The C -algebras of right-angled ArtinTits monoids Sren Eilers - - PowerPoint PPT Presentation

c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

university of copenhagen

The C ∗-algebras of right-angled Artin–Tits monoids

Søren Eilers

Centre for Symmetry and Deformation

SYM lecture, March 19, 2014 Slide 1/15

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

The Elliott Program

Goal

Classify nuclear C∗-algebras by K-theoretical invariants.

Slide 2/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

The Elliott Program

Goal

Classify nuclear C∗-algebras by K-theoretical invariants.

Progress bars

Simple C∗-algebras 91% Purely infinite C∗-algebras 57% C∗-algebras with finitely many ideals 8%

Slide 2/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

Graphs

We work with finite, simple, undirected graphs with no loops and call them Γ = (V , E), Γ′ = (V ′, E ′).

Definition

For Γ = (V , E) we let Γop = (V , E op) with E op = (V × V )\(E ∪ {(v, v) | v ∈ V }). We call Γ co-irreducible when Γop is irreducible.

Slide 3/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

Graphs

We work with finite, simple, undirected graphs with no loops and call them Γ = (V , E), Γ′ = (V ′, E ′).

Definition

For Γ = (V , E) we let Γop = (V , E op) with E op = (V × V )\(E ∪ {(v, v) | v ∈ V }). We call Γ co-irreducible when Γop is irreducible, and for non-co-irreducible graphs consider co-irreducible components: Γ = Γ1∗ Γ2∗ · · ·∗ Γn

Slide 3/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

Graphs (cont)

Examples

Slide 4/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

Graphs (cont)

Examples Definition (Euler characteristic)

χ(Γ) =

• K Γ-simplex

(−1)|K|

Slide 4/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

C ∗-algebras of Artin-Tits monoids

Definition (Crisp–Laca ’02)

Let Γ be a graph. The C∗-algebra associated to the Artin-Tits monoid of Γ is C∗(A+

Γ ) = C∗

• {sv}v∈V
• svsw = swsv

(v, w) ∈ E svs∗

w = s∗ wsv

(v, w) ∈ E s∗

v sw = δv,w · 1

(v, w) / ∈ E

• .

Slide 5/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

C ∗-algebras of Artin-Tits monoids

Definition (Crisp–Laca ’02)

Let Γ be a graph. The C∗-algebra associated to the Artin-Tits monoid of Γ is C∗(A+

Γ ) = C∗

• {sv}v∈V
• svsw = swsv

(v, w) ∈ E svs∗

w = s∗ wsv

(v, w) ∈ E s∗

v sw = δv,w · 1

(v, w) / ∈ E

• .

Observation

C∗(A+

Γ ) = C∗(A+ Γ1) ⊗ C∗(A+ Γ2) ⊗ · · · ⊗ C∗(A+ Γn)

when Γ = Γ1∗ Γ2∗ · · ·∗ Γn.

Slide 5/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

C ∗-algebras of Artin-Tits monoids (cont.)

Theorem (Cuntz–Echterhoff–Li)

For any Γ, K∗(C∗(A+

Γ )) = Z ⊕ 0.

Proof.

The Baum–Connes conjecture holds for the group AΓ since it has the Haagerup property.

Slide 6/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

The co-irreducible case

When Γ is co-irreducible with |Γ| > 1 and χ(Γ) = 0, we have

K C∗(A+

Γ )

O|χ(Γ)|+1

with K-theory Z

χ(Γ) Z

Z/χ(Γ)Z.

Slide 7/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

The co-irreducible case 2

When Γ is co-irreducible with |Γ| > 1 and χ(Γ) = 0, we have

K C∗(A+

Γ )

O1

with K-theory Z

Z Z.

Z

• Here O1 is the unique unital Kirchberg algebra with the

indicated K-theory and [1] = 1.

