Classification of Continuous Fields of C* Algebras

Classiﬁcation of Continuous Fields of C* Algebras

Prahlad Vaidyanathan (Joint work with M. Dadarlat)

Department of Mathematics IISER Bhopal

Sept 11, 2014 2 Continuous Fields of C* Algebras

3 E-theory for Continuous ﬁelds

4 Classiﬁcation Results

Outline

1 Elliott’s Classiﬁcation Program 3 E-theory for Continuous ﬁelds

4 Classiﬁcation Results

Outline

1 Elliott’s Classiﬁcation Program

2 Continuous Fields of C* Algebras 4 Classiﬁcation Results

Outline

1 Elliott’s Classiﬁcation Program

2 Continuous Fields of C* Algebras

3 E-theory for Continuous ﬁelds Outline

1 Elliott’s Classiﬁcation Program

2 Continuous Fields of C* Algebras

3 E-theory for Continuous ﬁelds

4 Classiﬁcation Results A ﬁnite dimensional C* algebra is of the form

Mn1 (C) ⊕ Mn2 (C) ⊕ ... ⊕ Mnk (C)

A nuclear C* algebra is one that can be approximated “nicely" by ﬁnite dimensional C* algebras.

Elliott’s Classiﬁcation Program C* algebras

A C* algebra is An algebra of bounded linear operators on a Hilbert space that is closed in the norm topology and closed under taking adjoints. A complex Banach algebra with an involution map a 7→ a∗ such that ka∗ak = kak2 A nuclear C* algebra is one that can be approximated “nicely" by ﬁnite dimensional C* algebras.

Elliott’s Classiﬁcation Program C* algebras

Mn1 (C) ⊕ Mn2 (C) ⊕ ... ⊕ Mnk (C) Elliott’s Classiﬁcation Program C* algebras

Mn1 (C) ⊕ Mn2 (C) ⊕ ... ⊕ Mnk (C)

A nuclear C* algebra is one that can be approximated “nicely" by ﬁnite dimensional C* algebras. The Elliott Program To classify nuclear C* algebras by their K-theory

Elliott’s Classiﬁcation Program K-theory

To each C* algebra A, one can associate two groups

K0(A) and K1(A) Elliott’s Classiﬁcation Program K-theory

To each C* algebra A, one can associate two groups

K0(A) and K1(A)

The Elliott Program To classify nuclear C* algebras by their K-theory A class of simple algebras for which the Elliott program has been successful is the class of Kirchberg algebras (Purely inﬁnite, simple, unital, nuclear algebras) Fact Every separable nuclear C* algebra is K-theoretically equivalent to a Kirchberg algebra.

Elliott’s Classiﬁcation Program Kirchberg Algebras

A C* algebra is called simple if it has no non-trivial closed two-sided ideals. Fact Every separable nuclear C* algebra is K-theoretically equivalent to a Kirchberg algebra.

Elliott’s Classiﬁcation Program Kirchberg Algebras

A C* algebra is called simple if it has no non-trivial closed two-sided ideals.

A class of simple algebras for which the Elliott program has been successful is the class of Kirchberg algebras (Purely inﬁnite, simple, unital, nuclear algebras) Elliott’s Classiﬁcation Program Kirchberg Algebras

A C* algebra is called simple if it has no non-trivial closed two-sided ideals.

A class of simple algebras for which the Elliott program has been successful is the class of Kirchberg algebras (Purely inﬁnite, simple, unital, nuclear algebras) Fact Every separable nuclear C* algebra is K-theoretically equivalent to a Kirchberg algebra. By the fact mentioned above, we may assume that all simple quotients of our C* algebras are Kirchberg algebras.

