uhf slicing and classification of nuclear c*-algebrascstar/workshop/slides/strung.pdfuhf slicing and...

UHF slicing and classification of nuclear C*-algebras

Karen R. Strung (joint work with Wilhelm Winter)

Mathematisches InstitutUniversitat Munster

Workshop on C*-Algebras andNoncommutative Dynamics

Sde Boker, IsraelMarch 2013

Karen R. Strung (University of Munster) UHF slicing March 14, 2013 1 / 18

Let A be the class of separable nuclear unital simple C ∗-algebras satisfying

1 A ∈ A =⇒ A is locally recursive subhomogeneous (RSH) where theRSH algebras can be chosen so that projections can be lifted along an(F , η)-connected decomposition,

2 A ∈ A =⇒ T (A) has finitely many extreme points, each of whichinduce the same state on K0(A).

Theorem (S.–Winter)

Let A,B ∈ A. Then

A⊗Z ∼= B ⊗Z ⇐⇒ Ell(A⊗Z) ∼= Ell(B ⊗Z)

Corollary

Let A,B ∈ A and suppose that A and B have finite decomposition rank.Then

A ∼= B ⇐⇒ Ell(A) ∼= Ell(B)

Let A,B ∈ A. Then

A⊗Z ∼= B ⊗Z ⇐⇒ Ell(A⊗Z) ∼= Ell(B ⊗Z)

Corollary

Let A,B ∈ A and suppose that A and B have finite decomposition rank.Then

A ∼= B ⇐⇒ Ell(A) ∼= Ell(B)

Key tools

1 Tensor with a UHF algebra to care of the lack of projections.UHF-stable classification can (often) be used to deduce Z-stableclassification (eg. Winter, Lin).

2 Tracial approximation for A⊗Q, for the universal UHF algebra Q(i.e. K0(Q) ∼= Q).

We will show that A ∈ A =⇒ A⊗Q is a tracially approximately intervalalgebra (TAI).

Then (Lin, 2009) =⇒ classification.

Key tools

Tracial approximationA is tracially approximately S:

F ⊂ε

(1−p)A(1−p)

p = 1B , B ∈ S

x ∈ A then x ≈ pxp + (1− p)x(1− p)whereτ(1− p) < ε for every τ ∈ T (A)andpxp ∈ε B.

I = (⊕Kk=1C ([0, 1])⊗Mnk

)⊕ (⊕Ll=1Mnl

F ⊂ε

(1−p)A(1−p)

p = 1B , B ∈ S

x ∈ A then x ≈ pxp + (1− p)x(1− p)

whereτ(1− p) < ε for every τ ∈ T (A)andpxp ∈ε B.

I = (⊕Kk=1C ([0, 1])⊗Mnk

)⊕ (⊕Ll=1Mnl

F ⊂ε

(1−p)A(1−p)

p = 1B , B ∈ S

x ∈ A then x ≈ pxp + (1− p)x(1− p)whereτ(1− p) < ε for every τ ∈ T (A)

andpxp ∈ε B.

I = (⊕Kk=1C ([0, 1])⊗Mnk

)⊕ (⊕Ll=1Mnl

F ⊂ε

(1−p)A(1−p)

p = 1B , B ∈ S

I = (⊕Kk=1C ([0, 1])⊗Mnk

)⊕ (⊕Ll=1Mnl

F ⊂ε

(1−p)A(1−p)

p = 1B , B ∈ S

I = (⊕Kk=1C ([0, 1])⊗Mnk

)⊕ (⊕Ll=1Mnl

Main theorem

Let A ∈ A. Then A⊗Q is TAI.

Recall: A is the class of separable nuclear unital simple C ∗-algebrassatisfying

1 A ∈ A =⇒ A is locally recursive subhomoeneous (RSH) where theRSH algebras can be chosen so that projections can be lifted along an(F , η)-connected decomposition,

Main theorem

Let A ∈ A. Then A⊗Q is TAI.

Recall: A is the class of separable nuclear unital simple C ∗-algebrassatisfying

Recursive subhomogeneous C ∗-algebras

[Phillips 2001] B is RSH if it can be written as an iterated pullback

B =(. . .((

C0 ⊕C(0)1

). . .)⊕

whereCl = C (Xl)⊗Mnl

for some compact metrizable Xl and

C(0)l = C (Ωl)⊗Mnl

for a closed subset Ωl ⊂ Xl .

