UHF slicing and classification of nuclear C*-algebras Karen R. - PowerPoint PPT Presentation

UHF slicing and classification of nuclear C*-algebras Karen R. Strung (joint work with Wilhelm Winter) Mathematisches Institut Universit¨ at M¨ unster Workshop on C*-Algebras and Noncommutative Dynamics Sde Boker, Israel March 2013 Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 1 / 18

Let A be the class of separable nuclear unital simple C ∗ -algebras satisfying Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 2 / 18

Let A be the class of separable nuclear unital simple C ∗ -algebras satisfying 1 A ∈ A = ⇒ A is locally recursive subhomogeneous (RSH) where the RSH algebras can be chosen so that projections can be lifted along an ( F , η )-connected decomposition, Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 2 / 18

Let A be the class of separable nuclear unital simple C ∗ -algebras satisfying 1 A ∈ A = ⇒ A is locally recursive subhomogeneous (RSH) where the RSH 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 which induce the same state on K 0 ( A ). Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 2 / 18

Theorem (S.–Winter) Let A , B ∈ A . Then A ⊗ Z ∼ ⇒ Ell( A ⊗ Z ) ∼ = B ⊗ Z ⇐ = Ell( B ⊗ Z ) Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 3 / 18

Theorem (S.–Winter) Let A , B ∈ A . Then A ⊗ Z ∼ ⇒ Ell( A ⊗ Z ) ∼ = B ⊗ Z ⇐ = Ell( B ⊗ Z ) Corollary Let A , B ∈ A and suppose that A and B have finite decomposition rank. Then A ∼ ⇒ Ell( A ) ∼ = B ⇐ = Ell( B ) Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 3 / 18

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 -stable classification (eg. Winter, Lin). Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 4 / 18

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 -stable classification (eg. Winter, Lin). 2 Tracial approximation for A ⊗ Q , for the universal UHF algebra Q (i.e. K 0 ( Q ) ∼ = Q ). Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 4 / 18

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 -stable classification (eg. Winter, Lin). 2 Tracial approximation for A ⊗ Q , for the universal UHF algebra Q (i.e. K 0 ( Q ) ∼ = Q ). We will show that A ∈ A = ⇒ A ⊗ Q is a tracially approximately interval algebra (TAI). Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 4 / 18

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 -stable classification (eg. Winter, Lin). 2 Tracial approximation for A ⊗ Q , for the universal UHF algebra Q (i.e. K 0 ( Q ) ∼ = Q ). We will show that A ∈ A = ⇒ A ⊗ Q is a tracially approximately interval algebra (TAI). Then (Lin, 2009) = ⇒ classification. Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 4 / 18

Tracial approximation A is tracially approximately S :  (1 − p ) A (1 − p )    F ⊂ ǫ   B   p = 1 B , B ∈ S Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 5 / 18

Tracial approximation A is tracially approximately S :  (1 − p ) A (1 − p )    F ⊂ ǫ   B   p = 1 B , B ∈ S x ∈ A then x ≈ pxp + (1 − p ) x (1 − p ) Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 5 / 18

Tracial approximation A is tracially approximately S :  (1 − p ) A (1 − p )    F ⊂ ǫ   B   p = 1 B , B ∈ S x ∈ A then x ≈ pxp + (1 − p ) x (1 − p ) where τ (1 − p ) < ǫ for every τ ∈ T ( A ) Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 5 / 18

Tracial approximation A is tracially approximately S :  (1 − p ) A (1 − p )    F ⊂ ǫ   B   p = 1 B , B ∈ S x ∈ A then x ≈ pxp + (1 − p ) x (1 − p ) where τ (1 − p ) < ǫ for every τ ∈ T ( A ) and pxp ∈ ǫ B . Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 5 / 18

Tracial approximation A is tracially approximately S :  (1 − p ) A (1 − p )    F ⊂ ǫ   B   p = 1 B , B ∈ S x ∈ A then x ≈ pxp + (1 − p ) x (1 − p ) where τ (1 − p ) < ǫ for every τ ∈ T ( A ) and pxp ∈ ǫ B . I = { ( ⊕ K k =1 C ([0 , 1]) ⊗ M n k ) ⊕ ( ⊕ L l =1 M n l ) } Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 5 / 18

Main theorem Theorem (S.–Winter) Let A ∈ A . Then A ⊗ Q is TAI. Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 6 / 18

