Interplay between C* and von Neumann...

INTERPLAY BETWEEN C˚ AND VON NEUMANN

ALGEBRAS

Stuart White

University of Glasgow

26 March 2013, British Mathematical Colloquium,University of Sheffield.

C˚ AND VON NEUMANN ALGEBRAS

C˚-algebras von Neuman algebras

Banach algebra Awith an involution ˚

satisfying the C˚-identity}x˚x} “ }x}2 for all x P A.

Weak operator closed˚-subalgebra M Ď BpHqWeak operator topology:xi Ñ x ô xpxi ´ xqξ, ηy Ñ 0for all ξ, η P H

Norm closed ˚-subalgebraA Ď BpHq

C˚-algebra M which isisometrically the dual spaceof some Banach space.

Abelian C˚-algebras of formC0pX q for X locally compact Hff

Abelian vNas of form L8pX , µqfor some measure space pX , µq.

C˚ AND VON NEUMANN ALGEBRAS

C˚-algebras von Neuman algebras

Banach algebra Awith an involution ˚

satisfying the C˚-identity}x˚x} “ }x}2 for all x P A.

Weak operator closed˚-subalgebra M Ď BpHqWeak operator topology:xi Ñ x ô xpxi ´ xqξ, ηy Ñ 0for all ξ, η P H

Norm closed ˚-subalgebraA Ď BpHq

C˚-algebra M which isisometrically the dual spaceof some Banach space.

Abelian C˚-algebras of formC0pX q for X locally compact Hff

Abelian vNas of form L8pX , µqfor some measure space pX , µq.

C˚ AND VON NEUMANN ALGEBRAS

C˚-algebras von Neuman algebras

Banach algebra Awith an involution ˚

satisfying the C˚-identity}x˚x} “ }x}2 for all x P A.

Weak operator closed˚-subalgebra M Ď BpHqWeak operator topology:xi Ñ x ô xpxi ´ xqξ, ηy Ñ 0for all ξ, η P H

Norm closed ˚-subalgebraA Ď BpHq

C˚-algebra M which isisometrically the dual spaceof some Banach space.

Abelian C˚-algebras of formC0pX q for X locally compact Hff

Abelian vNas of form L8pX , µqfor some measure space pX , µq.

C˚ AND VON NEUMANN ALGEBRAS

C˚-algebras von Neuman algebras

Banach algebra Awith an involution ˚

satisfying the C˚-identity}x˚x} “ }x}2 for all x P A.

Weak operator closed˚-subalgebra M Ď BpHqWeak operator topology:xi Ñ x ô xpxi ´ xqξ, ηy Ñ 0for all ξ, η P H

Norm closed ˚-subalgebraA Ď BpHq

C˚-algebra M which isisometrically the dual spaceof some Banach space.

Abelian C˚-algebras of formC0pX q for X locally compact Hff

Abelian vNas of form L8pX , µqfor some measure space pX , µq.

C˚ AND VON NEUMANN ALGEBRAS

C˚-algebras von Neuman algebras

Banach algebra Awith an involution ˚

satisfying the C˚-identity}x˚x} “ }x}2 for all x P A.

Weak operator closed˚-subalgebra M Ď BpHqWeak operator topology:xi Ñ x ô xpxi ´ xqξ, ηy Ñ 0for all ξ, η P H

Norm closed ˚-subalgebraA Ď BpHq

C˚-algebra M which isisometrically the dual spaceof some Banach space.

Abelian C˚-algebras of formC0pX q for X locally compact Hff

Abelian vNas of form L8pX , µqfor some measure space pX , µq.

EXAMPLES

Mn, BpHq;C0pX q Ă BpL2pX qq, L8pX q Ă BpL2pX qq;G discrete group group C˚ and von Neumann algebras;

H “ `2pGq, basis tδg : g P Gu. Define unitaries λg : HÑ H byλgpδhq “ δgh;

C˚r pGq “ Spantλg : g P Gu}; LG “ Spantλg : g P Gu

wot.

α : G ñ X Dynamical system CpX q ¸r G, L8pX q ¸G.

