Logic and operator algebras

Ilijas Farah

York University

Harvard Logic Colloquium, April 16, 2014

Complex Hilbert space `2, C*-algebras

`2 = {a ∈ CN :∑

n |an|2 <∞}.‖a‖ = (

∑|an|2)1/2.

(B(`2),+, ·,∗ , ‖ · ‖): the algebra of bounded linear operators on `2.

DefinitionC*-algebra is a Banach algebra with involution which is*-isomorphic to a norm-closed self-adjoint subalgebra of B(`2).

Examples

1. B(`2).

2. Mn(C), for n ∈ N.

3. C (X ) = {f : X → C|f is continuous} for any compactHausdorff space X .

Inductive limits and the CAR algebra

Mn(C) ↪→ M2n(C)

via

a 7→(a 00 a

).

M2∞(C) = lim−→M2n(C) =⊗n∈N

M2(C).

(where lim−→ means ‘completion of the direct limit.’)

UHF (uniformly hyperfinite) algebras

Lemma

1. Mn(C) unitally embeds into Mk(C) iff n divides k .

2. All unital emeddings of Mn(C) into Mk(C) are unitarilyconjugate.

3. Mn(C)⊗Mk(C) ∼= Mnk(C).

Theorem (Glimm, 1960)

UHF algebras⊗

i Mn(i)(C) and⊗

i Mm(i)(C) are isomorphic iffthere is an ‘obvious’ isomorphism. In particular,

M2∞ 6∼= M3∞ .

Elliott invariant

The Elliott invariant, Ell, is a functor from the category ofC*-algebras into a category of K-theoretic invariants.

LemmaLet A be a UHF algebra, and let

Γ := {m/n : m ∈ Z : Mn(C) embeds unitally into A}.

Then Ell(A) = (Γ, 1, Γ ∩Q+).

One of many definitions of nuclearity for C*-algebras

A C*-algebra is nuclear if for every C*-algebra B there is a uniqueC*-algebra norm on A⊗ B.

Elliott’s program

Conjecture (Elliott, 1990)

Infinite-dimensional, simple, nuclear, unital, separable algebras areclassified by Ell. Classification is strongly functorial:

A Ell(A)

B Ell(B)

ϕ f = Ell(ϕ)

For every morphism f : Ell(A)→ Ell(B) there exist morphismϕ : A→ B such that Ell(ϕ) = f .

Remarkably, this is true for a large class of C*-algebras

(Elliott, Rørdam, Kirchberg–Phillips, Elliott–Gong–Li, Winter,. . . )

Elliott’s program: Counterexamples

Theorem (Jiang–Su, 2000)

There exists an ∞-dimensional simple, nuclear, unital, separablealgebra Z such that Ell(Z) = Ell(C).

Theorem (Toms, 2008)

There are ∞-dimensional simple, nuclear, unital, separable algebrasA and B such that Ell(A) = Ell(B), moreover F (A) = F (B) forevery continuous homotopy-invariant functor F , but A 6∼= B.

A ∼= B ⊗Z.

Abstract classification

Almost every classical classification problem (not of ‘obviouslyset-theoretic nature’) in mathematics is concerned with definableequivalence relations on a Polish (separable, completely metrizable)space.If E and F are equivalence relations on Polish spaces, thenE ≤B F if there exists Borel-measurable f such that

x E y ⇔ f (x)F f (y).

Hjorth developed a tool for proving that an equivalence relation isnot classifiable by the isomorphism of countable structures.

Theorem (F.–Toms–Tornquist, 2013)

Isomorphism relation of simple, nuclear, unital, separable algebrasis not Borel-reducible to the isomorphism relation of countablestructures.

Theorem (F.–Toms–Tornquist, Gao–Kechris,Elliott–F.–Paulsen–Rosendal–Toms–Tornquist, Sabok)

The following isomorphism relations are Borel-equireducible.

1. Isomorphism relation of arbitrary separable C*-algebras.

2. Isomorphism relation of Elliott–classifiable simple, nuclear,unital, separable algebras.

3. Isomorphism relation of Elliott invariants.

4. The ≤B -maximal orbit equivalence relation of a Polish groupaction.

None of these relations is Borel-reducible to the isomorphismrelation of countable structures.

Logic of metric structures

Ben Yaacov–Berenstein–Henson–Usvyatsov, 2008.(Bounded) metric structure has a complete metric space (M, d) asits domain.All functions and predicates are uniformly continuous.Uniform continuity moduli are a part of the language.

classical logic logic of metric structures

>,⊥ [0,∞)∧,∨,↔ continuous f : R2 → [0,∞)∀,∃ supx , infx .

