TOPOLOGICAL OBSTRUCTIONS C OLOURING S IMPLE C - ALGEBRAS S TUART W - - PowerPoint PPT Presentation

COLOURING SIMPLE C˚-ALGEBRAS

Stuart White

University of Glasgow and WWU Münster

Abel Symposium 2015

BBSTWW := Bosa, Brown, Sato, Tikusis, W. , Winter

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 1 / 7

COLOURING VIA ORDER ZERO MAPS

VIEW POINT

Well behaved simple nuclear C˚-algebras exhibit “coloured” versions of properties of injective factors. infinite dim factor M. infinite dim simple C˚-algebra A D ˚-hms Mn Ñ M A could be projectionless

ù USE ORDER ZERO MAPS IN PLACE OF ˚-HMS

φ : A Ñ B is order zero if it is c.p. and preserves orthogonality. unital order zero maps are just ˚-homomorphisms

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

VIEW POINT

Well behaved simple nuclear C˚-algebras exhibit “coloured” versions of properties of injective factors. infinite dim factor M. infinite dim simple C˚-algebra A D ˚-hms Mn Ñ M A could be projectionless

ù USE ORDER ZERO MAPS IN PLACE OF ˚-HMS

φ : A Ñ B is order zero if it is c.p. and preserves orthogonality. unital order zero maps are just ˚-homomorphisms

COLOURING: PROPERTY EXPRESSED WITH A UNITAL ˚-HM

Replace ˚-hm by finitely many order zero maps, whose sum is unital (possibly replace exact statements by approximate statements). Number of summands ” number of colours

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

COLOURING: PROPERTY EXPRESSED WITH A UNITAL ˚-HM

Replace ˚-hm by finitely many order zero maps, whose sum is unital (possibly replace exact statements by approximate statements). Number of summands ” number of colours

HYPERFINITENESS AND NUCLEAR DIMENSION

VNA version One coloured version n-coloured version Hyperfinite AF

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

COLOURING: PROPERTY EXPRESSED WITH A UNITAL ˚-HM

Replace ˚-hm by finitely many order zero maps, whose sum is unital (possibly replace exact statements by approximate statements). Number of summands ” number of colours

HYPERFINITENESS AND NUCLEAR DIMENSION

VNA version One coloured version n-coloured version Hyperfinite AF Exist approximations M

ucp

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆

id

M

Fi

˚ hm

⑥ ⑥ ⑥

• ⑥

⑥ ⑥ ⑥

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

COLOURING: PROPERTY EXPRESSED WITH A UNITAL ˚-HM

Replace ˚-hm by finitely many order zero maps, whose sum is unital (possibly replace exact statements by approximate statements). Number of summands ” number of colours

HYPERFINITENESS AND NUCLEAR DIMENSION

VNA version One coloured version n-coloured version Hyperfinite AF Exist approximations Exist approximations M

ucp

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆

id

M

Fi

˚ hm

⑥ ⑥ ⑥

• ⑥

⑥ ⑥ ⑥

A

ccp

• ❄

❄ ❄ ❄ ❄ ❄ ❄ ❄

id

A

Fi

˚ hm

⑧ ⑧ ⑧

• ⑧

⑧ ⑧

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

COLOURING: PROPERTY EXPRESSED WITH A UNITAL ˚-HM

Replace ˚-hm by finitely many order zero maps, whose sum is unital (possibly replace exact statements by approximate statements). Number of summands ” number of colours

HYPERFINITENESS AND NUCLEAR DIMENSION

VNA version One coloured version n-coloured version Hyperfinite AF dimnucpAq ď n ´ 1 Exist approximations Exist approximations Exist approximations M

ucp

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆

id

M

Fi

˚ hm

⑥ ⑥ ⑥

• ⑥

⑥ ⑥ ⑥

A

ccp

• ❄

❄ ❄ ❄ ❄ ❄ ❄ ❄

id

A

Fi

˚ hm

⑧ ⑧ ⑧

• ⑧

⑧ ⑧

A

ccp

• ❄

❄ ❄ ❄ ❄ ❄ ❄ ❄

id

A

Fi

řn´1

j“0 ccp ord 0

⑧ ⑧

• ⑧

⑧ ⑧

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

HYPERFINITENESS AND NUCLEAR DIMENSION

VNA version One coloured version n-coloured version Hyperfinite AF dimnucpAq ď n ´ 1 Exist approximations Exist approximations Exist approximations M

ucp

• ❆

❆ ❆ ❆ ❆ ❆ ❆ ❆

id

M

Fi

˚ hm

⑥ ⑥ ⑥

• ⑥

⑥ ⑥ ⑥

A

ccp

• ❄

❄ ❄ ❄ ❄ ❄ ❄ ❄

id

A

Fi

˚ hm

⑧ ⑧ ⑧

• ⑧

⑧ ⑧

A

ccp

• ❄

❄ ❄ ❄ ❄ ❄ ❄ ❄

id

A

Fi

řn´1

j“0 ccp ord 0

⑧ ⑧

• ⑧

⑧ ⑧

SOME EXAMPLES OF COLOURINGS

Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . .

