INTERPLAY BETWEEN C AND VON NEUMANN

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1 INTERPLAY BETWEEN C AND VON NEUMANN ALGEBRAS Stuart White University of Glasgow 26 March 2013, British Mathematical Colloquium, University of Sheffield.

2 C AND VON NEUMANN ALGEBRAS C -algebras Banach algebra A with an involution satisfying the C -identity }x x} }x} 2 for all x P A. von Neuman algebras Weak operator closed -subalgebra M Ď BpHq Weak operator topology: x i Ñ x ô xpx i xqξ, ηy Ñ 0 for all ξ, η P H Norm closed -subalgebra A Ď BpHq C -algebra M which is isometrically the dual space of some Banach space. Abelian C -algebras of form C 0 pxq for X locally compact Hff Abelian vnas of form L 8 px, µq for some measure space px, µq.

3 C AND VON NEUMANN ALGEBRAS C -algebras Banach algebra A with an involution satisfying the C -identity }x x} }x} 2 for all x P A. von Neuman algebras Weak operator closed -subalgebra M Ď BpHq Weak operator topology: x i Ñ x ô xpx i xqξ, ηy Ñ 0 for all ξ, η P H Norm closed -subalgebra A Ď BpHq C -algebra M which is isometrically the dual space of some Banach space. Abelian C -algebras of form C 0 pxq for X locally compact Hff Abelian vnas of form L 8 px, µq for some measure space px, µq.

4 C AND VON NEUMANN ALGEBRAS C -algebras Banach algebra A with an involution satisfying the C -identity }x x} }x} 2 for all x P A. von Neuman algebras Weak operator closed -subalgebra M Ď BpHq Weak operator topology: x i Ñ x ô xpx i xqξ, ηy Ñ 0 for all ξ, η P H Norm closed -subalgebra A Ď BpHq C -algebra M which is isometrically the dual space of some Banach space. Abelian C -algebras of form C 0 pxq for X locally compact Hff Abelian vnas of form L 8 px, µq for some measure space px, µq.

5 C AND VON NEUMANN ALGEBRAS C -algebras Banach algebra A with an involution satisfying the C -identity }x x} }x} 2 for all x P A. von Neuman algebras Weak operator closed -subalgebra M Ď BpHq Weak operator topology: x i Ñ x ô xpx i xqξ, ηy Ñ 0 for all ξ, η P H Norm closed -subalgebra A Ď BpHq C -algebra M which is isometrically the dual space of some Banach space. Abelian C -algebras of form C 0 pxq for X locally compact Hff Abelian vnas of form L 8 px, µq for some measure space px, µq.

6 C AND VON NEUMANN ALGEBRAS C -algebras Banach algebra A with an involution satisfying the C -identity }x x} }x} 2 for all x P A. von Neuman algebras Weak operator closed -subalgebra M Ď BpHq Weak operator topology: x i Ñ x ô xpx i xqξ, ηy Ñ 0 for all ξ, η P H Norm closed -subalgebra A Ď BpHq C -algebra M which is isometrically the dual space of some Banach space. Abelian C -algebras of form C 0 pxq for X locally compact Hff Abelian vnas of form L 8 px, µq for some measure space px, µq.

7 EXAMPLES M n, BpHq; C 0 pxq Ă BpL 2 pxqq, L 8 pxq Ă BpL 2 pxqq; G discrete group group C and von Neumann algebras; H l 2 pgq, basis tδ g : g P Gu. Define unitaries λ g : H Ñ H by λ g pδ h q δ gh ; C r pgq Spantλ g : g P Gu } ; LG Spantλ g : g P Gu wot. α : G ñ X Dynamical system CpXq r G, L 8 pxq G. CpXq r G is a C -algebra generated by CpXq and copy of G; Unitaries pλ g q gpg with λ gh λ g λ h, λ g f λ g f α 1 g for g, h P G, f P CpXq. How much information about G or G ñ X is captured by their operator algebras?