CpX q ¸r G is a C˚-algebra generated by CpX q and copy of G;

Unitaries pλgqgPG with λgh “ λgλh, λg fλ˚g “ f ˝ α´1g for g,h P G,

f P CpX q.

How much information about G or G ñ X is captured by theiroperator algebras?

EXAMPLES

Mn, BpHq;C0pX q Ă BpL2pX qq, L8pX q Ă BpL2pX qq;G discrete group group C˚ and von Neumann algebras;

H “ `2pGq, basis tδg : g P Gu. Define unitaries λg : HÑ H byλgpδhq “ δgh;

C˚r pGq “ Spantλg : g P Gu}; LG “ Spantλg : g P Gu

wot.

α : G ñ X Dynamical system CpX q ¸r G, L8pX q ¸G.

CpX q ¸r G is a C˚-algebra generated by CpX q and copy of G;

Unitaries pλgqgPG with λgh “ λgλh, λg fλ˚g “ f ˝ α´1g for g,h P G,

f P CpX q.

How much information about G or G ñ X is captured by theiroperator algebras?

ANOTHER EXAMPLE

Consider M2 ãÑ M4 – M2 bM2; x ÞÑ x b 1.This preserves the normalised trace on M2 and M4. Carryon and form

M2 ãÑ M2 bM2 ãÑ Mb32 ãÑ ¨ ¨ ¨

to obtain direct limit ˚-algebraÄ8

n“1 M2.Perform GNS construction for canonical trace τ and takethe weak operator closure — obtain hyperfinite II1 factor R.Could also close in the norm topology to get theCAR-algebra M28 .

IMPORTANT CONSEQUENCE

R – RbR

ANOTHER EXAMPLE

Consider M2 ãÑ M4 – M2 bM2; x ÞÑ x b 1.This preserves the normalised trace on M2 and M4. Carryon and form

M2 ãÑ M2 bM2 ãÑ Mb32 ãÑ ¨ ¨ ¨

to obtain direct limit ˚-algebraÄ8

n“1 M2.Perform GNS construction for canonical trace τ and takethe weak operator closure — obtain hyperfinite II1 factor R.Could also close in the norm topology to get theCAR-algebra M28 .

IMPORTANT CONSEQUENCE

R – RbR

ANOTHER EXAMPLE

Consider M2 ãÑ M4 – M2 bM2; x ÞÑ x b 1.This preserves the normalised trace on M2 and M4. Carryon and form

M2 ãÑ M2 bM2 ãÑ Mb32 ãÑ ¨ ¨ ¨

to obtain direct limit ˚-algebraÄ8

n“1 M2.Perform GNS construction for canonical trace τ and takethe weak operator closure — obtain hyperfinite II1 factor R.Could also close in the norm topology to get theCAR-algebra M28 .

IMPORTANT CONSEQUENCE

R – RbR

II1 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 II1 if:it is infinite dimensional;D tracial state τ : M Ñ C1, i.e. τpxyq “ τpyxq, @x , y P M.

R;LG for countable discrete G with |thgh´1 : h P Gu| “ 8 forg ‰ e;LpX q ¸G for G ñ pX , µq is free, ergodic and probabilitymeasure preserving.

HYPERFINITENESS

DEFINITION (MURRAY AND VON NEUMANN)A vNa M is hyperfinite if every finite subset of M can bearbitrarily approximated arbitrarily in the weak˚-topology by afinite dimensional subalgebra of M.

Separably acting hyperfinite vnas arise are inductive limitsof finite dimensional vnas.

THEOREM (MURRAY AND VON NEUMANN)There is a unique (separably acting) hyperfinite II1 factor.

UNIFORM PERTURBATIONS

Given two operator algebras A,B Ă BpHq, write dpA,Bq for theHausdorff distance between the unit balls of A and B in theoperator norm.

For a unitary u, dpA,uAu˚q ď 2}u ´ 1H}.

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˚ “ Nand }u ´ 1H} ď 150dpM,Nq.

IDEA

Establish dimension independent perturbation results fornear containments of finite dimensional algebras.i.e. @ε ą 0, Dδ ą 0 such that if F is finite dim, F Ăδ N, thenD unitary u P W ˚pF ,Nq with uFu˚ Ď N and }u ´ 1H} ă ε.