This Project

Goal To classify non-simple C* algebras using K-theory K-theory of all quotients Maps between these K-theory groups This Project

Goal To classify non-simple C* algebras using K-theory K-theory of all quotients Maps between these K-theory groups By the fact mentioned above, we may assume that all simple quotients of our C* algebras are Kirchberg algebras. Θ is called the structure morphism. We usually drop Θ and write

f · a := Θ(f )(a)

Continuous Fields of C* Algebras C(X)-algebras

Let X be a compact metric space. A structure of a C(X)-algebra on a separable C* algebra A is a unital *-homomorphism

Θ: C(X) → ZM(A)

from C(X) to the center of the multiplier algebra of A. In other words, A has the structure of a non-degenerate C(X)-module. Continuous Fields of C* Algebras C(X)-algebras

Let X be a compact metric space. A structure of a C(X)-algebra on a separable C* algebra A is a unital *-homomorphism

Θ: C(X) → ZM(A)

from C(X) to the center of the multiplier algebra of A. In other words, A has the structure of a non-degenerate C(X)-module.

Θ is called the structure morphism. We usually drop Θ and write

f · a := Θ(f )(a) Write πx : A → A(x) for the natural quotient map. For a ∈ A, write

a(x) := πx (a)

A C(X)-algebra A is called a continuous ﬁeld of C* algebras over X if the function x 7→ ka(x)k is continuous on X for each a ∈ A

Continuous Fields of C* Algebras Deﬁnition and Examples Continuous Fields of C* Algebras

The C(X)-module structure allows us to localize at points of X.

For x ∈ X, the quotient

A A(x) := C0(X \{x}) · A is called the ﬁber of A at x. Continuous Fields of C* Algebras Deﬁnition and Examples Continuous Fields of C* Algebras

The C(X)-module structure allows us to localize at points of X.

For x ∈ X, the quotient

A A(x) := C0(X \{x}) · A is called the ﬁber of A at x. Write πx : A → A(x) for the natural quotient map. For a ∈ A, write

a(x) := πx (a)

A C(X)-algebra A is called a continuous ﬁeld of C* algebras over X if the function x 7→ ka(x)k is continuous on X for each a ∈ A If X = Prim(A) Hausdorff, the Dauns-Hoffman Theorem says that ∼ C(X) = ZM(A)

and A(x) is simple for all x ∈ X. Thus,

A unital separable C* algebra with a Hausdorff primitive spectrum is a continuous ﬁeld over its spectrum.

Continuous Fields of C* Algebras Deﬁnition and Examples The Primitive Spectrum

If A is a C* algebra A primitive ideal is the kernel of an irreducible representation.

The collection of primitive ideals, Prim(A), is a compact T0-space Thus,

A unital separable C* algebra with a Hausdorff primitive spectrum is a continuous ﬁeld over its spectrum.

Continuous Fields of C* Algebras Deﬁnition and Examples The Primitive Spectrum

If A is a C* algebra A primitive ideal is the kernel of an irreducible representation.

The collection of primitive ideals, Prim(A), is a compact T0-space

If X = Prim(A) Hausdorff, the Dauns-Hoffman Theorem says that ∼ C(X) = ZM(A)

and A(x) is simple for all x ∈ X. Continuous Fields of C* Algebras Deﬁnition and Examples The Primitive Spectrum

If A is a C* algebra A primitive ideal is the kernel of an irreducible representation.

The collection of primitive ideals, Prim(A), is a compact T0-space

If X = Prim(A) Hausdorff, the Dauns-Hoffman Theorem says that ∼ C(X) = ZM(A)

and A(x) is simple for all x ∈ X. Thus,

A unital separable C* algebra with a Hausdorff primitive spectrum is a continuous ﬁeld over its spectrum. If A is a continuous ﬁeld of C* algebras such that, for each x ∈ X, there is a neighbourhood V of x and a C* algebra D such that ∼ C0(V ) · A = C0(V , D)

then A is said to be locally trivial. Locally trivial fields are rare when the fibers are infinite dimensional C* algebras.