Main theorem Theorem (S.–Winter) Let A ∈ A . Then A ⊗ Q is TAI. Recall: A is the class of separable nuclear unital simple C ∗ -algebras satisfying 1 A ∈ A = ⇒ A is locally recursive subhomoeneous (RSH) where the RSH 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 which induce the same state on K 0 ( A ). Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 6 / 18

Recursive subhomogeneous C ∗ -algebras [Phillips 2001] B is RSH if it can be written as an iterated pullback � �� � � � B = . . . C 0 ⊕ C (0) C 1 ⊕ C (0) C 2 . . . ⊕ C (0) R C R , 1 2 where C l = C ( X l ) ⊗ M n l for some compact metrizable X l and C (0) = C (Ω l ) ⊗ M n l l for a closed subset Ω l ⊂ X l . Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 7 / 18

Recursive subhomogeneous C ∗ -algebras The l th stage B l is given by B l = B l − 1 ⊕ C (0) C l = { ( b , c ) ∈ B l − 1 ⊕ C l | φ ( b ) = ρ ( c ) } l where φ : B l − 1 → C (0) l is a unital ∗ -homomorphism, and ρ : C l → C (0) l is the restriction map. Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 8 / 18

The decomposition is not unique, so we keep track of it: [ B l , X l , Ω l , n l , φ l ] R l =1 . Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 9 / 18

The decomposition is not unique, so we keep track of it: [ B l , X l , Ω l , n l , φ l ] R l =1 . We say that projections can be lifted along this decomposition if: ∀ n ∈ N , ∀ l = 1 , . . . , R − 1 and for every projection p ∈ B l ⊗ M n , there exists a projection ˜ p ∈ B l +1 ⊗ M n lifting p . Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 9 / 18

The decomposition is not unique, so we keep track of it: [ B l , X l , Ω l , n l , φ l ] R l =1 . We say that projections can be lifted along this decomposition if: ∀ n ∈ N , ∀ l = 1 , . . . , R − 1 and for every projection p ∈ B l ⊗ M n , there exists a projection ˜ p ∈ B l +1 ⊗ M n lifting p . Proposition If dim( X l ) ≤ 1 for l = 2 , . . . , R then projections can be lifted along [ B l , X l , Ω l , n l , φ l ] R l =1 . Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 9 / 18

Idea of proof ( A ∈ A = ⇒ A TAI): Given F ⊂⊂ A ⊗ Q , ǫ > 0, need C ∈ I with τ (1 C ) bounded away from 0, ∀ τ , Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 10 / 18

Idea of proof ( A ∈ A = ⇒ A TAI): Given F ⊂⊂ A ⊗ Q , ǫ > 0, need C ∈ I with τ (1 C ) bounded away from 0, ∀ τ , and 1 C commutes up to ǫ with f ∈ F Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 10 / 18

Idea of proof ( A ∈ A = ⇒ A TAI): Given F ⊂⊂ A ⊗ Q , ǫ > 0, need C ∈ I with τ (1 C ) bounded away from 0, ∀ τ , and 1 C commutes up to ǫ with f ∈ F and that approximates 1 C F 1 C up to ǫ. Assume τ 0 , τ 1 are the only extreme tracial states. Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 10 / 18

Idea of proof ( A ∈ A = ⇒ A TAI): Given F ⊂⊂ A ⊗ Q , ǫ > 0, need C ∈ I with τ (1 C ) bounded away from 0, ∀ τ , and 1 C commutes up to ǫ with f ∈ F and that approximates 1 C F 1 C up to ǫ. Assume τ 0 , τ 1 are the only extreme tracial states. W.L.O.G., assume F = F 0 ⊗{ 1 Q } with F 0 ⊂⊂ RSH algebra B . Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 10 / 18

Idea of proof ( A ∈ A = ⇒ A TAI): Given F ⊂⊂ A ⊗ Q , ǫ > 0, need C ∈ I with τ (1 C ) bounded away from 0, ∀ τ , and 1 C commutes up to ǫ with f ∈ F and that approximates 1 C F 1 C up to ǫ. Assume τ 0 , τ 1 are the only extreme tracial states. W.L.O.G., assume F = F 0 ⊗{ 1 Q } with F 0 ⊂⊂ RSH algebra B . Find a tracially large interval: Take a ∈ ( A ⊗ Q ) + with τ 0 ( a ) ≈ 0 and τ ( a ) ≈ 1 (Brown–Toms 2007), then take C ∗ ( a , 1). Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 10 / 18