Th(A) {ϕ|ϕA = 0}.

LemmaEvery formula has a uniform continuity modulus.

Completeness, compactness, ultraproducts, Los’s theorem,Lindstrom-type theorems, EF-games,. . .everything works out as one would expect.

Theorem (Elliott–F.–Paulsen–Rosendal–Toms–Tornquist,2012)

For any separable metric language L, the isomorphism of separableL-models is Borel-reducible to an orbit equivalence relation of acontinuous action of a Polish group Iso(U) on a Polish space.

Classical logic

S∞=

Logic of metric structures

Iso(U)

Logic of metric structures was adapted to operator algebras byF.–Hart–Sherman.

Uniform continuity moduli of functions and predicates are attachedto bounded balls, and quantification is allowed only over thebounded balls.

In general, sorts over which one can quantify correspond tofunctors from the category of models into metric spaces withuniformly continuous functions that commute with ultraproducts.

Examples

1. (sup‖x‖≤1,‖y‖≤1 ‖xy − yx‖)A = 0 iff A is abelian.

2. (inf‖x‖≤1 |1− ‖x‖|+ ‖x2‖)A = 0 iff A is non-abelian.

3. Being nuclear is not axiomatizable.

4. Being simple is not axiomatizable.

Counterexamples to Elliott’s conjecture revisited

Theorem (Toms, 2009)

There are ∞-dimensional simple, nuclear, unital, separable algebrasAr for r ∈ [0, 1] such that Ell(As) = Ell(Ar ), but Ar 6∼= As if r 6= s.

Theorem (L. Robert)

No two of these algebras are elementarily equivalent.

Question (Strong Conjecture)

For simple, nuclear, unital, separable A and B, do Ell(A) = Ell(B)and Th(A) = Th(B) together imply A ∼= B?

Intertwining

Every known instance of Elliott’s conjecture is proved by lifting amorphism between the invariants.

A1 A2 A3 A4. . . A = limn An

B1 B2 B3 B4. . . B = limn Bn

Φ1 Φ2 Φ3 Φ4Ψ1 Ψ2 Ψ3

Φn, Ψn are partial *-homomorphisms. The n-th triangle commutesup to 2−n.

Then A ∼= B.

Jiang–Su algebra Z revisited

Revised Elliott’s Conjecture (Toms–Winter, 2007)

Infinite-dimensional, simple, nuclear, unital, separable, Z-stable(i.e., A⊗Z ∼= A) algebras are classified by Ell.

LemmaBeing Z-stable is ∀∃-axiomatizable, for separable algebras.

Therefore a positive answer to ‘Strong Conjecture’ implies apositive answer to the Revised Elliott’s Conjecture.

Omitting Types

DefinitionType p(x) is a set of conditions ϕγ(x) = rγ , for γ ∈ I .

It is realized by a in A if ϕγ(a)A = rγ for all γ.

Theorem (F.–Hart–Tikuisis–Robert–Lupini–Winter, 2014)

Each of the following classes of algebras: UHF, AF, AT, AI,nuclear, simple, nuclear dimension < n, decomposition rank < n(n ≤ ℵ0), . . . is characterized as the set of all algebras that omit asequence of types.

Given a complete theory T, one defines ‘Henkin forcing’ PT whoseconditions are of the form ϕ(d) < ε, consistent with T. Thegeneric model is denoted MG .

Topologies on the space of complete types in a completetheory T.

Logic topology

is defined as in the discrete case: basic open sets are conditionsin PT.

Metric topology

d(t, s) = inf{d(a, b) : (∃A |= T)t(a)A, s(b)A}.A type is isolated if none of its metric open neighbourhoods isnowhere dense in the logic topology.

Theorem (BYBHU, 2008)

Given a separable language L, complete L-theory T, a completetype p is omissible in a model of T if and only if it is not isolated.

Non-complete types over a complete theory

Lemma (Ben Yaacov, 2010)

There are types that are neither isolated nor omissible.

Theorem (F.–Magidor, 2014)

(1) There is a theory T in a separable language such that

{t : t is omissible in a model of T}

is a complete Σ12 set.

(2) There is a complete theory T in a separable language such that

{t : t is omissible in a model of T}

is Π11 hard.

Lemma (F–Magidor, 2014)

The set of (ground-model) types forced by PT to be omitted inMG is Π1

1(T).