Continuous Fields of C* Algebras Deﬁnition and Examples Examples of Continuous Fields

If D is any ﬁxed C* algebra, then

A = C(X, D) = {f : X → D such that f is continuous }

is called the trivial field with fiber D Continuous Fields of C* Algebras Definition and Examples Examples of Continuous Fields

If D is any ﬁxed C* algebra, then

A = C(X, D) = {f : X → D such that f is continuous }

is called the trivial field with fiber D If A is a continuous field of C* algebras such that, for each x ∈ X, there is a neighbourhood V of x and a C* algebra D such that ∼ C0(V ) · A = C0(V , D)

then A is said to be locally trivial. Locally trivial fields are rare when the fibers are infinite dimensional C* algebras. Continuous Fields of C* Algebras Definition and Examples A field with one singularity

Let D, H be two C* algebras, and γ : D → H an injective *-homomorphism. For x ∈ [0, 1] ﬁxed, deﬁne

Bx := {(f , g) ∈ C([0, x], H) ⊕ C([x, 1], D): f (r) = γ(g(x))}

We represent this pictorially by

H •0 •x O γ D •x •1

Then, Bx is a continuous field over [0, 1] which is locally trivial at all points of [0, 1] except x. Continuous Fields of C* Algebras Definition and Examples Elementary fields over [0, 1]

An elementary continuous field over [0, 1] is a field which is locally trivial at all but finitely many points. We depict them pictorially by

H1 H2 H3 Hk •0 •x1 •x2 •x3 •x4 ... •xn •1 O O O O O γ1 γ2 γ3 γ4 γn D1 D2 Dk−1 •x1 •x2 •x3 •x4 ... •xn Question Given two continuous ﬁelds A and B over the same space X, how do we know if there is a C(X)-linear *-isomorphism

∼ ϕ : A −→ B

Continuous Fields of C* Algebras The Classiﬁcation Problem

Let A and B be a continuous ﬁelds over a space X. A morphism of continuous ﬁelds is a *-homomorphism

ϕ : A → B

such that ϕ(f · a) = f · ϕ(a) ∀f ∈ C(X), a ∈ A Continuous Fields of C* Algebras The Classiﬁcation Problem

Let A and B be a continuous ﬁelds over a space X. A morphism of continuous ﬁelds is a *-homomorphism

ϕ : A → B

such that ϕ(f · a) = f · ϕ(a) ∀f ∈ C(X), a ∈ A

Question Given two continuous ﬁelds A and B over the same space X, how do we know if there is a C(X)-linear *-isomorphism

∼ ϕ : A −→ B E-theory for Continuous ﬁelds The idea behind E-theory

The collection HomX (A, B) of C(X)-linear *-homomorphisms is not a group, but can be replaced by a more ﬂexible invariant, the E-theory group.

EX (A, B) For each a ∈ A, t 7→ kϕt (a)k is a bounded continuous function from T to B ∗ ∗ ϕt (a + λb) − ϕt (a) − λϕt (b) → 0

ϕt (ab) − ϕt (a)ϕt (b) → 0

ϕt (f · a) − f · ϕt (a) → 0

E-theory for Continuous ﬁelds Asymptotic morphisms

Let A and B be two C(X)-algebras. An asymptotic morphism is a family ϕt : A → B, t ∈ T := [0, ∞)

such that {ϕt } behaves like a genuine C(X)-linear *-homomorphism as t → ∞. E-theory for Continuous ﬁelds Asymptotic morphisms

Let A and B be two C(X)-algebras. An asymptotic morphism is a family ϕt : A → B, t ∈ T := [0, ∞)

such that {ϕt } behaves like a genuine C(X)-linear *-homomorphism as t → ∞.