UNIFORM PERTURBATIONS

Given two operator algebras A,B Ă BpHq, write dpA,Bq for theHausdorff distance between the unit balls of A and B in theoperator norm.

For a unitary u, dpA,uAu˚q ď 2}u ´ 1H}.

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˚ “ Nand }u ´ 1H} ď 150dpM,Nq.

IDEA

Establish dimension independent perturbation results fornear containments of finite dimensional algebras.i.e. @ε ą 0, Dδ ą 0 such that if F is finite dim, F Ăδ N, thenD unitary u P W ˚pF ,Nq with uFu˚ Ď N and }u ´ 1H} ă ε.

AF ALGEBRAS

DEFINITION

C˚-algebra A is AF if every finite subset of A can be arbitrarilyapproximated in norm by finite dimensional subalgebras of A.

Separable AF algebras are inductive limits of finitedimensional algebras.eg M28 . Changing matrix sizes matters: M38 fl M28 .

CLASSIFICATION — ELLIOTT

Separable AF -algebras are classified by K0.

THEOREM (CHRISTENSEN 80)Let A,B Ă BpHq be C˚-algebras, with A separable and AF. IfdpA,Bq ă 10´9, then there exists a unitary u P W ˚pA,Bq withuAu˚ “ B.

Don’t get }u ´ 1H} small, in terms of dpA,Bq but cancontrol Adpuq ´ ιA in the point-norm topology.

AF ALGEBRAS

DEFINITION

C˚-algebra A is AF if every finite subset of A can be arbitrarilyapproximated in norm by finite dimensional subalgebras of A.

Separable AF algebras are inductive limits of finitedimensional algebras.eg M28 . Changing matrix sizes matters: M38 fl M28 .

CLASSIFICATION — ELLIOTT

Separable AF -algebras are classified by K0.

THEOREM (CHRISTENSEN 80)Let A,B Ă BpHq be C˚-algebras, with A separable and AF. IfdpA,Bq ă 10´9, then there exists a unitary u P W ˚pA,Bq withuAu˚ “ B.

Don’t get }u ´ 1H} small, in terms of dpA,Bq but cancontrol Adpuq ´ ιA in the point-norm topology.

AF ALGEBRAS

DEFINITION

C˚-algebra A is AF if every finite subset of A can be arbitrarilyapproximated in norm by finite dimensional subalgebras of A.

Separable AF algebras are inductive limits of finitedimensional algebras.eg M28 . Changing matrix sizes matters: M38 fl M28 .

CLASSIFICATION — ELLIOTT

Separable AF -algebras are classified by K0.

THEOREM (CHRISTENSEN 80)Let A,B Ă BpHq be C˚-algebras, with A separable and AF. IfdpA,Bq ă 10´9, then there exists a unitary u P W ˚pA,Bq withuAu˚ “ B.

Don’t get }u ´ 1H} small, in terms of dpA,Bq but cancontrol Adpuq ´ ιA in the point-norm topology.

AMENABLE VON NEUMANN ALGEBRAS

THEOREM (CONNES 75)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 ifφpxq ě 0 when x ě 0.φ is completely positive if each φpnq : MnpAq Ñ MnpBq ispositive.M is semidiscrete if idM can be approximated in the pointweak˚-topology by completely positive contractions (cpc)M Ñ F Ñ M, with with F finite dimensional.

AMENABLE VON NEUMANN ALGEBRAS

THEOREM (CONNES 75)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 ifφpxq ě 0 when x ě 0.φ is completely positive if each φpnq : MnpAq Ñ MnpBq ispositive.M is semidiscrete if idM can be approximated in the pointweak˚-topology by completely positive contractions (cpc)M Ñ F Ñ M, with with F finite dimensional.

AMENABLE VON NEUMANN ALGEBRAS

THEOREM (CONNES 75)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 ifφpxq ě 0 when x ě 0.φ is completely positive if each φpnq : MnpAq Ñ MnpBq ispositive.M is semidiscrete if idM can be approximated in the pointweak˚-topology by completely positive contractions (cpc)M Ñ F Ñ M, with with F finite dimensional.