8 EXAMPLES M n, BpHq; C 0 pxq Ă BpL 2 pxqq, L 8 pxq Ă BpL 2 pxqq; G discrete group group C and von Neumann algebras; H l 2 pgq, basis tδ g : g P Gu. Define unitaries λ g : H Ñ H by λ g pδ h q δ gh ; C r pgq Spantλ g : g P Gu } ; LG Spantλ g : g P Gu wot. α : G ñ X Dynamical system CpXq r G, L 8 pxq G. CpXq r G is a C -algebra generated by CpXq and copy of G; Unitaries pλ g q gpg with λ gh λ g λ h, λ g f λ g f α 1 g for g, h P G, f P CpXq. How much information about G or G ñ X is captured by their operator algebras?

9 ANOTHER EXAMPLE Consider M 2 ãñ M 4 M 2 b M 2 ; x ÞÑ x b 1. This preserves the normalised trace on M 2 and M 4. Carry on and form M 2 ãñ M 2 b M 2 ãñ M b3 2 ãñ to obtain direct limit -algebra Ä 8 n 1 M 2. Perform GNS construction for canonical trace τ and take the weak operator closure obtain hyperfinite II 1 factor R. Could also close in the norm topology to get the CAR-algebra M 2 8. IMPORTANT CONSEQUENCE R R b R

10 ANOTHER EXAMPLE Consider M 2 ãñ M 4 M 2 b M 2 ; x ÞÑ x b 1. This preserves the normalised trace on M 2 and M 4. Carry on and form M 2 ãñ M 2 b M 2 ãñ M b3 2 ãñ to obtain direct limit -algebra Ä 8 n 1 M 2. Perform GNS construction for canonical trace τ and take the weak operator closure obtain hyperfinite II 1 factor R. Could also close in the norm topology to get the CAR-algebra M 2 8. IMPORTANT CONSEQUENCE R R b R

11 ANOTHER EXAMPLE Consider M 2 ãñ M 4 M 2 b M 2 ; x ÞÑ x b 1. This preserves the normalised trace on M 2 and M 4. Carry on and form M 2 ãñ M 2 b M 2 ãñ M b3 2 ãñ to obtain direct limit -algebra Ä 8 n 1 M 2. Perform GNS construction for canonical trace τ and take the weak operator closure obtain hyperfinite II 1 factor R. Could also close in the norm topology to get the CAR-algebra M 2 8. IMPORTANT CONSEQUENCE R R b R

12 II 1 FACTORS DEFINITION A factor is a von Neumann algebra M with trivial centre: Z pmq C1. Every vna can be written as a direct integral of factors. Murray and von Neumann classified factors into types. DEFINITION A factor M is type II 1 if: it is infinite dimensional; D tracial state τ : M Ñ C1, i.e. τpxyq y P M. R; LG for countable discrete G with thgh 1 : h P Gu 8 for g e; LpXq G for G ñ px, µq is free, ergodic and probability measure preserving.

13 HYPERFINITENESS DEFINITION (MURRAY AND VON NEUMANN) A vna M is hyperfinite if every finite subset of M can be arbitrarily approximated arbitrarily in the weak -topology by a finite dimensional subalgebra of M. Separably acting hyperfinite vnas arise are inductive limits of finite dimensional vnas. THEOREM (MURRAY AND VON NEUMANN) There is a unique (separably acting) hyperfinite II 1 factor.

14 UNIFORM PERTURBATIONS Given two operator algebras A, B Ă BpHq, write dpa, Bq for the Hausdorff distance between the unit balls of A and B in the operator norm. For a unitary u, dpa, uau q ď 2}u 1 H }. THEOREM (CHRISTENSEN, JOHNSON, RAEBURN-TAYLOR 77) Let M, N Ă BpHq be vnas with M hyperfinite. dpm, Nq ă 1{101 ùñ D unitary u P W pm Y Nq s.t. umu N IDEA and }u 1 H } ď 150dpM, Nq. Establish dimension independent perturbation results for near containments of finite dimensional algebras. ą 0, Dδ ą 0 such that if F is finite dim, F Ă δ N, then D unitary u P W pf, Nq with ufu Ď N and }u 1 H } ă ε.

