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 Γ op = ( V , E op ) with E op = ( V × V ) \ ( E ∪ { ( v , v ) | v ∈ V } ) . We call Γ co-irreducible when Γ op 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 Γ op = ( V , E op ) with E op = ( V × V ) \ ( E ∪ { ( v , v ) | v ∈ V } ) . We call Γ co-irreducible when Γ op is irreducible, and for non-co-irreducible graphs consider co-irreducible components: Γ = Γ 1 ∗ Γ 2 ∗ · · · ∗ Γ n 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) � ( − 1 ) | K | χ (Γ) = K Γ -simplex 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 � s v s w = s w s v ( v , w ) ∈ E � � � � C ∗ ( A + Γ ) = C ∗ s v s ∗ w = s ∗ { s v } v ∈ V w s v ( v , w ) ∈ E . � � s ∗ � v s w = δ v , 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 � s v s w = s w s v ( v , w ) ∈ E � � � � C ∗ ( A + Γ ) = C ∗ s v s ∗ w = s ∗ { s v } v ∈ V w s v ( v , w ) ∈ E . � � s ∗ � v s w = δ v , w · 1 ( v , w ) / ∈ E � Observation C ∗ ( A + Γ ) = C ∗ ( A + Γ 1 ) ⊗ C ∗ ( A + Γ 2 ) ⊗ · · · ⊗ C ∗ ( A + Γ n ) when Γ = Γ 1 ∗ Γ 2 ∗ · · · ∗ Γ n . 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 A Γ 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 + � O | χ (Γ) | + 1 � 0 0 Γ ) with K -theory χ (Γ) � Z � Z /χ (Γ) Z . 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 + � O 1 � 0 0 Γ ) with K -theory 0 � Z � Z . Z Z Here O 1 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 � E � Q � 0 � K 0 with Q a UCT Kirchberg algebra are classified by their six-term exact sequence when moreover • K ∗ ( Q ) finitely generated • K 1 ( Q ) free • rank K 1 ( Q ) ≤ rank K 0 ( 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 + ⇒ χ (Γ) = χ (Γ ′ ) 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 = Γ 1 ∗ Γ 2 ∗ · · · ∗ Γ n , define t (Γ) = # { i | | Γ i | = 1 } N k (Γ) = # { i | χ (Γ i ) = 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 = Γ 1 ∗ Γ 2 ∗ · · · ∗ Γ n , define t (Γ) = # { i | | Γ i | = 1 } N k (Γ) = # { i | χ (Γ i ) = 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 N k (Γ) + N − k (Γ) = N k (Γ ′ ) + N − k (Γ ′ ) for all k k > 0 N k (Γ ′ ) mod 2 3 N 0 (Γ) > 0 or � k > 0 N k (Γ) ≡ � 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 N 0 = 1 N 0 = 1 N 0 = 1 N 0 = 1 N 0 = 1 N 0 = 1 N 1 = 1 N 1 = 1 N 1 = 1 N − 3 = 1 N − 2 = 1 N − 2 = 1 N − 1 = 2 N − 1 = 1 N − 1 = 1 N 1 = 1 t = 1 N − 1 = 1 t = 1 t = 1 t = 1 t = 1 N 0 = 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 E 2 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