For each a ∈ A, t 7→ kϕt (a)k is a bounded continuous function from T to B ∗ ∗ ϕt (a + λb) − ϕt (a) − λϕt (b) → 0

ϕt (ab) − ϕt (a)ϕt (b) → 0

ϕt (f · a) − f · ϕt (a) → 0 E-theory for Continuous ﬁelds Homotopy of Asymptotic morphisms

A homotopy of asymptotic morphisms ϕt , ψt : A → B is a C(X)-linear asymptotic morphism

Φt : A → C([0, 1], B)

such that Φt (a)(0) = ϕt (a), and Φt (a)(1) = ψt (a)

We denote the homotopy class of an asymptotic morphism ϕt : A → B by [ϕt ] and we write [A, B]X for the collection of homotopy classes of asymptotic morphisms from A to B. And by passing to suspensions (SB = C0((0, 1), B)); if ϕt , ψt : A → SB, we can compose them by ( ϕt (a)(2s): 0 ≤ s ≤ 1/2 ϕt · ψt : A → SB, ϕt · ψt (a)(s) = ψt (2s − 1): 1/2 ≤ s ≤ 1

The two additions coincide when we consider maps

A 7→ S(B ⊗ K)

where K denotes the compact operators.

E-theory for Continuous ﬁelds Addition of asymptotic morphisms

We may add two asymptotic morphisms ϕt , ψt : A → B by Passing to matrices ϕt (a) 0 ϕt ⊕ ψt : A → M2(B), a 7→ 0 ψt (a) The two additions coincide when we consider maps

A 7→ S(B ⊗ K)

where K denotes the compact operators.

E-theory for Continuous ﬁelds Addition of asymptotic morphisms

We may add two asymptotic morphisms ϕt , ψt : A → B by Passing to matrices ϕt (a) 0 ϕt ⊕ ψt : A → M2(B), a 7→ 0 ψt (a)

And by passing to suspensions (SB = C0((0, 1), B)); if ϕt , ψt : A → SB, we can compose them by ( ϕt (a)(2s): 0 ≤ s ≤ 1/2 ϕt · ψt : A → SB, ϕt · ψt (a)(s) = ψt (2s − 1): 1/2 ≤ s ≤ 1 E-theory for Continuous ﬁelds Addition of asymptotic morphisms

We may add two asymptotic morphisms ϕt , ψt : A → B by Passing to matrices ϕt (a) 0 ϕt ⊕ ψt : A → M2(B), a 7→ 0 ψt (a)

And by passing to suspensions (SB = C0((0, 1), B)); if ϕt , ψt : A → SB, we can compose them by ( ϕt (a)(2s): 0 ≤ s ≤ 1/2 ϕt · ψt : A → SB, ϕt · ψt (a)(s) = ψt (2s − 1): 1/2 ≤ s ≤ 1

The two additions coincide when we consider maps

A 7→ S(B ⊗ K)

where K denotes the compact operators. Functoriality Given a morphism ϕ : B → C, there is an induced group homomorphism ϕ∗ : EX (A, B) → EX (A, C)

Similarly, EX (·, B) is a contravariant functor.

E-theory for Continuous ﬁelds The E-theory group

For two separable C(X)-algebras A and B,

EX (A, B) = [SA, SB ⊗ K]X

Then, EX (A, B) is a group under the addition deﬁned above. Moreover, we have a map

HomX (A, B) → EX (A, B), ϕ 7→ [ϕ] E-theory for Continuous ﬁelds The E-theory group

For two separable C(X)-algebras A and B,

EX (A, B) = [SA, SB ⊗ K]X

Then, EX (A, B) is a group under the addition deﬁned above. Moreover, we have a map

HomX (A, B) → EX (A, B), ϕ 7→ [ϕ]

Functoriality Given a morphism ϕ : B → C, there is an induced group homomorphism ϕ∗ : EX (A, B) → EX (A, C)

Similarly, EX (·, B) is a contravariant functor. E-theory for Continuous ﬁelds The composition product