15 UNIFORM PERTURBATIONS Given two operator algebras A, B Ă BpHq, write dpa, Bq for the Hausdorff distance between the unit balls of A and B in the operator norm. For a unitary u, dpa, uau q ď 2}u 1 H }. THEOREM (CHRISTENSEN, JOHNSON, RAEBURN-TAYLOR 77) Let M, N Ă BpHq be vnas with M hyperfinite. dpm, Nq ă 1{101 ùñ D unitary u P W pm Y Nq s.t. umu N IDEA and }u 1 H } ď 150dpM, Nq. Establish dimension independent perturbation results for near containments of finite dimensional algebras. ą 0, Dδ ą 0 such that if F is finite dim, F Ă δ N, then D unitary u P W pf, Nq with ufu Ď N and }u 1 H } ă ε.

16 AF ALGEBRAS DEFINITION C -algebra A is AF if every finite subset of A can be arbitrarily approximated in norm by finite dimensional subalgebras of A. Separable AF algebras are inductive limits of finite dimensional algebras. eg M 2 8. Changing matrix sizes matters: M 3 8 fl M 2 8. CLASSIFICATION ELLIOTT Separable AF-algebras are classified by K 0. THEOREM (CHRISTENSEN 80) Let A, B Ă BpHq be C -algebras, with A separable and AF. If dpa, Bq ă 10 9, then there exists a unitary u P W pa, Bq with uau B. Don t get }u 1 H } small, in terms of dpa, Bq but can control Adpuq ι A in the point-norm topology.

17 AF ALGEBRAS DEFINITION C -algebra A is AF if every finite subset of A can be arbitrarily approximated in norm by finite dimensional subalgebras of A. Separable AF algebras are inductive limits of finite dimensional algebras. eg M 2 8. Changing matrix sizes matters: M 3 8 fl M 2 8. CLASSIFICATION ELLIOTT Separable AF-algebras are classified by K 0. THEOREM (CHRISTENSEN 80) Let A, B Ă BpHq be C -algebras, with A separable and AF. If dpa, Bq ă 10 9, then there exists a unitary u P W pa, Bq with uau B. Don t get }u 1 H } small, in terms of dpa, Bq but can control Adpuq ι A in the point-norm topology.

18 AF ALGEBRAS DEFINITION C -algebra A is AF if every finite subset of A can be arbitrarily approximated in norm by finite dimensional subalgebras of A. Separable AF algebras are inductive limits of finite dimensional algebras. eg M 2 8. Changing matrix sizes matters: M 3 8 fl M 2 8. CLASSIFICATION ELLIOTT Separable AF-algebras are classified by K 0. THEOREM (CHRISTENSEN 80) Let A, B Ă BpHq be C -algebras, with A separable and AF. If dpa, Bq ă 10 9, then there exists a unitary u P W pa, Bq with uau B. Don t get }u 1 H } small, in terms of dpa, Bq but can control Adpuq ι A in the point-norm topology.

19 THEOREM (CONNES 75) AMENABLE VON NEUMANN ALGEBRAS Let M be a (separably acting) von Neumann algebra. Then (amongst others) the following are equivalent. 1 M is injective (abstract categorical property) 2 M is semidiscrete (finite dimensional approximation property) 3 M is hyperfinite (inductive limit structure) Call these factors amenable. A linear map φ : A Ñ B between C -algebras is positive if φpxq ě 0 when x ě 0. φ is completely positive if each φ pnq : M n paq Ñ M n pbq is positive. M is semidiscrete if id M can be approximated in the point weak -topology by completely positive contractions (cpc) M Ñ F Ñ M, with with F finite dimensional.