Recursive subhomogeneous C ∗-algebras

The l th stage Bl is given by

Bl = Bl−1 ⊕C(0)l

Cl = (b, c) ∈ Bl−1 ⊕ Cl | φ(b) = ρ(c)

whereφ : Bl−1 → C

is a unital ∗-homomorphism, and

ρ : Cl → C(0)l

is the restriction map.

The decomposition is not unique, so we keep track of it:

[Bl ,Xl ,Ωl , nl , φl ]Rl=1.

We say that projections can be lifted along this decomposition if:

∀n ∈ N, ∀l = 1, . . . ,R − 1 and for every projection p ∈ Bl ⊗Mn, thereexists a projection p ∈ Bl+1 ⊗Mn lifting p.

Proposition

If dim(Xl) ≤ 1 for l = 2, . . . ,R then projections can be lifted along[Bl ,Xl ,Ωl , nl , φl ]

Proposition

Idea of proof (A ∈ A =⇒ A TAI):

Given F ⊂⊂ A⊗Q, ε > 0, need C ∈ I with τ(1C ) bounded away from 0,∀τ ,

and 1C commutes up to ε with f ∈ F and that approximates 1C F 1C

up to ε.

Assume τ0, τ1 are the only extreme tracial states.

W.L.O.G., assume F = F0⊗1Q with F0 ⊂⊂ RSH algebra B.

Find a tracially large interval: Take a ∈ (A⊗Q)+ with τ0(a) ≈ 0 andτ(a) ≈ 1 (Brown–Toms 2007), then take C ∗(a, 1).

Must move this interval into position (w.r.t. F): model an interval inB ⊗Q, use strict comparison.

Given F ⊂⊂ A⊗Q, ε > 0, need C ∈ I with τ(1C ) bounded away from 0,∀τ , and 1C commutes up to ε with f ∈ F

and that approximates 1C F 1C

up to ε.

Given F ⊂⊂ A⊗Q, ε > 0, need C ∈ I with τ(1C ) bounded away from 0,∀τ , and 1C commutes up to ε with f ∈ F and that approximates 1C F 1C

up to ε.

Interval model: excisors and bridges

Definition

B unital RSH with decomposition [Bl ,Xl ,Ωl , nl , φl ], F ⊂⊂ B+1 , η > 0. An

(F , η)-excisor (E , ρ, σ, κ) consists of

1 a finite dimensional algebra E = ⊕Rl=1El ,

2 a unital ∗-homomorphism ρ = ⊕Rl=1ρl : B → ⊕R

3 an isometric c.p. order zero map σ = ⊕Rl=1σl : ⊕R

l=1El → B ⊗Q suchthat

‖σ(1E )(b ⊗ 1Q) = σ ρ(b)‖ < η for all b ∈ F ,

4 a unital ∗-homomorphism κ : E → Q.

Definition

l=1El → B ⊗Q

suchthat

Definition

We say that (E , ρ, σ, κ) is compatible with the RSH decomposition if eachρl factorizes through

ρl // El

Blψl // C(Xl)⊗Mrl

for some compact Xl ⊂ Xl \ Ωl .

Definition

An (F , η)-bridge between (E0, ρ0, σ0, κ0) and (E1, ρ1, σ1, κ1) consists ofK ∈ N and (F , η)-excisors (Ej/K , ρj/K , σj/K , κj/K ), j = 1, . . . ,K − 1satisfying

‖κj/K ρj/K (b)− κ(j+1)/K ρ(j+1)/K (b)‖ < η

for all b ∈ F and j = 0, . . . ,K − 1.

In this case, write (E0, ρ0, σ0, κ0) ∼(F ,η) (E1, ρ1, σ1, κ1).

Definition

An (F , η)-bridge between (E0, ρ0, σ0, κ0) and (E1, ρ1, σ1, κ1) consists ofK ∈ N and (F , η)-excisors (Ej/K , ρj/K , σj/K , κj/K ), j = 1, . . . ,K − 1satisfying

‖κj/K ρj/K (b)− κ(j+1)/K ρ(j+1)/K (b)‖ < η

for all b ∈ F and j = 0, . . . ,K − 1.

In this case, write (E0, ρ0, σ0, κ0) ∼(F ,η) (E1, ρ1, σ1, κ1).

