UHF slicing and classification of nuclear C*-algebras

Transcription

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

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

3 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 Münster) UHF slicing March 14, / 18

4 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 Münster) UHF slicing March 14, / 18

5 Theorem (S. Winter) Let A, B A. Then A Z = B Z Ell(A Z) = Ell(B Z) Karen R. Strung (University of Münster) UHF slicing March 14, / 18

6 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) Karen R. Strung (University of Münster) UHF slicing March 14, / 18

7 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 Münster) UHF slicing March 14, / 18

8 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 Münster) UHF slicing March 14, / 18

9 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 Münster) UHF slicing March 14, / 18

10 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 Münster) UHF slicing March 14, / 18

11 Tracial approximation A is tracially approximately S: F ɛ (1 p)a(1 p) B p = 1 B, B S Karen R. Strung (University of Münster) UHF slicing March 14, / 18

12 Tracial approximation A is tracially approximately S: F ɛ (1 p)a(1 p) B p = 1 B, B S x A then x pxp + (1 p)x(1 p) Karen R. Strung (University of Münster) UHF slicing March 14, / 18

13 Tracial approximation A is tracially approximately S: F ɛ (1 p)a(1 p) B p = 1 B, B S x A then x pxp + (1 p)x(1 p) where τ(1 p) < ɛ for every τ T (A) Karen R. Strung (University of Münster) UHF slicing March 14, / 18

14 Tracial approximation A is tracially approximately S: F ɛ (1 p)a(1 p) B p = 1 B, B S x A then x pxp + (1 p)x(1 p) where τ(1 p) < ɛ for every τ T (A) and pxp ɛ B. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

15 Tracial approximation A is tracially approximately S: F ɛ (1 p)a(1 p) B p = 1 B, B S x A then x pxp + (1 p)x(1 p) where τ(1 p) < ɛ for every τ T (A) and pxp ɛ B. I = {( K k=1 C([0, 1]) M n k ) ( L l=1 M n l )} Karen R. Strung (University of Münster) UHF slicing March 14, / 18

16 Main theorem Theorem (S. Winter) Let A A. Then A Q is TAI. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

17 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 Münster) UHF slicing March 14, / 18

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

19 Recursive subhomogeneous C -algebras The l th stage B l is given by where B l = B l 1 C (0) l is a unital -homomorphism, and is the restriction map. C l = {(b, c) B l 1 C l φ(b) = ρ(c)} φ : B l 1 C (0) l ρ : C l C (0) l Karen R. Strung (University of Münster) UHF slicing March 14, / 18

20 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 Münster) UHF slicing March 14, / 18

21 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 Münster) UHF slicing March 14, / 18

22 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 Münster) UHF slicing March 14, / 18

23 Idea of proof (A A = A TAI): Given F A Q, ɛ > 0, need C I with τ(1 C ) bounded away from 0, τ, Karen R. Strung (University of Münster) UHF slicing March 14, / 18

24 Idea of proof (A A = A TAI): Given F A Q, ɛ > 0, need C I with τ(1 C ) bounded away from 0, τ, and 1 C commutes up to ɛ with f F Karen R. Strung (University of Münster) UHF slicing March 14, / 18

25 Idea of proof (A A = A TAI): Given F A Q, ɛ > 0, need C I with τ(1 C ) bounded away from 0, τ, and 1 C commutes up to ɛ with f F and that approximates 1 C F 1 C up to ɛ. Assume τ 0, τ 1 are the only extreme tracial states. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

26 Idea of proof (A A = A TAI): Given F A Q, ɛ > 0, need C I with τ(1 C ) bounded away from 0, τ, and 1 C commutes up to ɛ with f F and that approximates 1 C F 1 C up to ɛ. 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 Münster) UHF slicing March 14, / 18

27 Idea of proof (A A = A TAI): Given F A Q, ɛ > 0, need C I with τ(1 C ) bounded away from 0, τ, and 1 C commutes up to ɛ with f F and that approximates 1 C F 1 C up to ɛ. 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 Münster) UHF slicing March 14, / 18

28 Idea of proof (A A = A TAI): Given F A Q, ɛ > 0, need C I with τ(1 C ) bounded away from 0, τ, and 1 C commutes up to ɛ with f F and that approximates 1 C F 1 C up to ɛ. 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). Must move this interval into position (w.r.t. F): model an interval in B Q, use strict comparison. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

29 Interval model: excisors and bridges Definition B unital RSH with decomposition [B l, X l, Ω l, n l, φ l ], F B 1 +, η > 0. An (F, η)-excisor (E, ρ, σ, κ) consists of Karen R. Strung (University of Münster) UHF slicing March 14, / 18