The composition map

HomX (A, B) × HomX (B, C) → HomX (A, C)

extends to E-theory. Composition product

EX (A, B) × EX (B, C) → EX (A, C) (α, β) 7→ β ◦ α Theorem (Kirchberg (2000)) Let A and B be separable continuous ﬁelds over X each of whose ﬁbers are stable Kirchberg algebras. If α ∈ EX (A, B) is invertible, then ∃ an isomorphism ϕ : A → B such that

α = [ϕ]

E-theory for Continuous ﬁelds Kirchberg’s theorem

An element α ∈ EX (A, B) is called invertible if ∃β ∈ EX (B, A) such that

β ◦ α = [idA], and α ◦ β = [idB]

If ϕ : A → B is an isomorphism, then α := [ϕ] is invertible. E-theory for Continuous ﬁelds Kirchberg’s theorem

An element α ∈ EX (A, B) is called invertible if ∃β ∈ EX (B, A) such that

β ◦ α = [idA], and α ◦ β = [idB]

If ϕ : A → B is an isomorphism, then α := [ϕ] is invertible. Theorem (Kirchberg (2000)) Let A and B be separable continuous ﬁelds over X each of whose ﬁbers are stable Kirchberg algebras. If α ∈ EX (A, B) is invertible, then ∃ an isomorphism ϕ : A → B such that

α = [ϕ] E-theory for Continuous ﬁelds Towards Classiﬁcation

Question

Can we effectively compute EX (A, B)?

Can we determine when EX (A, B) has an invertible element? The goal of this project was to answer these two questions in the case where X = [0, 1] Classiﬁcation Results An approximation theorem

Recall that an elementary continuous field over [0, 1] is one that is locally trivial at all but finitely many points. Theorem (Dadarlat-Elliott (2007)) Let A be a continuous field over [0, 1] each of whose fibers are inductive limits of unital semi-projective Kirchberg algebras, then A can be represented as an inductive limit ∼ A = lim An n→∞

where each An is an elementary continuous field for each n ∈ N Classification Results Fibered morphisms between Elementary fields

Suppose A, B are two elementary ﬁelds over [0, 1]. We can construct a morphism of continuous ﬁelds in the following way :

H •0 •x O γ D •x •1

ϕ1 ϕ2 •0 •x H0 O µ

•x •1 D0

0 0 where ϕ1 : H → H and ϕ2 : D → D are such that ϕ1 ◦ γ = µ ◦ ϕ2 Classiﬁcation Results

In general, a ﬁbered morphism would look like

E1 E2 E3 Ek •0 •x1 •x2 •x3 •x4 ... •xn •1 O O O O O γ1 γ2 γ3 γ4 γn D1 D2 Dk−1 •x1 •x2 •x3 •x4 ... •xn

...

ϕ1 ψ1 ϕ2 ψ2 ϕ3 ψk−1 ϕk •0 •x •x •x •x ... •x •1 E0 1 2 E0 3 4 E0 n E0 1 O O 2 O O 3 O k µ1 µ2 µ3 µ4 µn 0 0 D0 D1 D2 k−1 •x1 •x2 •x3 •x4 ... •xn with ϕ1 ◦ γ1 = µ1 ◦ ψ1, etc. We show that, under certain conditions on the ﬁbers, all morphisms look like this. Classiﬁcation Results E-theory for Elementary Fields over [0, 1]

Let F denote the class of separable C* algebras D satisfying the UCT, and such that K0(D) is free of finite rank, and K1(D) = 0. Let C(F) denote the class of elementary continuous fields over [0, 1] whose fibers are in F Theorem (Dadarlat - V (2014))

If A, B ∈ C(F), then, for every α ∈ EX (A, B), there exists a ﬁbered morphism ϕ : A → B such that

α = [ϕ] If I ⊂ J, then we get an induced quotient map A(J) → A(I), and hence an induced map K0(A(J)) → K0(A(I))

This collection {K0(A(I)) : I ⊂ [0, 1] closed subinterval}, together with these maps, has the structure of a sheaf on the family of closed subintervals of [0, 1], denoted by K0(A)