MORE ON CONNES THEOREM

COROLLARY (CONNES)There is a unique (separably acting) injective II1 factor.

Connes didn’t classify injective II1 factors: he showed theabstract property of injectivity implies the existence of aninductive limit structure.Factors with this inductive limit structure had already beenclassified by Murray and von Neumann.

IMPORTANT STEP IN CONNES’ PROOF

First show that an injective II1 factor M tensorially absorbs thehyperfinite II1 factor, i.e. M – M bR.

II1 factors M with M – M bR where characterised byMcDuff.M – M bR iff the central sequence algebra Mω XM 1 isnon-abelian iff Mk ãÑ Mω XM 1 for all k .

MORE ON CONNES THEOREM

COROLLARY (CONNES)There is a unique (separably acting) injective II1 factor.

Connes didn’t classify injective II1 factors: he showed theabstract property of injectivity implies the existence of aninductive limit structure.Factors with this inductive limit structure had already beenclassified by Murray and von Neumann.

IMPORTANT STEP IN CONNES’ PROOF

First show that an injective II1 factor M tensorially absorbs thehyperfinite II1 factor, i.e. M – M bR.

II1 factors M with M – M bR where characterised byMcDuff.M – M bR iff the central sequence algebra Mω XM 1 isnon-abelian iff Mk ãÑ Mω XM 1 for all k .

AMENABLE C˚-ALGEBRAS

DEFINITION

A C˚-algebra A is nuclear if for every C˚-algebra B there is onlyone way of completing the algebraic tensor product Ad B toobtain a C˚-algebra.

DEFINITION

A C˚-algebra A has the completely positive factorisationproperty if idA : A Ñ A can be approximated by completelypositive 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.

AMENABLE C˚-ALGEBRAS

DEFINITION

A C˚-algebra A is nuclear if for every C˚-algebra B there is onlyone way of completing the algebraic tensor product Ad B toobtain a C˚-algebra.

DEFINITION

A C˚-algebra A has the completely positive factorisationproperty if idA : A Ñ A can be approximated by completelypositive 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.

AMENABLE C˚-ALGEBRAS

DEFINITION

A C˚-algebra A is nuclear if for every C˚-algebra B there is onlyone way of completing the algebraic tensor product Ad B toobtain a C˚-algebra.

DEFINITION

A C˚-algebra A has the completely positive factorisationproperty if idA : A Ñ A can be approximated by completelypositive 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.

A˚˚ semidiscreteHahn Banach argument

ùñ A has cpap

A˚˚ semidiscretecurrently requires Connes

ðù A has cpap

Due to Connes theorem, can witness semidiscreteness ofA˚˚ with maps A˚˚ Ñ F φ

Ñ A˚˚ with φ a ˚-homomorphism.

THEOREM (HIRSHBERG, KIRCHBERG, W.)Let A be a nuclear C˚-algebra. Then idA : A Ñ A can beapproximated by cpc maps A Ñ F φ

Ñ A where φ is a convexcombination 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 commuteswith φpAq.

A˚˚ semidiscreteHahn Banach argument

ùñ A has cpap

A˚˚ semidiscretecurrently requires Connes

ðù A has cpap

Due to Connes theorem, can witness semidiscreteness ofA˚˚ with maps A˚˚ Ñ F φ

Ñ A˚˚ with φ a ˚-homomorphism.

THEOREM (HIRSHBERG, KIRCHBERG, W.)Let A be a nuclear C˚-algebra. Then idA : A Ñ A can beapproximated by cpc maps A Ñ F φ

Ñ A where φ is a convexcombination 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 commuteswith φpAq.

A˚˚ semidiscreteHahn Banach argument

ùñ A has cpap

A˚˚ semidiscretecurrently requires Connes

ðù A has cpap

Due to Connes theorem, can witness semidiscreteness ofA˚˚ with maps A˚˚ Ñ F φ

Ñ A˚˚ with φ a ˚-homomorphism.

