ) if, for any finite F ⊆ M and any ǫ > 0, there is a trace 0 unitary u such that ux − xu2 < ǫ for all x ∈ F. R has (Γ) (easy) while L(F2) does not (M-vN), so R ∼ = L(F2). Equivalently, M has property (Γ) if and only if M′ ∩ MU = C (M has nontrivial relative commutant). Note that (Γ) is axiomatizable by the sentences σn := sup

• x

infy

• yy∗ − 12 + |tr(y)| +
• [xi, y]2
• .

Therefore, R and L(F2) are not elementarily equivalent!

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 13 / 34

Isomorphic ultrapowers

Property (Γ)

For a while, Murray and von Neumann could not figure out whether R and L(F2) were isomorphic or not. They finally figured out a property that distinguished them. Say that a II1 factor M has property (Γ) if, for any finite F ⊆ M and any ǫ > 0, there is a trace 0 unitary u such that ux − xu2 < ǫ for all x ∈ F. R has (Γ) (easy) while L(F2) does not (M-vN), so R ∼ = L(F2). Equivalently, M has property (Γ) if and only if M′ ∩ MU = C (M has nontrivial relative commutant). Note that (Γ) is axiomatizable by the sentences σn := sup

• x

infy

• yy∗ − 12 + |tr(y)| +
• [xi, y]2
• .

Therefore, R and L(F2) are not elementarily equivalent!

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 13 / 34

Isomorphic ultrapowers

Property (Γ)

For a while, Murray and von Neumann could not figure out whether R and L(F2) were isomorphic or not. They finally figured out a property that distinguished them. Say that a II1 factor M has property (Γ) if, for any finite F ⊆ M and any ǫ > 0, there is a trace 0 unitary u such that ux − xu2 < ǫ for all x ∈ F. R has (Γ) (easy) while L(F2) does not (M-vN), so R ∼ = L(F2). Equivalently, M has property (Γ) if and only if M′ ∩ MU = C (M has nontrivial relative commutant). Note that (Γ) is axiomatizable by the sentences σn := sup

• x

infy

• yy∗ − 12 + |tr(y)| +
• [xi, y]2
• .

Therefore, R and L(F2) are not elementarily equivalent!

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 13 / 34

Isomorphic ultrapowers

Property (Γ)

For a while, Murray and von Neumann could not figure out whether R and L(F2) were isomorphic or not. They finally figured out a property that distinguished them. Say that a II1 factor M has property (Γ) if, for any finite F ⊆ M and any ǫ > 0, there is a trace 0 unitary u such that ux − xu2 < ǫ for all x ∈ F. R has (Γ) (easy) while L(F2) does not (M-vN), so R ∼ = L(F2). Equivalently, M has property (Γ) if and only if M′ ∩ MU = C (M has nontrivial relative commutant). Note that (Γ) is axiomatizable by the sentences σn := sup

• x

infy

• yy∗ − 12 + |tr(y)| +
• [xi, y]2
• .

Therefore, R and L(F2) are not elementarily equivalent!

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 13 / 34

Isomorphic ultrapowers

McDuff factors

Theorem (McDuff) For any separable II1 factor M, M′ ∩ MU takes one of the following forms:

1 C; 2 an abelian von Neumann algebra = C; 3 a II1 factor.

We call a II1 factor McDuff if case (3) holds. This is equivalent to M ⊗ R ∼ = M. In case (2), M′ ∩ MU is independent of the choice of U. McDuff asked whether M′ ∩ MU is independent of the choice of U in case M is McDuff.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 14 / 34

Isomorphic ultrapowers

McDuff factors

Theorem (McDuff) For any separable II1 factor M, M′ ∩ MU takes one of the following forms:

1 C; 2 an abelian von Neumann algebra = C; 3 a II1 factor.

We call a II1 factor McDuff if case (3) holds. This is equivalent to M ⊗ R ∼ = M. In case (2), M′ ∩ MU is independent of the choice of U. McDuff asked whether M′ ∩ MU is independent of the choice of U in case M is McDuff.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 14 / 34

