Logic and operator algebras Ilijas Farah York University Harvard Logic Colloquium, April 16, 2014
Complex Hilbert space ℓ 2 , C*-algebras = { a ∈ C N : � n | a n | 2 < ∞} . ℓ 2 = ( � | a n | 2 ) 1 / 2 . � a � ( B ( ℓ 2 ) , + , · , ∗ , � · � ): the algebra of bounded linear operators on ℓ 2 . Definition C*-algebra is a Banach algebra with involution which is *-isomorphic to a norm-closed self-adjoint subalgebra of B ( ℓ 2 ). Examples 1. B ( ℓ 2 ) . 2. M n ( C ) , for n ∈ N . 3. C ( X ) = { f : X → C | f is continuous } for any compact Hausdorff space X.
Inductive limits and the CAR algebra M n ( C ) ֒ → M 2 n ( C ) via � a � 0 a �→ . 0 a � M 2 ∞ ( C ) = lim → M 2 n ( C ) = M 2 ( C ) . − n ∈ N (where lim → means ‘completion of the direct limit.’) −
UHF (uniformly hyperfinite) algebras Lemma 1. M n ( C ) unitally embeds into M k ( C ) iff n divides k. 2. All unital emeddings of M n ( C ) into M k ( C ) are unitarily conjugate. 3. M n ( C ) ⊗ M k ( C ) ∼ = M nk ( C ) . Theorem (Glimm, 1960) UHF algebras � i M n ( i ) ( C ) and � i M m ( i ) ( C ) are isomorphic iff there is an ‘obvious’ isomorphism. In particular, M 2 ∞ �∼ = M 3 ∞ .
Elliott invariant The Elliott invariant, Ell, is a functor from the category of C*-algebras into a category of K-theoretic invariants. Lemma Let A be a UHF algebra, and let Γ := { m / n : m ∈ Z : M n ( 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 unique C*-algebra norm on A ⊗ B .
Elliott’s program Conjecture (Elliott, 1990) Infinite-dimensional, simple, nuclear, unital, separable algebras are classified by Ell . Classification is strongly functorial: Ell( A ) A ϕ f = Ell( ϕ ) Ell( B ) B For every morphism f : Ell( A ) → Ell( B ) there exist morphism ϕ : 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, separable algebra Z such that Ell( Z ) = Ell( C ) . Theorem (Toms, 2008) There are ∞ -dimensional simple, nuclear, unital, separable algebras A and B such that Ell( A ) = Ell( B ) , moreover F ( A ) = F ( B ) for every continuous homotopy-invariant functor F, but A �∼ = B. A ∼ = B ⊗ Z .
Abstract classification Almost every classical classification problem (not of ‘obviously set-theoretic nature’) in mathematics is concerned with definable equivalence relations on a Polish (separable, completely metrizable) space. If E and F are equivalence relations on Polish spaces, then E ≤ 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 is not classifiable by the isomorphism of countable structures.
Theorem (F.–Toms–T¨ ornquist, 2013) Isomorphism relation of simple, nuclear, unital, separable algebras is not Borel-reducible to the isomorphism relation of countable structures. Theorem (F.–Toms–T¨ ornquist, Gao–Kechris, Elliott–F.–Paulsen–Rosendal–Toms–T¨ ornquist, 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 group action. None of these relations is Borel-reducible to the isomorphism relation of countable structures.
Logic of metric structures Ben Yaacov–Berenstein–Henson–Usvyatsov, 2008. (Bounded) metric structure has a complete metric space ( M , d ) as its 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 : R 2 → [0 , ∞ ) ∧ , ∨ , ↔ ∀ , ∃ sup x , inf x . { ϕ | ϕ A = 0 } . Th( A ) Lemma Every formula has a uniform continuity modulus. Completeness, compactness, ultraproducts, � Los’s theorem, Lindstr¨ om-type theorems, EF-games,. . . everything works out as one would expect.
Theorem (Elliott–F.–Paulsen–Rosendal–Toms–T¨ ornquist, 2012) For any separable metric language L, the isomorphism of separable L-models is Borel-reducible to an orbit equivalence relation of a continuous action of a Polish group Iso( U ) on a Polish space. Classical logic = Logic of metric structures S ∞ Iso( U )
Logic of metric structures was adapted to operator algebras by F.–Hart–Sherman. Uniform continuity moduli of functions and predicates are attached to bounded balls, and quantification is allowed only over the bounded balls. In general, sorts over which one can quantify correspond to functors from the category of models into metric spaces with uniformly 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 �| + � x 2 � ) 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 algebras A r for r ∈ [0 , 1] such that Ell( A s ) = Ell( A r ) , but A r �∼ = A s if r � = 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 a morphism between the invariants. . . . A 1 A 2 A 3 A 4 A = lim n A n Φ 1 Ψ 1 Φ 2 Ψ 2 Φ 3 Ψ 3 Φ 4 . . . B 1 B 2 B 3 B 4 B = lim n B n Φ n , Ψ n are partial *-homomorphisms. The n -th triangle commutes up 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. Lemma Being Z -stable is ∀∃ -axiomatizable, for separable algebras. Therefore a positive answer to ‘Strong Conjecture’ implies a positive answer to the Revised Elliott’s Conjecture.
Omitting Types Definition Type p (¯ x ) is a set of conditions ϕ γ (¯ x ) = r γ , for γ ∈ I . a ) A = r γ for all γ . It is realized by ¯ a in A if ϕ 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 ≤ ℵ 0 ), . . . is characterized as the set of all algebras that omit a sequence of types. Given a complete theory T , one defines ‘Henkin forcing’ P T whose conditions are of the form ϕ ( ¯ d ) < ε , consistent with T . The generic model is denoted M G .
Topologies on the space of complete types in a complete theory T . Logic topology is defined as in the discrete case: basic open sets are conditions in P T . Metric topology = T ) t ( a ) A , s ( b ) A } . d ( t , s ) = inf { d ( a , b ) : ( ∃ A | A type is isolated if none of its metric open neighbourhoods is nowhere dense in the logic topology. Theorem (BYBHU, 2008) Given a separable language L, complete L-theory T , a complete type 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 Σ 1 2 set. (2) There is a complete theory T in a separable language such that { t : t is omissible in a model of T } is Π 1 1 hard.
Lemma (F–Magidor, 2014) The set of (ground-model) types forced by P T to be omitted in M G is Π 1 1 ( T ) . Theorem (F.–Magidor, 2014) There is a separable complete theory T and an omissible type t (¯ x ) which is realized in P T -generic model.
Uniform sequences of types A sequence of types t n (¯ x ) for n ∈ N is uniform if there are formulas ϕ j (¯ x ), with the same modulus of uniform continuity, such that t n (¯ x ) = { ϕ j (¯ x ) ≥ 1 / n : j ∈ N } , for all n . Equivalently, the interpretation of the L ω 1 ,ω formula inf j ϕ j (¯ x ) is a uniformly continuous function in every model of the theory. Theorem (F.–Magidor, 2014) A uniform sequence of types { t n } is omissible in a model of a complete theory T if and only if for every n type t n 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 ≤ ℵ 0 ), . . . is characterized as the set of all algebras that omit a uniform sequence of types. Corollary Sets of theories of UHF, AF, AT, AI, nuclear,. . . algebras are Borel. Proposition The ultraproduct � U M n ( C ) is not elementarily equivalent to a nuclear C*-algebra. C ∗ r ( F ∞ ) is not elementarily equivalent to a nuclear C*-algebra.
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