Slide 8/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

Classifying C ∗-algebras with 1 ideal

Progress bars

Stable, purely infinite 98% [Rørdam ’94] Unital, purely infinite 98% [E–Restorff ’04] Stable, mixed AF/PI 41% [E–Restorff–Ruiz ’09] Unital, mixed AF/PI 7%

Slide 9/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

Theorem (E–Restorff–Ruiz)

Unital C∗-algebras E of the form

K E Q

with Q a UCT Kirchberg algebra are classified by their six-term exact sequence when moreover

• K∗(Q) finitely generated
• K1(Q) free
• rank K1(Q) ≤ rank K0(Q)

Slide 10/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

The co-irreducible case 3

Theorem (E–Li–Ruiz)

When Γ, Γ′ are co-irreducible with |Γ|, |Γ′| > 1 we have C∗(A+

Γ ) ≃ C∗(A+ Γ′) ⇐

⇒ χ(Γ) = χ(Γ′)

Slide 11/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

Classifying C ∗-algebras with finitely many ideals

Progress bars

Stable, purely infinite 32% Unital, purely infinite 15%

[Arklint, Bentmann, Katsura, Köhler, Meyer, Nest, Restorff, Ruiz]

Stable, mixed AF/PI 3% Unital, mixed AF/PI 1%

Slide 12/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

The general case

Definition

When Γ = Γ1 ∗ Γ2 ∗ · · ·∗ Γn, define t(Γ) = #{i | |Γi| = 1} Nk(Γ) = #{i | χ(Γi) = k}

Slide 13/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

The general case

Definition

When Γ = Γ1 ∗ Γ2 ∗ · · ·∗ Γn, define t(Γ) = #{i | |Γi| = 1} Nk(Γ) = #{i | χ(Γi) = k}

Theorem (E–Li–Ruiz)

For general graphs Γ, Γ′ we have C∗(A+

Γ ) ≃ C∗(A+ Γ′)

precisely when

1 t(Γ) = t(Γ′) 2 Nk(Γ) + N−k(Γ) = Nk(Γ′) + N−k(Γ′) for all k 3 N0(Γ) > 0 or k>0 Nk(Γ) ≡ k>0 Nk(Γ′) mod 2

Slide 13/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

n = 5

N−4 = 1 N−3 = 1 N−2 = 1 N−2 = 1 N−2 = 1 N−1 = 1 N−1 = 1 N−1 = 1 N−1 = 1 N−1 = 1 N−1 = 1 N−1 = 1 N0 = 1 N0 = 1 N0 = 1 N0 = 1 N0 = 1 N0 = 1 N1 = 1 N1 = 1 N1 = 1 N−3 = 1 N−2 = 1 N−2 = 1 N−1 = 2 N−1 = 1 N−1 = 1 N1 = 1 t = 1 N−1 = 1 t = 1 t = 1 t = 1 t = 1 N0 = 1 N−2 = 1 N−1 = 2 N−1 = 1 N−1 = 1 t = 5 t = 1 t = 2 t = 1 t = 2 t = 3

Slide 14/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

Semiprojectivity

It is well known that T and E2 are semiprojective. But

Observation (Enders)

T ⊗ A is only semiprojective when A is finite-dimensional.

Slide 15/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

Semiprojectivity

It is well known that T and E2 are semiprojective. But

Observation (Enders)

T ⊗ A is only semiprojective when A is finite-dimensional.

Theorem

When t(Γ) = 0, C∗(A+

Γ ) is semiprojective if

• |k|=1

Nk ≤ 1

Slide 15/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014

u n i v e r s i t y o f c o p e n h a g e n c e n t r e f o r s y m m e t r y a n d d e f o r m a t i o n

Semiprojectivity

It is well known that T and E2 are semiprojective. But

Observation (Enders)

T ⊗ A is only semiprojective when A is finite-dimensional.

Theorem

When t(Γ) = 0, C∗(A+

Γ ) is semiprojective if

• |k|=1

Nk ≤ 1 This is exactly the extent of our knowledge in this direction.

Slide 15/15 — Søren Eilers — The C∗-algebras of right-angled Artin–Tits monoids — SYM lecture, March 19, 2014