20 THEOREM (CONNES 75) AMENABLE VON NEUMANN ALGEBRAS Let M be a (separably acting) von Neumann algebra. Then (amongst others) the following are equivalent. 1 M is injective (abstract categorical property) 2 M is semidiscrete (finite dimensional approximation property) 3 M is hyperfinite (inductive limit structure) Call these factors amenable. A linear map φ : A Ñ B between C -algebras is positive if φpxq ě 0 when x ě 0. φ is completely positive if each φ pnq : M n paq Ñ M n pbq is positive. M is semidiscrete if id M can be approximated in the point weak -topology by completely positive contractions (cpc) M Ñ F Ñ M, with with F finite dimensional.

21 THEOREM (CONNES 75) AMENABLE VON NEUMANN ALGEBRAS Let M be a (separably acting) von Neumann algebra. Then (amongst others) the following are equivalent. 1 M is injective (abstract categorical property) 2 M is semidiscrete (finite dimensional approximation property) 3 M is hyperfinite (inductive limit structure) Call these factors amenable. A linear map φ : A Ñ B between C -algebras is positive if φpxq ě 0 when x ě 0. φ is completely positive if each φ pnq : M n paq Ñ M n pbq is positive. M is semidiscrete if id M can be approximated in the point weak -topology by completely positive contractions (cpc) M Ñ F Ñ M, with with F finite dimensional.

22 MORE ON CONNES THEOREM COROLLARY (CONNES) There is a unique (separably acting) injective II 1 factor. Connes didn t classify injective II 1 factors: he showed the abstract property of injectivity implies the existence of an inductive limit structure. Factors with this inductive limit structure had already been classified by Murray and von Neumann. IMPORTANT STEP IN CONNES PROOF First show that an injective II 1 factor M tensorially absorbs the hyperfinite II 1 factor, i.e. M M b R. II 1 factors M with M M b R where characterised by McDuff. M M b R iff the central sequence algebra M ω X M 1 is non-abelian iff M k ãñ M ω X M 1 for all k.

23 MORE ON CONNES THEOREM COROLLARY (CONNES) There is a unique (separably acting) injective II 1 factor. Connes didn t classify injective II 1 factors: he showed the abstract property of injectivity implies the existence of an inductive limit structure. Factors with this inductive limit structure had already been classified by Murray and von Neumann. IMPORTANT STEP IN CONNES PROOF First show that an injective II 1 factor M tensorially absorbs the hyperfinite II 1 factor, i.e. M M b R. II 1 factors M with M M b R where characterised by McDuff. M M b R iff the central sequence algebra M ω X M 1 is non-abelian iff M k ãñ M ω X M 1 for all k.

24 AMENABLE C -ALGEBRAS DEFINITION A C -algebra A is nuclear if for every C -algebra B there is only one way of completing the algebraic tensor product A d B to obtain a C -algebra. DEFINITION A C -algebra A has the completely positive factorisation property if id A : A Ñ A can be approximated by completely positive contractions A Ñ F Ñ A in the point norm topology. THEOREM (CONNES, (KIRCHBERG, CHOI-EFFROS)) Let A be a C -algebra. TFAE: 1 A is nuclear; 2 A is semidiscrete; 3 A has the completely positive approximation property.

25 AMENABLE C -ALGEBRAS DEFINITION A C -algebra A is nuclear if for every C -algebra B there is only one way of completing the algebraic tensor product A d B to obtain a C -algebra. DEFINITION A C -algebra A has the completely positive factorisation property if id A : A Ñ A can be approximated by completely positive contractions A Ñ F Ñ A in the point norm topology. THEOREM (CONNES, (KIRCHBERG, CHOI-EFFROS)) Let A be a C -algebra. TFAE: 1 A is nuclear; 2 A is semidiscrete; 3 A has the completely positive approximation property.