THEOREM (HIRSHBERG, KIRCHBERG, W.)Let A be a nuclear C˚-algebra. Then idA : A Ñ A can beapproximated by cpc maps A Ñ F φ

Ñ A where φ is a convexcombination 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 commuteswith φpAq.

THEOREM (HIRSHBERG, KIRCHBERG, W. 2011)Let A be a nuclear C˚-algebra. Then idA : A Ñ A can beapproximated by cpc maps A Ñ F φ

Ñ A where φ is a convexcombination of cpc order zero maps.

COROLLARY (HKW, BUILDING ON RESULTS OFCHRISTENSEN, SINCLAIR, SMITH, W, WINTER)Let A be separable and nuclear, and suppose that A Ăγ B forγ ă 1{210000. Then A ãÑ B.

If in addition dpA,Bq small, then D unitary u P W ˚pA,Bqwith uAu˚ “ B.

DEFINITION (WINTER, ZACHARIAS, 2009)A C˚-algebra has nuclear dimension at most n if one can findcp factorisations A contractive

Ñ F φÑ A such that φ is a sum of at

most n ` 1 cpc order zero maps.

dimnucpCpX qq “ dimpX q; dimnucpAq “ 0 ô A is AF.

THEOREM (HIRSHBERG, KIRCHBERG, W. 2011)Let A be a nuclear C˚-algebra. Then idA : A Ñ A can beapproximated by cpc maps A Ñ F φ

Ñ A where φ is a convexcombination of cpc order zero maps.

COROLLARY (HKW, BUILDING ON RESULTS OFCHRISTENSEN, SINCLAIR, SMITH, W, WINTER)Let A be separable and nuclear, and suppose that A Ăγ B forγ ă 1{210000. Then A ãÑ B.

If in addition dpA,Bq small, then D unitary u P W ˚pA,Bqwith uAu˚ “ B.

DEFINITION (WINTER, ZACHARIAS, 2009)A C˚-algebra has nuclear dimension at most n if one can findcp factorisations A contractive

Ñ F φÑ A such that φ is a sum of at

most n ` 1 cpc order zero maps.

dimnucpCpX qq “ dimpX q; dimnucpAq “ 0 ô A is AF.

THEOREM (HIRSHBERG, KIRCHBERG, W. 2011)Let A be a nuclear C˚-algebra. Then idA : A Ñ A can beapproximated by cpc maps A Ñ F φ

Ñ A where φ is a convexcombination of cpc order zero maps.

COROLLARY (HKW, BUILDING ON RESULTS OFCHRISTENSEN, SINCLAIR, SMITH, W, WINTER)Let A be separable and nuclear, and suppose that A Ăγ B forγ ă 1{210000. Then A ãÑ B.

If in addition dpA,Bq small, then D unitary u P W ˚pA,Bqwith uAu˚ “ B.

DEFINITION (WINTER, ZACHARIAS, 2009)A C˚-algebra has nuclear dimension at most n if one can findcp factorisations A contractive

Ñ F φÑ A such that φ is a sum of at

most n ` 1 cpc order zero maps.

dimnucpCpX qq “ dimpX q; dimnucpAq “ 0 ô A is AF.

CLASSIFICATION PROGRAMME

CONNES, HAAGERUP

Classification of separably acting injective factors.

The analogous class of C˚-algebras are the simple, separableand unital C˚-algebras.

ELLIOTT’S CLASSIFICATION PROGRAMME: INITIAL AIM

Classify all simple, separable, nuclear C˚-algebras A byK -theory: data about homotopy equivalence classes ofprojections and unitaries in matrices over A

CLASSIFICATION PROGRAMME

CONNES, HAAGERUP

Classification of separably acting injective factors.

The analogous class of C˚-algebras are the simple, separableand unital C˚-algebras.

ELLIOTT’S CLASSIFICATION PROGRAMME: INITIAL AIM

Classify all simple, separable, nuclear C˚-algebras A byK -theory: data about homotopy equivalence classes ofprojections and unitaries in matrices over A

PURELY INFINITE ALGEBRAS

DEFINITION

A simple C˚-algebra A is purely infinite if for all x ‰ 0 thereexists a,b P A such that 1 “ axb.