(F , η)-connected decomposition

[Bl ,Xl ,Ωl , nl , φl ]Rl=1 the RSH decomposition.

For every l = 1, . . . ,R and every x ∈ Xl we can define an (F , η)-excisor.

The decomposition is (F , η)-connected if, for any l = 1, . . . ,R and anyx , y ∈ Xl , we can always find an (F , η)-bridge between their corresponding(F , η)-excisors.

(F , η)-connected decomposition

[Bl ,Xl ,Ωl , nl , φl ]Rl=1 the RSH decomposition.

For every l = 1, . . . ,R and every x ∈ Xl we can define an (F , η)-excisor.

The decomposition is (F , η)-connected if, for any l = 1, . . . ,R and anyx , y ∈ Xl , we can always find an (F , η)-bridge between their corresponding(F , η)-excisors.

Interval model: large trace

With a suitable calculus for (F , η)-excisors, we can find (E0, ρ0, σ0, κ0) and(E1, ρ0, σ0, κ0) with τi (σ(1Ei

)) large and τi (σj(1Ej)) small, i 6= j ∈ 0, 1.

It remains to find an (F , η)-bridge through the decomposition.

To do this, we use linear algebra based on equations which we can read offthe RSH decomposition.

This is where we require that projections can be lifted and that eachtracial state induces the same state on K0.

What we get...

With two extreme tracial states τ0, τ1, we find (F , η)-excisors

(E0, ρ0, σ0, κ0) ∼(F ,η) (E1, ρ1, σ1, κ)

such that(τi ⊗ τQ)(σi (1Ei

)) ≥ 1/3.

Now use strict comparison to move the elements in a partition of unity ofthe actual interval under this model.This will be an interval which is large in trace, and the condition

for (F , η)-excisors allows us to properly approximate elements in the finitesubset.

A⊗Q is TAI.

What we get...

(E0, ρ0, σ0, κ0) ∼(F ,η) (E1, ρ1, σ1, κ)

)) ≥ 1/3.

A⊗Q is TAI.

What we get...

(E0, ρ0, σ0, κ0) ∼(F ,η) (E1, ρ1, σ1, κ)

)) ≥ 1/3.

Now use strict comparison to move the elements in a partition of unity ofthe actual interval under this model.

This will be an interval which is large in trace, and the condition

A⊗Q is TAI.

What we get...

(E0, ρ0, σ0, κ0) ∼(F ,η) (E1, ρ1, σ1, κ)

)) ≥ 1/3.

A⊗Q is TAI.

What we get...

(E0, ρ0, σ0, κ0) ∼(F ,η) (E1, ρ1, σ1, κ)

)) ≥ 1/3.

A⊗Q is TAI.

Main theorem

Let A,B ∈ A. Then

A⊗Z ∼= B ⊗Z ⇐⇒ Ell(A⊗Z) ∼= Ell(B ⊗Z).

Where A is the class of separable nuclear unital simple C ∗-algebrassatisfying

Main theorem

Let A,B ∈ A. Then

A⊗Z ∼= B ⊗Z ⇐⇒ Ell(A⊗Z) ∼= Ell(B ⊗Z).

Where A is the class of separable nuclear unital simple C ∗-algebrassatisfying

Some consequences

1. Elliott 1996 – Simple approximately SH algebras constructed byattaching 1-dimensional spaces to the circle. Theorem =⇒ classificationwhen restricted to finitely many extreme tracial states, each inducing sameK0-state.

2. Lin–Matui 2005 – A := C (X × T) o Z. Restricting to finitely manytraces each inducing same state on K0, theorem =⇒ Ax ⊗Q is TAI,then (S.-Winter 2010) =⇒ A⊗Q TAI =⇒ classification.

Some consequences

1. Elliott 1996 – Simple approximately SH algebras constructed byattaching 1-dimensional spaces to the circle. Theorem =⇒ classificationwhen restricted to finitely many extreme tracial states, each inducing sameK0-state.

2. Lin–Matui 2005 – A := C (X × T) o Z. Restricting to finitely manytraces each inducing same state on K0, theorem =⇒ Ax ⊗Q is TAI,then (S.-Winter 2010) =⇒ A⊗Q TAI =⇒ classification.

uhf slicing and classification of nuclear c*-algebrascstar/workshop/slides/strung.pdfuhf slicing and...

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