26 AMENABLE C -ALGEBRAS DEFINITION A C -algebra A is nuclear if for every C -algebra B there is only one way of completing the algebraic tensor product A d B to obtain a C -algebra. DEFINITION A C -algebra A has the completely positive factorisation property if id A : A Ñ A can be approximated by completely positive contractions A Ñ F Ñ A in the point norm topology. THEOREM (CONNES, (KIRCHBERG, CHOI-EFFROS)) Let A be a C -algebra. TFAE: 1 A is nuclear; 2 A is semidiscrete; 3 A has the completely positive approximation property.

27 A Hahn Banach argument semidiscrete ùñ A has cpap A semidiscrete currently requires Connes ðù A has cpap Due to Connes theorem, can witness semidiscreteness of A with maps A Ñ F φ Ñ A with φ a -homomorphism. THEOREM (HIRSHBERG, KIRCHBERG, W.) Let A be a nuclear C -algebra. Then id A : A Ñ A can be approximated by cpc maps A Ñ F φ Ñ A where φ is a convex combination of cpc order zero maps. A could be projectionless: no -hms F Ñ A. Order zero maps, next best thing. φ : F Ñ A is order zero if φ preserves orthogonality, i.e. e, f ě 0, ef 0 ùñ φpeqφpf q 0. All cpc order zero maps φ obtained by compressing -homomorphism by a positive element which commutes with φpaq.

28 A Hahn Banach argument semidiscrete ùñ A has cpap A semidiscrete currently requires Connes ðù A has cpap Due to Connes theorem, can witness semidiscreteness of A with maps A Ñ F φ Ñ A with φ a -homomorphism. THEOREM (HIRSHBERG, KIRCHBERG, W.) Let A be a nuclear C -algebra. Then id A : A Ñ A can be approximated by cpc maps A Ñ F φ Ñ A where φ is a convex combination of cpc order zero maps. A could be projectionless: no -hms F Ñ A. Order zero maps, next best thing. φ : F Ñ A is order zero if φ preserves orthogonality, i.e. e, f ě 0, ef 0 ùñ φpeqφpf q 0. All cpc order zero maps φ obtained by compressing -homomorphism by a positive element which commutes with φpaq.

29 A Hahn Banach argument semidiscrete ùñ A has cpap A semidiscrete currently requires Connes ðù A has cpap Due to Connes theorem, can witness semidiscreteness of A with maps A Ñ F φ Ñ A with φ a -homomorphism. THEOREM (HIRSHBERG, KIRCHBERG, W.) Let A be a nuclear C -algebra. Then id A : A Ñ A can be approximated by cpc maps A Ñ F φ Ñ A where φ is a convex combination of cpc order zero maps. A could be projectionless: no -hms F Ñ A. Order zero maps, next best thing. φ : F Ñ A is order zero if φ preserves orthogonality, i.e. e, f ě 0, ef 0 ùñ φpeqφpf q 0. All cpc order zero maps φ obtained by compressing -homomorphism by a positive element which commutes with φpaq.

30 THEOREM (HIRSHBERG, KIRCHBERG, W. 2011) Let A be a nuclear C -algebra. Then id A : A Ñ A can be approximated by cpc maps A Ñ F φ Ñ A where φ is a convex combination of cpc order zero maps. COROLLARY (HKW, BUILDING ON RESULTS OF CHRISTENSEN, SINCLAIR, SMITH, W, WINTER) Let A be separable and nuclear, and suppose that A Ă γ B for γ ă 1{ Then A ãñ B. If in addition dpa, Bq small, then D unitary u P W pa, Bq with uau B. DEFINITION (WINTER, ZACHARIAS, 2009) A C -algebra has nuclear dimension at most n if one can find cp factorisations A contractive Ñ F Ñ φ A such that φ is a sum of at most n ` 1 cpc order zero maps. dim nuc pcpxqq dimpxq; dim nuc paq 0 ô A is AF.

