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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

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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

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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

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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

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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 K0(A).

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 2 / 18

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Theorem (S.–Winter)

Let A, B ∈ A. Then A ⊗ Z ∼ = B ⊗ Z ⇐ ⇒ Ell(A ⊗ Z) ∼ = Ell(B ⊗ Z)

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 3 / 18

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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 M¨ unster) UHF slicing March 14, 2013 3 / 18

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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

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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. K0(Q) ∼ = Q).

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 4 / 18

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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. K0(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

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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. K0(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

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Tracial approximation

A is tracially approximately S: F ⊂ǫ     (1−p)A(1−p) B     p = 1B, B ∈ S

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 5 / 18

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Tracial approximation

A is tracially approximately S: F ⊂ǫ     (1−p)A(1−p) B     p = 1B, 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

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Tracial approximation

A is tracially approximately S: F ⊂ǫ     (1−p)A(1−p) B     p = 1B, 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

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Tracial approximation

A is tracially approximately S: F ⊂ǫ     (1−p)A(1−p) B     p = 1B, 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

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Tracial approximation

A is tracially approximately S: F ⊂ǫ     (1−p)A(1−p) B     p = 1B, 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=1C([0, 1]) ⊗ Mnk) ⊕ (⊕L l=1Mnl)}

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 5 / 18

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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

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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 K0(A).

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 6 / 18

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Recursive subhomogeneous C ∗-algebras

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

. . .
C0 ⊕C (0)

C1

⊕C (0)

C2

. . .
⊕C (0)

R CR,

where Cl = C(Xl) ⊗ Mnl for some compact metrizable Xl and C (0)

l

= C(Ωl) ⊗ Mnl for a closed subset Ωl ⊂ Xl.

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 7 / 18

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Recursive subhomogeneous C ∗-algebras

The lth stage Bl is given by Bl = Bl−1 ⊕C (0)

Cl = {(b, c) ∈ Bl−1 ⊕ Cl | φ(b) = ρ(c)} where φ : Bl−1 → C (0)

l

is a unital ∗-homomorphism, and ρ : Cl → C (0)

l

is the restriction map.

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 8 / 18

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The decomposition is not unique, so we keep track of it: [Bl, Xl, Ωl, nl, φl]R

l=1.

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 9 / 18

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The decomposition is not unique, so we keep track of it: [Bl, Xl, Ωl, nl, φl]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 ∈ Bl ⊗ Mn, there exists a projection ˜ p ∈ Bl+1 ⊗ Mn lifting p.

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 9 / 18

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The decomposition is not unique, so we keep track of it: [Bl, Xl, Ωl, nl, φl]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 ∈ Bl ⊗ Mn, there exists 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]R

l=1.

Karen R. Strung (University of M¨ unster) UHF slicing March 14, 2013 9 / 18

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