Isomorphic ultrapowers

McDuff factors

Theorem (McDuff) For any separable II1 factor M, M′ ∩ MU takes one of the following forms:

1 C; 2 an abelian von Neumann algebra = C; 3 a II1 factor.

We call a II1 factor McDuff if case (3) holds. This is equivalent to M ⊗ R ∼ = M. In case (2), M′ ∩ MU is independent of the choice of U. McDuff asked whether M′ ∩ MU is independent of the choice of U in case M is McDuff.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 14 / 34

Isomorphic ultrapowers

Assuming (CH)

If we assume the Continuum Hypothesis, then MU and MV are saturated models of the same theory, so they are isomorphic (even over M). Consequently, M′ ∩ MU ∼ = M′ ∩ MV.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 15 / 34

Isomorphic ultrapowers

Assuming ¬(CH)

Recall that a (continuous) theory T is said to be unstable if there is a formula ϕ(x; y) and a sequence (ai) from a model M of T such that ϕ(ai; aj) = 0 if i < j and ϕ(aj; ai) = 1 if i ≥ j. Using the fact that II1 factors embed arbitrarily large matrix algebras, one can prove that every II1 factor is unstable. Given an ultrafilter U on N, let κ(U) denote the coinitiality of NրN/U. Using the instability of M, one can “encode” κ(U) into MU, so MU ∼ = MV ⇒ κ(U) = κ(V). Now use the fact (due to Dow and Shelah independently) that ¬(CH) implies that there exist U and V such that κ(U) = ℵ1 and κ(V) = ℵ2.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 16 / 34

Isomorphic ultrapowers

Assuming ¬(CH)

Recall that a (continuous) theory T is said to be unstable if there is a formula ϕ(x; y) and a sequence (ai) from a model M of T such that ϕ(ai; aj) = 0 if i < j and ϕ(aj; ai) = 1 if i ≥ j. Using the fact that II1 factors embed arbitrarily large matrix algebras, one can prove that every II1 factor is unstable. Given an ultrafilter U on N, let κ(U) denote the coinitiality of NրN/U. Using the instability of M, one can “encode” κ(U) into MU, so MU ∼ = MV ⇒ κ(U) = κ(V). Now use the fact (due to Dow and Shelah independently) that ¬(CH) implies that there exist U and V such that κ(U) = ℵ1 and κ(V) = ℵ2.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 16 / 34

Isomorphic ultrapowers

Assuming ¬(CH)

Recall that a (continuous) theory T is said to be unstable if there is a formula ϕ(x; y) and a sequence (ai) from a model M of T such that ϕ(ai; aj) = 0 if i < j and ϕ(aj; ai) = 1 if i ≥ j. Using the fact that II1 factors embed arbitrarily large matrix algebras, one can prove that every II1 factor is unstable. Given an ultrafilter U on N, let κ(U) denote the coinitiality of NրN/U. Using the instability of M, one can “encode” κ(U) into MU, so MU ∼ = MV ⇒ κ(U) = κ(V). Now use the fact (due to Dow and Shelah independently) that ¬(CH) implies that there exist U and V such that κ(U) = ℵ1 and κ(V) = ℵ2.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 16 / 34

Isomorphic ultrapowers

Assuming ¬(CH)

Recall that a (continuous) theory T is said to be unstable if there is a formula ϕ(x; y) and a sequence (ai) from a model M of T such that ϕ(ai; aj) = 0 if i < j and ϕ(aj; ai) = 1 if i ≥ j. Using the fact that II1 factors embed arbitrarily large matrix algebras, one can prove that every II1 factor is unstable. Given an ultrafilter U on N, let κ(U) denote the coinitiality of NրN/U. Using the instability of M, one can “encode” κ(U) into MU, so MU ∼ = MV ⇒ κ(U) = κ(V). Now use the fact (due to Dow and Shelah independently) that ¬(CH) implies that there exist U and V such that κ(U) = ℵ1 and κ(V) = ℵ2.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 16 / 34