31 THEOREM (HIRSHBERG, KIRCHBERG, W. 2011) Let A be a nuclear C -algebra. Then id A : A Ñ A can be approximated by cpc maps A Ñ F φ Ñ A where φ is a convex combination of cpc order zero maps. COROLLARY (HKW, BUILDING ON RESULTS OF CHRISTENSEN, SINCLAIR, SMITH, W, WINTER) Let A be separable and nuclear, and suppose that A Ă γ B for γ ă 1{ Then A ãñ B. If in addition dpa, Bq small, then D unitary u P W pa, Bq with uau B. DEFINITION (WINTER, ZACHARIAS, 2009) A C -algebra has nuclear dimension at most n if one can find cp factorisations A contractive Ñ F Ñ φ A such that φ is a sum of at most n ` 1 cpc order zero maps. dim nuc pcpxqq dimpxq; dim nuc paq 0 ô A is AF.

32 THEOREM (HIRSHBERG, KIRCHBERG, W. 2011) Let A be a nuclear C -algebra. Then id A : A Ñ A can be approximated by cpc maps A Ñ F φ Ñ A where φ is a convex combination of cpc order zero maps. COROLLARY (HKW, BUILDING ON RESULTS OF CHRISTENSEN, SINCLAIR, SMITH, W, WINTER) Let A be separable and nuclear, and suppose that A Ă γ B for γ ă 1{ Then A ãñ B. If in addition dpa, Bq small, then D unitary u P W pa, Bq with uau B. DEFINITION (WINTER, ZACHARIAS, 2009) A C -algebra has nuclear dimension at most n if one can find cp factorisations A contractive Ñ F Ñ φ A such that φ is a sum of at most n ` 1 cpc order zero maps. dim nuc pcpxqq dimpxq; dim nuc paq 0 ô A is AF.

33 CLASSIFICATION PROGRAMME CONNES, HAAGERUP Classification of separably acting injective factors. The analogous class of C -algebras are the simple, separable and unital C -algebras. ELLIOTT S CLASSIFICATION PROGRAMME: INITIAL AIM Classify all simple, separable, nuclear C -algebras A by K -theory: data about homotopy equivalence classes of projections and unitaries in matrices over A

34 CLASSIFICATION PROGRAMME CONNES, HAAGERUP Classification of separably acting injective factors. The analogous class of C -algebras are the simple, separable and unital C -algebras. ELLIOTT S CLASSIFICATION PROGRAMME: INITIAL AIM Classify all simple, separable, nuclear C -algebras A by K -theory: data about homotopy equivalence classes of projections and unitaries in matrices over A

35 PURELY INFINITE ALGEBRAS DEFINITION A simple C -algebra A is purely infinite if for all x 0 there exists a, b P A such that 1 axb. Very strong infiniteness condition. Implies that all non-zero projections are equivalent; Equivalently: every hereditary subalgebra has an infinite projection. e.g Cuntz algebras O n universal C -algebras generated by n isometries s 1,..., s n with ř n i 1 s is i 1. THEOREM (KIRCHBERG, KIRCHBERG-PHILIPS) Purely infinite simple separable nuclear C -algebras (satisfying the UCT) are classified by K -theory.

36 PURELY INFINITE ALGEBRAS DEFINITION A simple C -algebra A is purely infinite if for all x 0 there exists a, b P A such that 1 axb. Very strong infiniteness condition. Implies that all non-zero projections are equivalent; Equivalently: every hereditary subalgebra has an infinite projection. e.g Cuntz algebras O n universal C -algebras generated by n isometries s 1,..., s n with ř n i 1 s is i 1. THEOREM (KIRCHBERG, KIRCHBERG-PHILIPS) Purely infinite simple separable nuclear C -algebras (satisfying the UCT) are classified by K -theory.