30 Interval model: excisors and bridges Definition B unital RSH with decomposition [B l, X l, Ω l, n l, φ l ], F B 1 +, η > 0. An (F, η)-excisor (E, ρ, σ, κ) consists of 1 a finite dimensional algebra E = R l=1 E l, Karen R. Strung (University of Münster) UHF slicing March 14, / 18

31 Interval model: excisors and bridges Definition B unital RSH with decomposition [B l, X l, Ω l, n l, φ l ], F B 1 +, η > 0. An (F, η)-excisor (E, ρ, σ, κ) consists of 1 a finite dimensional algebra E = R l=1 E l, 2 a unital -homomorphism ρ = R l=1 ρ l : B R l=1 E l Karen R. Strung (University of Münster) UHF slicing March 14, / 18

32 Interval model: excisors and bridges Definition B unital RSH with decomposition [B l, X l, Ω l, n l, φ l ], F B 1 +, η > 0. An (F, η)-excisor (E, ρ, σ, κ) consists of 1 a finite dimensional algebra E = R l=1 E l, 2 a unital -homomorphism ρ = R l=1 ρ l : B R l=1 E l 3 an isometric c.p. order zero map σ = R l=1 σ l : R l=1 E l B Q Karen R. Strung (University of Münster) UHF slicing March 14, / 18

33 Interval model: excisors and bridges Definition B unital RSH with decomposition [B l, X l, Ω l, n l, φ l ], F B 1 +, η > 0. An (F, η)-excisor (E, ρ, σ, κ) consists of 1 a finite dimensional algebra E = R l=1 E l, 2 a unital -homomorphism ρ = R l=1 ρ l : B R l=1 E l 3 an isometric c.p. order zero map σ = R l=1 σ l : R l=1 E l B Q such that σ(1 E )(b 1 Q ) = σ ρ(b) < η for all b F, Karen R. Strung (University of Münster) UHF slicing March 14, / 18

34 Interval model: excisors and bridges Definition B unital RSH with decomposition [B l, X l, Ω l, n l, φ l ], F B 1 +, η > 0. An (F, η)-excisor (E, ρ, σ, κ) consists of 1 a finite dimensional algebra E = R l=1 E l, 2 a unital -homomorphism ρ = R l=1 ρ l : B R l=1 E l 3 an isometric c.p. order zero map σ = R l=1 σ l : R l=1 E l B Q such that σ(1 E )(b 1 Q ) = σ ρ(b) < η for all b F, 4 a unital -homomorphism κ : E Q. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

35 Interval model: excisors and bridges We say that (E, ρ, σ, κ) is compatible with the RSH decomposition if each ρ l factorizes through ρ l B E l ψ l ˇρ l for some compact ˇX l X l \ Ω l. B l ˇψ l C( ˇX l ) M rl Karen R. Strung (University of Münster) UHF slicing March 14, / 18

36 Interval model: excisors and bridges Definition An (F, η)-bridge between (E 0, ρ 0, σ 0, κ 0 ) and (E 1, ρ 1, σ 1, κ 1 ) consists of K N and (F, η)-excisors (E j/k, ρ j/k, σ j/k, κ j/k ), j = 1,..., K 1 satisfying κ j/k ρ j/k (b) κ (j+1)/k ρ (j+1)/k (b) < η for all b F and j = 0,..., K 1. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

37 Interval model: excisors and bridges Definition An (F, η)-bridge between (E 0, ρ 0, σ 0, κ 0 ) and (E 1, ρ 1, σ 1, κ 1 ) consists of K N and (F, η)-excisors (E j/k, ρ j/k, σ j/k, κ j/k ), j = 1,..., K 1 satisfying κ j/k ρ j/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 (E 0, ρ 0, σ 0, κ 0 ) (F,η) (E 1, ρ 1, σ 1, κ 1 ). Karen R. Strung (University of Münster) UHF slicing March 14, / 18

38 (F, η)-connected decomposition [B l, X l, Ω l, n l, φ l ] R l=1 the RSH decomposition. For every l = 1,..., R and every x X l we can define an (F, η)-excisor. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

39 (F, η)-connected decomposition [B l, X l, Ω l, n l, φ l ] R l=1 the RSH decomposition. For every l = 1,..., R and every x X l we can define an (F, η)-excisor. The decomposition is (F, η)-connected if, for any l = 1,..., R and any x, y X l, we can always find an (F, η)-bridge between their corresponding (F, η)-excisors. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

