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Faculty of Science

Closure properties for the class of Cuntz-Krieger algebras

Sara Arklint

Department of Mathematical Sciences

Canadian Operator Symposium, May 27-31, 2013 Slide 1/5

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

Corners of Cuntz-Krieger algebras

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

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

Corners of Cuntz-Krieger algebras

Theorem (A-Ruiz)

Let E be a countable directed graph. TFAE:

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

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

Corners of Cuntz-Krieger algebras

Theorem (A-Ruiz)

Let E be a countable directed graph. TFAE:

• C ∗(E) is a Cuntz-Krieger algebra,

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

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

Corners of Cuntz-Krieger algebras

Theorem (A-Ruiz)

Let E be a countable directed graph. TFAE:

• C ∗(E) is a Cuntz-Krieger algebra,
• E is finite with no sinks,

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

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

Corners of Cuntz-Krieger algebras

Theorem (A-Ruiz)

Let E be a countable directed graph. TFAE:

• C ∗(E) is a Cuntz-Krieger algebra,
• E is finite with no sinks,
• C ∗(E) is unital and rank K0(C ∗(E)) = rank K1(C ∗(E)).

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

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

Corners of Cuntz-Krieger algebras

Theorem (A-Ruiz)

Let E be a countable directed graph. TFAE:

• C ∗(E) is a Cuntz-Krieger algebra,
• E is finite with no sinks,
• C ∗(E) is unital and rank K0(C ∗(E)) = rank K1(C ∗(E)).

Theorem (A-Ruiz)

Let A be a unital C ∗-algebra and assume that A is stably isomorphic to a Cuntz-Krieger algebra. Then A is a Cuntz-Krieger algebra.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

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

Corners of Cuntz-Krieger algebras

Theorem (A-Ruiz)

Let E be a countable directed graph. TFAE:

• C ∗(E) is a Cuntz-Krieger algebra,
• E is finite with no sinks,
• C ∗(E) is unital and rank K0(C ∗(E)) = rank K1(C ∗(E)).

Theorem (A-Ruiz)

Let A be a unital C ∗-algebra and assume that A is stably isomorphic to a Cuntz-Krieger algebra. Then A is a Cuntz-Krieger algebra.

Corollary (A-Ruiz)

Corners of Cuntz-Krieger algebras are Cuntz-Krieger algebras.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

• A is unital, purely infinite, nuclear, separable, and of real rank zero,

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

• A is unital, purely infinite, nuclear, separable, and of real rank zero,
• A has finitely many ideals,

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

• A is unital, purely infinite, nuclear, separable, and of real rank zero,
• A has finitely many ideals,
• for all I J A, the group K∗(J/I) is finitely generated, the group

K1(J/I) is free, and rank K0(J/I) = rank K1(J/I),

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

• A is unital, purely infinite, nuclear, separable, and of real rank zero,
• A has finitely many ideals,
• for all I J A, the group K∗(J/I) is finitely generated, the group

K1(J/I) is free, and rank K0(J/I) = rank K1(J/I),

• the simple subquotients of A are in the bootstrap class.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

• A is unital, purely infinite, nuclear, separable, and of real rank zero,
• A has finitely many ideals,
• for all I J A, the group K∗(J/I) is finitely generated, the group

K1(J/I) is free, and rank K0(J/I) = rank K1(J/I),

• the simple subquotients of A are in the bootstrap class.

Observation

Consider a unital extension I ֒ → A ։ B.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

• A is unital, purely infinite, nuclear, separable, and of real rank zero,
• A has finitely many ideals,
• for all I J A, the group K∗(J/I) is finitely generated, the group

K1(J/I) is free, and rank K0(J/I) = rank K1(J/I),

• the simple subquotients of A are in the bootstrap class.

Observation

Consider a unital extension I ֒ → A ։ B. If A is a purely infinite Cuntz-Krieger algebra, then

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

• A is unital, purely infinite, nuclear, separable, and of real rank zero,
• A has finitely many ideals,
• for all I J A, the group K∗(J/I) is finitely generated, the group

K1(J/I) is free, and rank K0(J/I) = rank K1(J/I),

• the simple subquotients of A are in the bootstrap class.

