Injectivity hyperfiniteness Connes 3 ingredients A (separably - - PDF document

COLOURING SIMPLE C˚-ALGEBRAS

Stuart White, University of Glasgow and WWU Münster. 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

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 ⑧ 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 . . .

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

1

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˚

nq

« wnp1M b θnpxqqw˚

n

Thus x « φnpwnp1M b θnpxqqw˚

nq P φnpwnp1M b Fnqw˚ nq.

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 1 (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 one 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). 2

Coloured equivalence K-theoretic obstructions to approx inner flips e.g. AF with inner flip are UHF. Definition 1 (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. 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. 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. 3

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). 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 ques-

tion of Rosenberg).

• Answers Blackadar-Kirchberg in UCT case with a faithful trace
• Completes classification of simple, separable, unital, nuclear C˚-algebras of fi-

nite nuclear dimension with the UCT (via Elliott, Gong, Lin, Niu). 4