Theorem (F.–Magidor, 2014)

There is a separable complete theory T and an omissible type t(x)which is realized in PT-generic model.

Uniform sequences of types

A sequence of types tn(x) for n ∈ N is uniform if there are formulasϕj(x), with the same modulus of uniform continuity, such that

tn(x) = {ϕj(x) ≥ 1/n : j ∈ N}, for all n.

Equivalently, the interpretation of the Lω1,ω formula inf j ϕj(x) is auniformly continuous function in every model of the theory.

Theorem (F.–Magidor, 2014)

A uniform sequence of types {tn} is omissible in a model of acomplete theory T if and only if for every n type tn is not isolated.

Theorem (F.–Hart–Tikuisis–Robert–Lupini–Winter, 2014)

Each of the following classes of algebras: UHF, AF, AT, AI,nuclear, simple, nuclear dimension < n, decomposition rank < n(n ≤ ℵ0), . . . is characterized as the set of all algebras that omit auniform sequence of types.

Corollary

Sets of theories of UHF, AF, AT, AI, nuclear,. . . algebras are Borel.

Proposition

The ultraproduct∏U Mn(C) is not elementarily equivalent to a

nuclear C*-algebra.C ∗r (F∞) is not elementarily equivalent to a nuclear C*-algebra.

Strongly self-absorbing (s.s.a.) C*-algebras

Definition (Toms–Winter, after McDuff/Connes)

A separable algebra A is s.s.a. if

1. A ∼= A⊗ A,

2. The inner automorphism group is dense in Aut(A).

LemmaIf A is s.s.a., then

1. A ∼=⊗ℵ0 A.

2. (Effros–Rosenberg, 1978) A is simple and nuclear.

3. A is a prime model of its theory.

4. If B ≡ A then every endomorphism f : A→ B is elementary.

All s.s.a. algebras

Proposition

If D and E are s.s.a. algebras then TFAE.

1. E ⊗ D ∼= E .

2. D is isomorphic to a subalgebra of E .

3. Th∃(D) ⊆ Th∃(E ).

O2: s∗s = t∗t = 1, ss∗ + tt∗ = 1

O∞⊗ UHF

O∞ UHF

?

Z

Relative commutants

If A is a C*-algebra, identify A with its diagonal image in AU , let

A′ ∩ AU = {b ∈ AU : (∀a ∈ A)ab = ba}.

Theorem (McDuff for II1 factors, Toms–Winter 2007)

If D is s.s.a. and A is separable then A⊗ D ∼= A iff D embeds intoA′ ∩ AU .

All ultrafilters are nonprincipal ultrafilters on NQuestion (McDuff 1970, Kirchberg, 2004)

Assume A is separable. Does A′ ∩ AU depend on U?

Theorem (Ge–Hadwin, F., F.–Hart–Sherman, F.–Shelah)

If CH fails and A is infinite-dimensional, then there are 22ℵ0

nonisomorphic ultrapowers of A and nonisomorphic relativecommutants of A.

Theorem (F.–Hart–Sherman, F.–Shelah 2011)

For a separable model A the following are equivalent.

1. Th(A) is not stable.

2. ¬CH implies A has nonisomorphic ultrapowers.

3. ¬CH implies A has 22ℵ0 nonisomorphic ultrapowers.

Corollary

For A as aboveCH ⇔ all ultrapowers of A are isomorphic.

Theorem (F.–Hart–Robert–Tikuisis, 2014)

Assume D is s.s.a.. Then D ′ ∩ DU ≺ DU .

Corollary

1. Every embedding of D into D ′ ∩ DU is elementary.

2. All embeddings of D into D ′ ∩ DU are unitarily conjugate.

3. CH implies D ′ ∩ DU ∼= DU .

4. CH implies⊗ℵ1 D embeds into DU so that its relative

commutant in DU is trivial.

Problems

1. Construct interesting C*-algebras by using omitting typestheorem.

2. Develop theory of Borel-reductions between ‘Polishcategories.’ (Some preliminary results by Lupini.)

3. Further develop model theory of II1 factors (only threedifferent theories of II1 factors are known!).

4. If T is a complete metric theory and types s and t areseparately omissible in models of T, are they jointly omissiblein a model of T?(F.–Magidor: There are a complete separable theory T andtypes tn for n ∈ N such that for every m ∈ N types tn, forn ≤ m, are simultaneously omissible but tn, for n ∈ N are notsimultaneously omissible.)