37 ABSOPTION THEOREMS: KEY STEPS IN KIRCHBERG-PHILIIPS CLASSIFICATION THEOREM (KIRCHBERG) Let A be simple, separable and nuclear. Then A is purely infinite ðñ A A b O 8. RECALL To prove his theorem, Connes needed to show that an injective II 1 factor has M M b R. THEOREM (KIRCHBERG) Let A be simple, separable unital and nuclear. Then O 2 A b O 2. The tensorial absorption O 2 O 2 b O 2 (due to Elliott) plays a significant role in this.

38 THE FINITE SETTING Oddly, in the C -setting the purely infinite case appears to be easier than the finite case. REASONS Higher dimensional topological phenoma can occur in simple C -algebras. The condition of pure infiniteness prevents this: in a precise sense these algebras have low topological dimension. In the finite case, need traces as well as K -theory. In late 90 s Jiang-Su constructed Z, an infinite dimensional C -algebra with the same Elliott data as the complex numbers. For reasonable A, A and A b Z have same Elliott data. Rørdam, Toms used a Chern class obstruction of Villadsen to produce vast counter examples to Elliott s conjecture.

39 THE FINITE SETTING Oddly, in the C -setting the purely infinite case appears to be easier than the finite case. REASONS Higher dimensional topological phenoma can occur in simple C -algebras. The condition of pure infiniteness prevents this: in a precise sense these algebras have low topological dimension. In the finite case, need traces as well as K -theory. In late 90 s Jiang-Su constructed Z, an infinite dimensional C -algebra with the same Elliott data as the complex numbers. For reasonable A, A and A b Z have same Elliott data. Rørdam, Toms used a Chern class obstruction of Villadsen to produce vast counter examples to Elliott s conjecture.

40 THE FINITE SETTING Oddly, in the C -setting the purely infinite case appears to be easier than the finite case. REASONS Higher dimensional topological phenoma can occur in simple C -algebras. The condition of pure infiniteness prevents this: in a precise sense these algebras have low topological dimension. In the finite case, need traces as well as K -theory. In late 90 s Jiang-Su constructed Z, an infinite dimensional C -algebra with the same Elliott data as the complex numbers. For reasonable A, A and A b Z have same Elliott data. Rørdam, Toms used a Chern class obstruction of Villadsen to produce vast counter examples to Elliott s conjecture.

41 ESTABLISHING Z-STABILITY We expect that simple, separable nuclear C -algebras should be classifable when they absorb Z tensorially. THEOREM (WINTER 2010) Let A be simple, separable unital and nuclear and have finite nuclear dimension. Then A A b Z. Can view this as obtaining tensorial absoption from a topological property. Absorbing Z gives room to maneouver in classification theorems. THEOREM (TOMS, WINTER 2009) The class C tcpxq Z : X finite dimensional metrisable, Z ñ X minimal, uniquely ergodicu is classified by K -theory.

42 ESTABLISHING Z-STABILITY We expect that simple, separable nuclear C -algebras should be classifable when they absorb Z tensorially. THEOREM (WINTER 2010) Let A be simple, separable unital and nuclear and have finite nuclear dimension. Then A A b Z. Can view this as obtaining tensorial absoption from a topological property. Absorbing Z gives room to maneouver in classification theorems. THEOREM (TOMS, WINTER 2009) The class C tcpxq Z : X finite dimensional metrisable, Z ñ X minimal, uniquely ergodicu is classified by K -theory.

43 Further, Z-stability is analogous to absorbing R in a very striking way. Recall that a II 1 factor M has M M b R if and only if M k ãñ M ω X M 1. For separable A, A A b Z if and only if the central sequence algebra A ω X A 1 is sufficiently non-commutative. For separable unital A, A A b Z if and only if, for some k ě 2, there exists a large cpc order zero map φ : M k Ñ A ω X A 1. Losely, large means that 1 A φp1 k q is dominated by φpe 11 q.