Observation

Consider a unital extension I ֒ → A ։ B. If A is a purely infinite Cuntz-Krieger algebra, then

1 B is a purely infinite Cuntz-Krieger algebra,

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

• A is unital, purely infinite, nuclear, separable, and of real rank zero,
• A has finitely many ideals,
• for all I J A, the group K∗(J/I) is finitely generated, the group

K1(J/I) is free, and rank K0(J/I) = rank K1(J/I),

• the simple subquotients of A are in the bootstrap class.

Observation

Consider a unital extension I ֒ → A ։ B. If A is a purely infinite Cuntz-Krieger algebra, then

1 B is a purely infinite Cuntz-Krieger algebra, 2 I is stably isomorphic to a purely infinite Cuntz-Krieger algebra,

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

• A is unital, purely infinite, nuclear, separable, and of real rank zero,
• A has finitely many ideals,
• for all I J A, the group K∗(J/I) is finitely generated, the group

K1(J/I) is free, and rank K0(J/I) = rank K1(J/I),

• the simple subquotients of A are in the bootstrap class.

Observation

Consider a unital extension I ֒ → A ։ B. If A is a purely infinite Cuntz-Krieger algebra, then

1 B is a purely infinite Cuntz-Krieger algebra, 2 I is stably isomorphic to a purely infinite Cuntz-Krieger algebra, 3 K0(B) → K1(I) vanishes.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Extensions of purely infinite Cuntz-Krieger algebras

Definition

A C ∗-algebra A looks like a purely infinite Cuntz-Krieger algebra if

• A is unital, purely infinite, nuclear, separable, and of real rank zero,
• A has finitely many ideals,
• for all I J A, the group K∗(J/I) is finitely generated, the group

K1(J/I) is free, and rank K0(J/I) = rank K1(J/I),

• the simple subquotients of A are in the bootstrap class.

Observation

Consider a unital extension I ֒ → A ։ B. If A is a purely infinite Cuntz-Krieger algebra, then

1 B is a purely infinite Cuntz-Krieger algebra, 2 I is stably isomorphic to a purely infinite Cuntz-Krieger algebra, 3 K0(B) → K1(I) vanishes.

If 1–3 holds, then A looks like a purely infinite Cuntz-Krieger algebra.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

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

Classification of purely infinite Cuntz-Krieger algebras

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 4/5

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

Classification of purely infinite Cuntz-Krieger algebras

Theorem (Restorff)

Let A and B be purely infinite Cuntz-Krieger algebras with Prim(A) ∼ = Prim(B). Then FKR(A) ∼ = FKR(B) implies A ⊗ K ∼ = B ⊗ K.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 4/5

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

Classification of purely infinite Cuntz-Krieger algebras

Theorem (Restorff)

Let A and B be purely infinite Cuntz-Krieger algebras with Prim(A) ∼ = Prim(B). Then FKR(A) ∼ = FKR(B) implies A ⊗ K ∼ = B ⊗ K.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 4/5

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

Classification of purely infinite Cuntz-Krieger algebras

Theorem (Restorff)

Let A and B be purely infinite Cuntz-Krieger algebras with Prim(A) ∼ = Prim(B). Then FKR(A) ∼ = FKR(B) implies A ⊗ K ∼ = B ⊗ K.

Example (Reduced filtered K-theory FKR )

For a C ∗-algebra A with ideal lattice A K0(I)

K0(Jn)

J1

② ②

J2,

❋ ❋

its FKR(A) consists of I

❊ ❊ ① ①

K0(Jn/I)

• K1(I),

n ∈ {1, 2}.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 4/5

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

Classification of purely infinite Cuntz-Krieger algebras

Theorem (Restorff)

Let A and B be purely infinite Cuntz-Krieger algebras with Prim(A) ∼ = Prim(B). Then FKR(A) ∼ = FKR(B) implies A ⊗ K ∼ = B ⊗ K.

Example (Reduced filtered K-theory FKR )

For a C ∗-algebra A with ideal lattice A K0(I)

K0(Jn)

J1

② ②

J2,

❋ ❋

its FKR(A) consists of I

❊ ❊ ① ①

K0(Jn/I)

• K1(I),

n ∈ {1, 2}.

Theorem (A-Bentmann-Katsura)

Let A be a C ∗-algebra that looks like a purely infinite Cuntz-Krieger

• algebra. Then there exists a purely infinite Cuntz-Krieger algebra B with

Prim(A) ∼ = Prim(B) and FKR(A) ∼ = FKR(B).