Isomorphic ultrapowers

Assuming ¬(CH)

Recall that a (continuous) theory T is said to be unstable if there is a formula ϕ(x; y) and a sequence (ai) from a model M of T such that ϕ(ai; aj) = 0 if i < j and ϕ(aj; ai) = 1 if i ≥ j. Using the fact that II1 factors embed arbitrarily large matrix algebras, one can prove that every II1 factor is unstable. Given an ultrafilter U on N, let κ(U) denote the coinitiality of NրN/U. Using the instability of M, one can “encode” κ(U) into MU, so MU ∼ = MV ⇒ κ(U) = κ(V). Now use the fact (due to Dow and Shelah independently) that ¬(CH) implies that there exist U and V such that κ(U) = ℵ1 and κ(V) = ℵ2.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 16 / 34

Isomorphic ultrapowers

Assuming ¬(CH)

Theorem (FHS) Assuming ¬(CH), for any II1 factor M there exist two nonisomorphic ultrapowers of M. By altering the definition of instability of a relative commutant, the same ideas can be used to prove that if M is McDuff, then M has nonisomorphic relative commutants M′ ∩ MU. Also, the same arguments show that there are nonisomorphic matrix ultraproducts

U Mn(C).

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 17 / 34

Connes Embedding Problem

1

Introduction to von Neumann algebras

2

Isomorphic ultrapowers

3

Connes Embedding Problem

4

Model companions

5

Computability theory

6

Future directions

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 18 / 34

Connes Embedding Problem

Connes’ Embedding Problem

In 1976, Connes proved that L(F2) is Rω-embeddable. He then remarked “Apparently such an embedding ought to exist for all II1 factors...” This remark is now known as the Connes Embedding Problem (CEP) and is the central question in II1 factor theory. It has zillions

• f equivalent reformulations.

For example, it is known that L(G) is Rω-embeddable if and only if G is hyperlinear. So settling the CEP for group von Neumann algebras would settle the question of whether or not all groups are hyperlinear (a serious question in group theory).

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 19 / 34

Connes Embedding Problem

CEP (continued)

Model theory 101: CEP is the statement: for any II1 factor M, Th∀(M) = Th∀(R). Call a separable II1 factor A locally universal if every separable II1 factor is Aω-embeddable. (So CEP asks whether or not R is locally universal.) Theorem (“Poor Man’s CEP”-FHS) There is a locally universal II1 factor. Proof. Amalgamate to your heart’s desire.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 20 / 34

Connes Embedding Problem

CEP (continued)

Model theory 101: CEP is the statement: for any II1 factor M, Th∀(M) = Th∀(R). Call a separable II1 factor A locally universal if every separable II1 factor is Aω-embeddable. (So CEP asks whether or not R is locally universal.) Theorem (“Poor Man’s CEP”-FHS) There is a locally universal II1 factor. Proof. Amalgamate to your heart’s desire.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 20 / 34

Connes Embedding Problem

CEP (continued)

Model theory 101: CEP is the statement: for any II1 factor M, Th∀(M) = Th∀(R). Call a separable II1 factor A locally universal if every separable II1 factor is Aω-embeddable. (So CEP asks whether or not R is locally universal.) Theorem (“Poor Man’s CEP”-FHS) There is a locally universal II1 factor. Proof. Amalgamate to your heart’s desire.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 20 / 34

Model companions

1

Introduction to von Neumann algebras

2

Isomorphic ultrapowers

3

Connes Embedding Problem

4

Model companions

5

Computability theory

6

Future directions

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 21 / 34

Model companions

Could R be rosy?