40 Interval model: large trace With a suitable calculus for (F, η)-excisors, we can find (E 0, ρ 0, σ 0, κ 0 ) and (E 1, ρ 0, σ 0, κ 0 ) with τ i (σ(1 Ei )) large and τ i (σ j (1 Ej )) small, i j {0, 1}. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

41 Interval model: large trace With a suitable calculus for (F, η)-excisors, we can find (E 0, ρ 0, σ 0, κ 0 ) and (E 1, ρ 0, σ 0, κ 0 ) with τ i (σ(1 Ei )) large and τ i (σ j (1 Ej )) small, i j {0, 1}. It remains to find an (F, η)-bridge through the decomposition. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

42 Interval model: large trace With a suitable calculus for (F, η)-excisors, we can find (E 0, ρ 0, σ 0, κ 0 ) and (E 1, ρ 0, σ 0, κ 0 ) with τ i (σ(1 Ei )) large and τ i (σ j (1 Ej )) small, i 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 off the RSH decomposition. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

43 Interval model: large trace With a suitable calculus for (F, η)-excisors, we can find (E 0, ρ 0, σ 0, κ 0 ) and (E 1, ρ 0, σ 0, κ 0 ) with τ i (σ(1 Ei )) large and τ i (σ j (1 Ej )) small, i 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 off the RSH decomposition. This is where we require that projections can be lifted and that each tracial state induces the same state on K 0. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

44 What we get... With two extreme tracial states τ 0, τ 1, we find (F, η)-excisors (E 0, ρ 0, σ 0, κ 0 ) (F,η) (E 1, ρ 1, σ 1, κ) Karen R. Strung (University of Münster) UHF slicing March 14, / 18

45 What we get... With two extreme tracial states τ 0, τ 1, we find (F, η)-excisors (E 0, ρ 0, σ 0, κ 0 ) (F,η) (E 1, ρ 1, σ 1, κ) such that (τ i τ Q )(σ i (1 Ei )) 1/3. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

46 What we get... With two extreme tracial states τ 0, τ 1, we find (F, η)-excisors such that (E 0, ρ 0, σ 0, κ 0 ) (F,η) (E 1, ρ 1, σ 1, κ) (τ i τ Q )(σ i (1 Ei )) 1/3. Now use strict comparison to move the elements in a partition of unity of the actual interval under this model. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

47 What we get... With two extreme tracial states τ 0, τ 1, we find (F, η)-excisors such that (E 0, ρ 0, σ 0, κ 0 ) (F,η) (E 1, ρ 1, σ 1, κ) (τ i τ Q )(σ i (1 Ei )) 1/3. Now use strict comparison to move the elements in a partition of unity of the actual interval under this model. This will be an interval which is large in trace, and the condition σ(1 E )(b 1 Q ) = σ ρ(b) < η for all b F, for (F, η)-excisors allows us to properly approximate elements in the finite subset. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

48 What we get... With two extreme tracial states τ 0, τ 1, we find (F, η)-excisors such that (E 0, ρ 0, σ 0, κ 0 ) (F,η) (E 1, ρ 1, σ 1, κ) (τ i τ Q )(σ i (1 Ei )) 1/3. Now use strict comparison to move the elements in a partition of unity of the actual interval under this model. This will be an interval which is large in trace, and the condition σ(1 E )(b 1 Q ) = σ ρ(b) < η for all b F, for (F, η)-excisors allows us to properly approximate elements in the finite subset. A Q is TAI. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

49 Main theorem Theorem (S. Winter) Let A, B A. Then A Z = B Z Ell(A Z) = Ell(B Z). Karen R. Strung (University of Münster) UHF slicing March 14, / 18

50 Main theorem Theorem (S. Winter) 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 -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 Münster) UHF slicing March 14, / 18

51 Some consequences 1. Elliott 1996 Simple approximately SH algebras constructed by attaching 1-dimensional spaces to the circle. Theorem = classification when restricted to finitely many extreme tracial states, each inducing same K 0 -state. Karen R. Strung (University of Münster) UHF slicing March 14, / 18

52 Some consequences 1. Elliott 1996 Simple approximately SH algebras constructed by attaching 1-dimensional spaces to the circle. Theorem = classification when restricted to finitely many extreme tracial states, each inducing same K 0 -state. 2. Lin Matui 2005 A := C(X T) Z. Restricting to finitely many traces each inducing same state on K 0, theorem = A {x} Q is TAI, then (S.-Winter 2010) = A Q TAI = classification. Karen R. Strung (University of Münster) UHF slicing March 14, / 18