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 4/5

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

Applying classification to extensions of Cuntz-Krieger algebras

Theorem (A-Bentmann-Katsura)

Let A be a C ∗-algebra that look like a purely infinite Cuntz-Krieger

• algebra. Then there exists a purely infinite Cuntz-Krieger algebra B with

Prim(A) ∼ = Prim(B) and FKR (A) ∼ = FKR (B).

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 5/5

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

Applying classification to extensions of Cuntz-Krieger algebras

Theorem (A-Bentmann-Katsura)

Let A be a C ∗-algebra that look like a purely infinite Cuntz-Krieger

• algebra. Then there exists a purely infinite Cuntz-Krieger algebra B with

Prim(A) ∼ = Prim(B) and FKR (A) ∼ = FKR (B).

Theorem (Kirchberg, Meyer-Nest, Bentmann-Köhler)

Let A and B be Kirchberg X-algebras with X an accordion space. Then FK (A) ∼ = FK (B) implies A ⊗ K ∼ = B ⊗ K.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 5/5

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

Applying classification to extensions of Cuntz-Krieger algebras

Theorem (A-Bentmann-Katsura)

Let A be a C ∗-algebra that look like a purely infinite Cuntz-Krieger

• algebra. Then there exists a purely infinite Cuntz-Krieger algebra B with

Prim(A) ∼ = Prim(B) and FKR (A) ∼ = FKR (B).

Theorem (Kirchberg, Meyer-Nest, Bentmann-Köhler)

Let A and B be Kirchberg X-algebras with X an accordion space. Then FK (A) ∼ = FK (B) implies A ⊗ K ∼ = B ⊗ K.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 5/5

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

Applying classification to extensions of Cuntz-Krieger algebras

Theorem (A-Bentmann-Katsura)

Let A be a C ∗-algebra that look like a purely infinite Cuntz-Krieger

• algebra. Then there exists a purely infinite Cuntz-Krieger algebra B with

Prim(A) ∼ = Prim(B) and FKR (A) ∼ = FKR (B).

Theorem (Kirchberg, Meyer-Nest, Bentmann-Köhler)

Let A and B be Kirchberg X-algebras with X an accordion space. Then FK (A) ∼ = FK (B) implies A ⊗ K ∼ = B ⊗ K.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 5/5

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

Applying classification to extensions of Cuntz-Krieger algebras

Theorem (A-Bentmann-Katsura)

Let A be a C ∗-algebra that look like a purely infinite Cuntz-Krieger

• algebra. Then there exists a purely infinite Cuntz-Krieger algebra B with

Prim(A) ∼ = Prim(B) and FKR (A) ∼ = FKR (B).

Theorem (Kirchberg, Meyer-Nest, Bentmann-Köhler)

Let A and B be Kirchberg X-algebras with X an accordion space. Then FK (A) ∼ = FK (B) implies A ⊗ K ∼ = B ⊗ K.

Theorem (A-Bentmann-Katsura)

Let A and B be C ∗-algebras that looks like purely infinite Cuntz-Krieger algebras and assume that Prim(A) and Prim(B) are homeomorphic accordion spaces. Then FKR(A) ∼ = FKR(B) implies FK(A) ∼ = FK(B).

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 5/5

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

Applying classification to extensions of Cuntz-Krieger algebras

Theorem (A-Bentmann-Katsura)

Let A be a C ∗-algebra that look like a purely infinite Cuntz-Krieger

• algebra. Then there exists a purely infinite Cuntz-Krieger algebra B with

Prim(A) ∼ = Prim(B) and FKR (A) ∼ = FKR (B).

Theorem (Kirchberg, Meyer-Nest, Bentmann-Köhler)

Let A and B be Kirchberg X-algebras with X an accordion space. Then FK (A) ∼ = FK (B) implies A ⊗ K ∼ = B ⊗ K.

Theorem (A-Bentmann-Katsura)

Let A and B be C ∗-algebras that looks like purely infinite Cuntz-Krieger algebras and assume that Prim(A) and Prim(B) are homeomorphic accordion spaces. Then FKR(A) ∼ = FKR(B) implies FK(A) ∼ = FK(B).

Corollary

Let A be a C ∗-algebra with Prim(A) an accordion space. Then A is a purely infinite Cuntz-Krieger algebra if and only if it looks like one.

Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 5/5