OK, so no theory of II1 factors is stable. Nor are they (model-theoretically) simple. (Folklore-Hart) Could they admit some “nice” notion of independence (and thus be rosy)? A natural candidate exists using conditional expectation in analogy with ordinary probability theory. When trying to verify the axioms for being an independence relation, we realized it would be useful to know if Th(R) had QE.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 22 / 34

Model companions

Bad News

Theorem (G., Hart, Sinclair) Th(R) does not have QE. The proof uses nontrivial results of Nate Brown concerning embeddings of algebras into Rω. The same proof actually shows that Th(S) does not have QE if S is locally universal and McDuff.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 23 / 34

Model companions

A reminder on model companions

Recall that a theory T is model complete if any embedding between models of T is elementary. If T ′ is a theory, then a model complete theory T is a model companion for T ′ if any model of T ′ embeds in a model of T and vic-versa (that is, if T ′

∀ = T∀). A theory can have at most one

model companion. If T ′ is universal, then T ′ has a model companion T if and only if the class of its existentially closed structures is elementary; in this case T is their theory. Relevant to this discussion: any ec vNa is a McDuff II1 factor. Why? Any vNa M embeds into a II1 factor: M ⊆ M ∗ L(Z). Any II1 factor N embeds into a McDuff II1 factor: N ⊆ N ⊗ R. Being a McDuff II1 factor is ∀∃-axiomatizable (FHS).

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 24 / 34

Model companions

A reminder on model companions

Recall that a theory T is model complete if any embedding between models of T is elementary. If T ′ is a theory, then a model complete theory T is a model companion for T ′ if any model of T ′ embeds in a model of T and vic-versa (that is, if T ′

∀ = T∀). A theory can have at most one

model companion. If T ′ is universal, then T ′ has a model companion T if and only if the class of its existentially closed structures is elementary; in this case T is their theory. Relevant to this discussion: any ec vNa is a McDuff II1 factor. Why? Any vNa M embeds into a II1 factor: M ⊆ M ∗ L(Z). Any II1 factor N embeds into a McDuff II1 factor: N ⊆ N ⊗ R. Being a McDuff II1 factor is ∀∃-axiomatizable (FHS).

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 24 / 34

Model companions

Really bad news

Theorem (G., Hart, Sinclair-2012) TvNa does not have a model companion. Theorem (Farah, G., Hart-2013) Th∀(R) does not have a model companion.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 25 / 34

Model companions

TvNa does not have a model companion

Suppose that that the theory of vNas did have a model companion, say T. Let S be a separable model of T. We already know that S is McDuff. It can be shown that S is also locally universal. It follows that T does not admit quantifier-elimination. But a basic model theoretic fact says that a model-companion of a theory with the amalgamation property has quantifier-elimination, a contradiction.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 26 / 34

Model companions

Th∀(R) does not have a model companion

The only possible model of Th∀(R) that could be model-complete (even ∀∃-axiomatizable) is R. (Draw crude diagram on board!) This already tells us that CEP implies that there is no model-complete theory of II1 factors. In fact, a weaker version of CEP , namely that Th∀(R) having the amalgamation property, tells us that Th(R) is not model complete. Recently, we were able to show (unconditionally) that Th(R) is not model complete. The proof uses an analysis of automorphisms of e.c. models of Th∀(R) together with a nontrivial theorem of Kenley Jung characterizing R as the only Rω-embeddable II1 factor for which all embeddings into Rω are unitarily conjugate. (We showed that any nonstandard model of Th(R) also had this property.)

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 27 / 34

Model companions

Th∀(R) does not have a model companion

The only possible model of Th∀(R) that could be model-complete (even ∀∃-axiomatizable) is R. (Draw crude diagram on board!) This already tells us that CEP implies that there is no model-complete theory of II1 factors. In fact, a weaker version of CEP , namely that Th∀(R) having the amalgamation property, tells us that Th(R) is not model complete. Recently, we were able to show (unconditionally) that Th(R) is not model complete. The proof uses an analysis of automorphisms of e.c. models of Th∀(R) together with a nontrivial theorem of Kenley Jung characterizing R as the only Rω-embeddable II1 factor for which all embeddings into Rω are unitarily conjugate. (We showed that any nonstandard model of Th(R) also had this property.)