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

COLOURING VIA ORDER ZERO MAPS

SOME EXAMPLES OF COLOURINGS

hyperfinite ú AF ú dimnucpAq Rohklin Theorems ú Rohklin Poperty ú Rohklin dim McDuff (M – M b R) ú A – A b UHF ú A – A b Z a.u. equivalence ú a.u. equivalence ú ? . . . ú . . . ú . . . Why?

TOPOLOGICAL OBSTRUCTIONS

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 2 / 7

INJECTIVITY ù ñ HYPERFINITENESS

CONNES’ 3 INGREDIENTS

A (separably acting) injective II1 factor M

1

is McDuff: M – M b R (get sequence of isomorphisms φn : M b R – Ñ M such that φnpx b 1Rq Ñ x)

2

has approximately inner flip, (D sequence pvnq of unitaries in M b M with vnpx b yqv˚

n Ñ y b x);

3

has an embedding θ : M ã Ñ Rω (can assume θ represented by θn : M Ñ Fn Ă R s.t. pidM b θnqpvnq « wn, for a unitary wn)

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 3 / 7

INJECTIVITY ù ñ HYPERFINITENESS

CONNES’ 3 INGREDIENTS

A (separably acting) injective II1 factor M

1

is McDuff: M – M b R (get sequence of isomorphisms φn : M b R – Ñ M such that φnpx b 1Rq Ñ x)

2

has approximately inner flip, (D sequence pvnq of unitaries in M b M with vnpx b yqv˚

n Ñ y b x);

3

has an embedding θ : M ã Ñ Rω (can assume θ represented by θn : M Ñ Fn Ă R s.t. pidM b θnqpvnq « wn, for a unitary wn) M

1Mbθ

• pidMb1Rqω

pM b Rqω

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 3 / 7

INJECTIVITY ù ñ HYPERFINITENESS

CONNES’ 3 INGREDIENTS

A (separably acting) injective II1 factor M

1

is McDuff: M – M b R (get sequence of isomorphisms φn : M b R – Ñ M such that φnpx b 1Rq Ñ x)

2

has approximately inner flip, (D sequence pvnq of unitaries in M b M with vnpx b yqv˚

n Ñ y b x);

3

has an embedding θ : M ã Ñ Rω (can assume θ represented by θn : M Ñ Fn Ă R s.t. pidM b θnqpvnq « wn, for a unitary wn) M

1Mbθ

• pidMb1Rqω

pM b Rqω

Then px b 1Rωq « pidM b θnqpvnqp1M b θnpxqqpidM b θnqpv˚

n q

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 3 / 7

INJECTIVITY ù ñ HYPERFINITENESS

CONNES’ 3 INGREDIENTS

A (separably acting) injective II1 factor M

1

is McDuff: M – M b R (get sequence of isomorphisms φn : M b R – Ñ M such that φnpx b 1Rq Ñ x)

2

has approximately inner flip, (D sequence pvnq of unitaries in M b M with vnpx b yqv˚

n Ñ y b x);

3

has an embedding θ : M ã Ñ Rω (can assume θ represented by θn : M Ñ Fn Ă R s.t. pidM b θnqpvnq « wn, for a unitary wn) M

1Mbθ

• pidMb1Rqω

pM b Rqω

Then px b 1Rωq « pidM b θnqpvnqp1M b θnpxqqpidM b θnqpv˚

n q

« wnp1M b θnpxqqw˚

n

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 3 / 7

INJECTIVITY ù ñ HYPERFINITENESS

CONNES’ 3 INGREDIENTS

A (separably acting) injective II1 factor M

1

is McDuff: M – M b R (get sequence of isomorphisms φn : M b R – Ñ M such that φnpx b 1Rq Ñ x)

2

has approximately inner flip, (D sequence pvnq of unitaries in M b M with vnpx b yqv˚

n Ñ y b x);

3

has an embedding θ : M ã Ñ Rω (can assume θ represented by θn : M Ñ Fn Ă R s.t. pidM b θnqpvnq « wn, for a unitary wn) Then px b 1Rωq « pidM b θnqpvnqp1M b θnpxqqpidM b θnqpv˚

n q

« wnp1M b θnpxqqw˚

n

Thus x « φnpwnp1M b θnpxqqw˚

n q P φnpwnp1M b Fnqw˚ n q.