Very strong infiniteness condition. Implies that all non-zeroprojections are equivalent;Equivalently: every hereditary subalgebra has an infiniteprojection.e.g Cuntz algebras On — universal C˚-algebras generatedby n isometries s1, . . . , sn with

řni“1 sis˚i “ 1.

THEOREM (KIRCHBERG, KIRCHBERG-PHILIPS)Purely infinite simple separable nuclear C˚-algebras (satisfyingthe UCT) are classified by K -theory.

PURELY INFINITE ALGEBRAS

DEFINITION

A simple C˚-algebra A is purely infinite if for all x ‰ 0 thereexists a,b P A such that 1 “ axb.

Very strong infiniteness condition. Implies that all non-zeroprojections are equivalent;Equivalently: every hereditary subalgebra has an infiniteprojection.e.g Cuntz algebras On — universal C˚-algebras generatedby n isometries s1, . . . , sn with

řni“1 sis˚i “ 1.

THEOREM (KIRCHBERG, KIRCHBERG-PHILIPS)Purely infinite simple separable nuclear C˚-algebras (satisfyingthe UCT) are classified by K -theory.

ABSOPTION THEOREMS: KEY STEPS IN

KIRCHBERG-PHILIIPS CLASSIFICATION

THEOREM (KIRCHBERG)Let A be simple, separable and nuclear. Then

A is purely infinite ðñ A – AbO8.

RECALL

To prove his theorem, Connes needed to show that an injectiveII1 factor has M – M bR.

THEOREM (KIRCHBERG)Let A be simple, separable unital and nuclear. Then

O2 – AbO2.

The tensorial absorption O2 – O2 bO2 (due to Elliott) plays asignificant role in this.

THE FINITE SETTING

Oddly, in the C˚-setting the purely infinite case appears to beeasier than the finite case.

REASONS

Higher dimensional topological phenoma can occur insimple C˚-algebras.The condition of pure infiniteness prevents this: in aprecise sense these algebras have low topologicaldimension.

In the finite case, need traces as well as K -theory.In late 90’s Jiang-Su constructed Z, an infinite dimensionalC˚-algebra with the same Elliott data as the complexnumbers.For “reasonable” A, A and Ab Z have same Elliott data.Rørdam, Toms used a Chern class obstruction of Villadsento produce vast counter examples to Elliott’s conjecture.

THE FINITE SETTING

Oddly, in the C˚-setting the purely infinite case appears to beeasier than the finite case.

REASONS

Higher dimensional topological phenoma can occur insimple C˚-algebras.The condition of pure infiniteness prevents this: in aprecise sense these algebras have low topologicaldimension.

In the finite case, need traces as well as K -theory.In late 90’s Jiang-Su constructed Z, an infinite dimensionalC˚-algebra with the same Elliott data as the complexnumbers.For “reasonable” A, A and Ab Z have same Elliott data.Rørdam, Toms used a Chern class obstruction of Villadsento produce vast counter examples to Elliott’s conjecture.

THE FINITE SETTING

Oddly, in the C˚-setting the purely infinite case appears to beeasier than the finite case.

REASONS

Higher dimensional topological phenoma can occur insimple C˚-algebras.The condition of pure infiniteness prevents this: in aprecise sense these algebras have low topologicaldimension.

In the finite case, need traces as well as K -theory.In late 90’s Jiang-Su constructed Z, an infinite dimensionalC˚-algebra with the same Elliott data as the complexnumbers.For “reasonable” A, A and Ab Z have same Elliott data.Rørdam, Toms used a Chern class obstruction of Villadsento produce vast counter examples to Elliott’s conjecture.

ESTABLISHING Z -STABILITY

We expect that simple, separable nuclear C˚-algebras shouldbe classifable when they absorb Z tensorially.

THEOREM (WINTER 2010)Let A be simple, separable unital and nuclear and have finitenuclear dimension. Then A – Ab Z.

Can view this as obtaining tensorial absoption from atopological property.Absorbing Z gives room to maneouver in classificationtheorems.

THEOREM (TOMS, WINTER 2009)The class C “ tCpX q ¸ Z : X finite dimensional metrisable,Z ñ X minimal, uniquely ergodicu is classified by K -theory.