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 27 / 34

Computability theory

1

Introduction to von Neumann algebras

2

Isomorphic ultrapowers

3

Connes Embedding Problem

4

Model companions

5

Computability theory

6

Future directions

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 28 / 34

Computability theory

A computability-theoretic consequence of CEP

Suppose that M is a vNa. We say that Th∀(M) is computable if there is an algorithm such that, upon inputs universal sentence σ and (dyadic rational) ǫ > 0, returns an interval I ⊆ R of length ≤ ǫ such that σM ∈ I. Theorem (G., Hart) CEP ⇒ Th∀(R) is computable. The proof uses a proof theory for continuous logic developed by Ben-Yaacov and Petersen.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 29 / 34

Computability theory

A computability-theoretic consequence of CEP

Suppose that M is a vNa. We say that Th∀(M) is computable if there is an algorithm such that, upon inputs universal sentence σ and (dyadic rational) ǫ > 0, returns an interval I ⊆ R of length ≤ ǫ such that σM ∈ I. Theorem (G., Hart) CEP ⇒ Th∀(R) is computable. The proof uses a proof theory for continuous logic developed by Ben-Yaacov and Petersen.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 29 / 34

Computability theory

A converse

Theorem (G., Hart) CEP is equivalent to: for every vNa M that contains R, we have Th∀(M) is computable. Proof. If CEP fails, then there is a universal sentence σ and a II1 factor M such that σR = 0 and σM > 0. For t ∈ [0, 1], let Mt := tR + (1 − t)M. The map t → σMt : [0, 1] → [0, 1] is continuous and has range bigger than a point, whence has size continuum. But there can only be countably many algorithms for computing universal theories.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 30 / 34

Computability theory

More computability-theoretic consequences of CEP

Suppose that (M, X) is a separable II1 factor with countable dense set X = (xn). Certainly, for any universal sentence σ = supx ϕ(x) and any ǫ > 0, there is n ∈ ω such that ϕM(xn) > σM − ǫ; call such n good for (M, X, σ, ǫ). Theorem (G., Hart) Assume CEP . Then there is a computable partial function f : N × N × D>0 ⇀ N such that, if m is the Gödel code for a universal sentence σ and k is the Gödel code for a recursively presented II1 factor (M, X), then there is n ≤ f(m, k, ǫ) good for (M, X, σ, ǫ). Is this evidence that CEP is false?

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 31 / 34

Future directions

1

Introduction to von Neumann algebras

2

Isomorphic ultrapowers

3

Connes Embedding Problem

4

Model companions

5

Computability theory

6

Future directions

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 32 / 34

Future directions

Things we’re working on now

Finding many theories of II1 factors (we know 3 right now) Other possible quantifier-simplifications (via augmenting the language) Complexity of axiomatizations The relation between ≡ and ⊗, ∗, ⋊α Other properties of ec models Theories of pairs of algebras Changing the category/language

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 33 / 34

Future directions

References

• N. Brown, Topological dynamical systems associated to II1

factors, Adv. Math. 227 (2011), 1665-1699.

• I. Farah, I. Goldbring, B. Hart, and D. Sherman, Existentially

closed II1 factors, preprint on arXiv.

• I. Farah, B. Hart, and D. Sherman, Model theory of operator

algebras: I, II, & III.

• I. Goldbring and B. Hart, A computability theoretic reformulation of

the Connes Embedding Problem, submitted.

• I. Goldbring, B. Hart, T. Sinclair, The theory of tracial von

Neumann algebras does not have a model companion, JSL Volume 78.

Isaac Goldbring (UIC) Model theory of tracial vNas Harvard-MIT Logic Seminar 34 / 34