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 3 / 7

C˚-VERSIONS OF CONNES’ ARGUMENT

CONNES’ 3 INGREDIENTS

A (separably acting) injective II1 factor M

1

is McDuff: M – M b R;

2

has approximately inner flip;

3

has an embedding θ : M ã Ñ Rω.

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 4 / 7

C˚-VERSIONS OF CONNES’ ARGUMENT

CONNES’ 3 INGREDIENTS

A (separably acting) injective II1 factor M

1

is McDuff: M – M b R;

2

has approximately inner flip;

3

has an embedding θ : M ã Ñ Rω.

PROPOSITION (EFROS, ROSENBERG (78))

Let A be a separable unital C˚-algebra such that:

1

A absorbs the universal UHF algebra Q, (A – A b Q);

2

A has approximately inner flip ( ù ñ nuclearity, simplicity, at most

• ne trace).

3

A is quasidiagonal (for A sep, unital, nuclear, QD ô A unital ã Ñ Qω); Then A is AF (and in fact A – Q).

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 4 / 7

COLOURED EQUIVALENCE

K -THEORETIC OBSTRUCTIONS TO APPROX INNER FLIPS

e.g. AF with inner flip are UHF .

DEFINITION (BBSTWW 15)

Let φ, ψ : A Ñ B be unital ˚-hms. Say φ and ψ are n-coloured equivalent if Du1, ¨ ¨ ¨ , un P B such that: φpxq “ řn

i“1 uiψpxqu˚ i ,

ψpxq “ řn

i“1 u˚ i φpxqui

ru˚

i ui, ψpAqs “ ruiu˚ i , φpAqs “ 0

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 5 / 7

COLOURED EQUIVALENCE

DEFINITION (BBSTWW 15)

Let φ, ψ : A Ñ B be unital ˚-hms. Say φ and ψ are n-coloured equivalent if Du1, ¨ ¨ ¨ , un P B such that: φpxq “ řn

i“1 uiψpxqu˚ i ,

ψpxq “ řn

i“1 u˚ i φpxqui

ru˚

i ui, ψpAqs “ ruiu˚ i , φpAqs “ 0

THEOREM (MATUI, SATO 13)

Let A be simple, unital, separable, nuclear, UHF-stable with unique

• trace. Then A has a 2-coloured approx inner flip.

If A also QD, then decomposition rank A at most 1 Simple, unital, separable, unclear, QD, Z-stable C˚-algebras with unique trace have decomposition rank at most 3.

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 5 / 7

COLOURED EQUIVALENCE

DEFINITION (BBSTWW 15)

Let φ, ψ : A Ñ B be unital ˚-hms. Say φ and ψ are n-coloured equivalent if Du1, ¨ ¨ ¨ , un P B such that: φpxq “ řn

i“1 uiψpxqu˚ i ,

ψpxq “ řn

i“1 u˚ i φpxqui

ru˚

i ui, ψpAqs “ ruiu˚ i , φpAqs “ 0

THEOREM (BBSTWW, 15)

Let A be unital, sep and nuclear. Let B be unital, simple, separable, Z-stable such that QTpBq “ TpBq and TpBq has compact boundary. Let φ, ψ : A Ñ Bω be unital ˚-hms such that φpaq, ψpaq is full in Bω for each non-zero a P A`.Then φ, ψ 2-coloured equivalent ô @τ P TpBωq, τ ˝ φ “ τ ˝ ψ Coloured version of φ, ψ :

inj, sep predual

B Ñ

II1 factor

Mω are unitarily equivalent iff τMω ˝ φ “ τMω.

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 5 / 7

COLOURED QUASIDIAGONALITY

THEOREM (VOICULESCU)

Quasidiagonality a homotopy invariant. The cone C0pp0, 1s, Aq on any C˚-algebra is QD. Exist order zero maps A Ñ Qω.

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 6 / 7

COLOURED QUASIDIAGONALITY

THEOREM (VOICULESCU)

Quasidiagonality a homotopy invariant. The cone C0pp0, 1s, Aq on any C˚-algebra is QD. Exist order zero maps A Ñ Qω.