44 Further, Z-stability is analogous to absorbing R in a very striking way. Recall that a II 1 factor M has M M b R if and only if M k ãñ M ω X M 1. For separable A, A A b Z if and only if the central sequence algebra A ω X A 1 is sufficiently non-commutative. For separable unital A, A A b Z if and only if, for some k ě 2, there exists a large cpc order zero map φ : M k Ñ A ω X A 1. Losely, large means that 1 A φp1 k q is dominated by φpe 11 q.

45 Further, Z-stability is analogous to absorbing R in a very striking way. Recall that a II 1 factor M has M M b R if and only if M k ãñ M ω X M 1. For separable A, A A b Z if and only if the central sequence algebra A ω X A 1 is sufficiently non-commutative. For separable unital A, A A b Z if and only if, for some k ě 2, there exists a large cpc order zero map φ : M k Ñ A ω X A 1. Losely, large means that 1 A φp1 k q is dominated by φpe 11 q.

46 TENSORIAL ABSORPTION FROM ABSTRACT PROPERTIES GOAL (PART OF A CONJECTURE OF TOMS AND WINTER) Let A be simple unital separable and nuclear. Then A has strict comparison ô A A b Z. ð is Rørdam. Strict comparison is a condition which says that we can determine the order on positive elements by looking at their ranks. The analogous condition holds for all II 1 factors. This would be a vast generalisation of Kirchberg s O 8 -absorption theorem as when A is simple, separable and traceless Rørdam has shown: A A b Z ô A A b O 8 A purely infinite ô A has strict comparson

47 TENSORIAL ABSORPTION FROM ABSTRACT PROPERTIES GOAL (PART OF A CONJECTURE OF TOMS AND WINTER) Let A be simple unital separable and nuclear. Then A has strict comparison ô A A b Z. ð is Rørdam. Strict comparison is a condition which says that we can determine the order on positive elements by looking at their ranks. The analogous condition holds for all II 1 factors. This would be a vast generalisation of Kirchberg s O 8 -absorption theorem as when A is simple, separable and traceless Rørdam has shown: A A b Z ô A A b O 8 A purely infinite ô A has strict comparson

48 TENSORIAL ABSORPTION FROM ABSTRACT PROPERTIES GOAL (PART OF A CONJECTURE OF TOMS AND WINTER) Let A be simple unital separable and nuclear. Then A has strict comparison ô A A b Z. ð is Rørdam. Strict comparison is a condition which says that we can determine the order on positive elements by looking at their ranks. The analogous condition holds for all II 1 factors. This would be a vast generalisation of Kirchberg s O 8 -absorption theorem as when A is simple, separable and traceless Rørdam has shown: A A b Z ô A A b O 8 A purely infinite ô A has strict comparson

49 THEOREM (KIRCHBERG-RØRDAM, SATO, TOMS-W-WINTER, BUILDING ON MATUI-SATO) CURRENT STATUS Let A be simple separable unital and nuclear and suppose that T paq is non-empty, and has a compact extreme boundary of finite covering dimension. Then A has strict comparison ô A A b Z. Proofs use interplay between von Neumann and C -algebras directly: For each extreme trace τ, the weak closure M τ of A in the GNS representation is an injective II 1 factor. By Connes, M τ M τ b R so have M k ãñ M ω τ X M 1 τ. Can use the surjectivity of A ω X A 1 Ñ M ω τ X M 1 τ to obtain order zero maps M k Ñ A ω X A 1 large wrt τ. The condition on T paq is used to glue these together to obtain one large order zero map M k Ñ A ω X A 1.

Elliott s program and descriptive set theory I

Elliott s program and descriptive set theory I Ilijas Farah LC 2012, Manchester, July 12 a, a, a, a, the, the, the, the. I shall need this exercise later, someone please solve it Exercise If A = limna