ESTABLISHING Z -STABILITY

We expect that simple, separable nuclear C˚-algebras shouldbe classifable when they absorb Z tensorially.

THEOREM (WINTER 2010)Let A be simple, separable unital and nuclear and have finitenuclear dimension. Then A – Ab Z.

Can view this as obtaining tensorial absoption from atopological property.Absorbing Z gives room to maneouver in classificationtheorems.

THEOREM (TOMS, WINTER 2009)The class C “ tCpX q ¸ Z : X finite dimensional metrisable,Z ñ X minimal, uniquely ergodicu is classified by K -theory.

Further, Z-stability is analogous to absorbing R in a verystriking way. Recall that a II1 factor M has M – M bR if andonly if Mk ãÑ Mω XM 1.

For separable A, A – Ab Z if and only if the centralsequence algebra Aω X A1 is sufficiently non-commutative.For separable unital A, A – Ab Z if and only if, for somek ě 2, there exists a “large” cpc order zero mapφ : Mk Ñ Aω X A1.Losely, large means that 1A´φp1k q is dominated by φpe11q.

Further, Z-stability is analogous to absorbing R in a verystriking way. Recall that a II1 factor M has M – M bR if andonly if Mk ãÑ Mω XM 1.

For separable A, A – Ab Z if and only if the centralsequence algebra Aω X A1 is sufficiently non-commutative.For separable unital A, A – Ab Z if and only if, for somek ě 2, there exists a “large” cpc order zero mapφ : Mk Ñ Aω X A1.Losely, large means that 1A´φp1k q is dominated by φpe11q.

Further, Z-stability is analogous to absorbing R in a verystriking way. Recall that a II1 factor M has M – M bR if andonly if Mk ãÑ Mω XM 1.

For separable A, A – Ab Z if and only if the centralsequence algebra Aω X A1 is sufficiently non-commutative.For separable unital A, A – Ab Z if and only if, for somek ě 2, there exists a “large” cpc order zero mapφ : Mk Ñ Aω X A1.Losely, large means that 1A´φp1k q is dominated by φpe11q.

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 – Ab Z.

ð is Rørdam.Strict comparison is a condition which says that we candetermine the order on positive elements by looking at theirranks. The analogous condition holds for all II1 factors.This would be a vast generalisation of Kirchberg’sO8-absorption theorem as when A is simple, separableand traceless Rørdam has shown:

A – Ab Z ô A – AbO8A purely infinite ô A has strict comparson

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 – Ab Z.

ð is Rørdam.Strict comparison is a condition which says that we candetermine the order on positive elements by looking at theirranks. The analogous condition holds for all II1 factors.This would be a vast generalisation of Kirchberg’sO8-absorption theorem as when A is simple, separableand traceless Rørdam has shown:

A – Ab Z ô A – AbO8A purely infinite ô A has strict comparson

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 – Ab Z.

ð is Rørdam.Strict comparison is a condition which says that we candetermine the order on positive elements by looking at theirranks. The analogous condition holds for all II1 factors.This would be a vast generalisation of Kirchberg’sO8-absorption theorem as when A is simple, separableand traceless Rørdam has shown:

A – Ab Z ô A – AbO8A purely infinite ô A has strict comparson

CURRENT STATUS

THEOREM (KIRCHBERG-RØRDAM, SATO,TOMS-W-WINTER, BUILDING ON MATUI-SATO)Let A be simple separable unital and nuclear and suppose thatT pAq is non-empty, and has a compact extreme boundary offinite covering dimension. Then

A has strict comparison ô A – Ab Z.

Proofs use interplay between von Neumann and C˚-algebrasdirectly:

For each extreme trace τ , the weak closure Mτ of A in theGNS representation is an injective II1 factor.By Connes, Mτ – Mτ bR so have Mk ãÑ Mω

τ XM 1τ .

Can use the surjectivity of Aω X A1 Ñ Mωτ XM 1

τ to obtainorder zero maps Mk Ñ Aω X A1 large wrt τ .The condition on T pAq is used to glue these together toobtain one large order zero map Mk Ñ Aω X A1.