THEOREM (SATO, W., WINTER 14)

Let A be a nuclear C˚-algebra and τA a trace on A. Then there exists an order zero map φ : A Ñ Qω, which is a ˚-hm modulo traces and τA “ τQω ˝ Φ. Uses injectivity ù ñ hyperfiniteness in an essential way. Can assemble two of these maps to get “2-coloured quasidiagonality of τA”. Z-stable, simple, separable, unital, nuclear C˚-algebras with unique trace have nuclear dimension ď 3.

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 6 / 7

COLOURED QUASIDIAGONALITY

THEOREM (SATO, W., WINTER 14)

Let A be a nuclear C˚-algebra and τA a trace on A. Then there exists an order zero map φ : A Ñ Qω, which is a ˚-hm modulo traces and τA “ τQω ˝ Φ. Combining coloured quasidiagonality and coloured equivalence, and counting the colours very carefully gives:

THEOREM (BBSTWW, ’15)

Let A be simple, separable, unital, nuclear, Z-stable such that TpAq has compact boundary.

1

dimnucpAq “ 1 (unless A is AF , when it is zero);

2

If all traces on A are quasidiagonal, then A has decomposition rank 1 (unless A is AF , when it is zero).

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 6 / 7

WHEN 2 BECOME 1

COLOURS CIRCUMVENT TOPOLOGICAL OBSTRUCTIONS

When there are no topological obstructions, do we really need colours?

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 7 / 7

WHEN 2 BECOME 1

COLOURS CIRCUMVENT TOPOLOGICAL OBSTRUCTIONS

When there are no topological obstructions, do we really need colours?

CONJECTURE (BLACKADAR, KIRCHBERG)

Stably finite unital nuclear C˚-algebras are quasidiagonal.

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 7 / 7

WHEN 2 BECOME 1

COLOURS CIRCUMVENT TOPOLOGICAL OBSTRUCTIONS

When there are no topological obstructions, do we really need colours?

CONJECTURE (BLACKADAR, KIRCHBERG)

Stably finite unital nuclear C˚-algebras are quasidiagonal.

THEOREM (TIKUISIS, W., WINTER)

A faithful trace τA on separable, unital and nuclear C˚-algebra A in the UCT class is quasidigaonal: D ˚-hm Φ : A Ñ Qω such that τA “ τQω ˝ Φ.

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 7 / 7

WHEN 2 BECOME 1

COLOURS CIRCUMVENT TOPOLOGICAL OBSTRUCTIONS

When there are no topological obstructions, do we really need colours?

CONJECTURE (BLACKADAR, KIRCHBERG)

Stably finite unital nuclear C˚-algebras are quasidiagonal.

THEOREM (TIKUISIS, W., WINTER)

A faithful trace τA on separable, unital and nuclear C˚-algebra A in the UCT class is quasidigaonal: D ˚-hm Φ : A Ñ Qω such that τA “ τQω ˝ Φ. Discrete amenable groups have quasidiagonal C˚-algebras (answering a question of Rosenberg).

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 7 / 7

WHEN 2 BECOME 1

COLOURS CIRCUMVENT TOPOLOGICAL OBSTRUCTIONS

When there are no topological obstructions, do we really need colours?

CONJECTURE (BLACKADAR, KIRCHBERG)

Stably finite unital nuclear C˚-algebras are quasidiagonal.

THEOREM (TIKUISIS, W., WINTER)

A faithful trace τA on separable, unital and nuclear C˚-algebra A in the UCT class is quasidigaonal: D ˚-hm Φ : A Ñ Qω such that τA “ τQω ˝ Φ. Discrete amenable groups have quasidiagonal C˚-algebras (answering a question of Rosenberg). Answers Blackadar-Kirchberg in UCT case with a faithful trace

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 7 / 7

WHEN 2 BECOME 1

COLOURS CIRCUMVENT TOPOLOGICAL OBSTRUCTIONS

When there are no topological obstructions, do we really need colours?

CONJECTURE (BLACKADAR, KIRCHBERG)

Stably finite unital nuclear C˚-algebras are quasidiagonal.

THEOREM (TIKUISIS, W., WINTER)

A faithful trace τA on separable, unital and nuclear C˚-algebra A in the UCT class is quasidigaonal: D ˚-hm Φ : A Ñ Qω such that τA “ τQω ˝ Φ. Discrete amenable groups have quasidiagonal C˚-algebras (answering a question of Rosenberg). Answers Blackadar-Kirchberg in UCT case with a faithful trace Completes classification of simple, separable, unital, nuclear C˚-algebras of finite nuclear dimension with the UCT (via Elliott, Gong, Lin, Niu).

STUART WHITE (GLASGOW) COLOURING SIMPLE C˚-ALGEBRAS 7 / 7