Classiﬁcation Results Deﬁnition

Let A be a C[0, 1]-algebra in F, then for each closed sub-interval I ⊂ [0, 1], we consider the K0-group

A K0(A(I)) := K0 C0(X \ I) · A These groups capture the local structure of A via K-theory. This collection {K0(A(I)) : I ⊂ [0, 1] closed subinterval}, together with these maps, has the structure of a sheaf on the family of closed subintervals of [0, 1], denoted by K0(A)

Classiﬁcation Results Deﬁnition

Let A be a C[0, 1]-algebra in F, then for each closed sub-interval I ⊂ [0, 1], we consider the K0-group

A K0(A(I)) := K0 C0(X \ I) · A These groups capture the local structure of A via K-theory.

If I ⊂ J, then we get an induced quotient map A(J) → A(I), and hence an induced map K0(A(J)) → K0(A(I)) Classiﬁcation Results Deﬁnition

Let A be a C[0, 1]-algebra in F, then for each closed sub-interval I ⊂ [0, 1], we consider the K0-group

A K0(A(I)) := K0 C0(X \ I) · A These groups capture the local structure of A via K-theory.

If I ⊂ J, then we get an induced quotient map A(J) → A(I), and hence an induced map K0(A(J)) → K0(A(I))

This collection {K0(A(I)) : I ⊂ [0, 1] closed subinterval}, together with these maps, has the structure of a sheaf on the family of closed subintervals of [0, 1], denoted by K0(A) If I ⊂ J, then the following diagram commutes

ϕI K0(A(I)) −−−−→ K0(B(I))     y y ϕJ K0(A(J)) −−−−→ K0(B(J))

Classiﬁcation Results E-theory and K-theory

If ϕ : A → B is a morphism of C[0, 1]-algebras, then for each closed interval I ⊂ [0, 1], we get a map

ϕI : A(I) → B(I)

and hence an induced map ϕI : K0(A(I)) → K0(B(I)). Classiﬁcation Results E-theory and K-theory

If ϕ : A → B is a morphism of C[0, 1]-algebras, then for each closed interval I ⊂ [0, 1], we get a map

ϕI : A(I) → B(I)

and hence an induced map ϕI : K0(A(I)) → K0(B(I)).

If I ⊂ J, then the following diagram commutes

ϕI K0(A(I)) −−−−→ K0(B(I))     y y ϕJ K0(A(J)) −−−−→ K0(B(J)) Classiﬁcation Results E-theory and K-theory

Thus, we get a map

HomX (A, B) → Hom(K0(A), K0(B))

which further extends to a natural homomorphism

Γ: EX (A, B) → Hom(K0(A), K0(B)) In particular, ∼ ∼ A = B ⇔ K0(A) = K0(B)

Classiﬁcation Results A Classiﬁcation Theorem

Theorem (Dadarlat-V (2014)) Let A and B be separable continuous fields over [0, 1] whose fibers are Kirchberg algebras satisfying the Universal Coefficient theorem with torsion-free K0 groups and zero K1 groups, then

Γ: EX (A, B) → Hom(K0(A), K0(B))

is an isomorphism. Classiﬁcation Results A Classiﬁcation Theorem

Γ: EX (A, B) → Hom(K0(A), K0(B))

is an isomorphism.

In particular, ∼ ∼ A = B ⇔ K0(A) = K0(B) Use the approximation result to write

A = lim An, and B = lim Bk

where An, Bk are elementary ﬁelds. Use a continuity property of K0(·), together with Kirchberg’s theorem and Elliott’s intertwining argument.

Classiﬁcation Results The idea behind the proof

First prove that, when A and B are both elementary ﬁelds, the map

Γ: EX (A, B) → Hom(K0(A), K0(B))

is an isomorphism. Use a continuity property of K0(·), together with Kirchberg’s theorem and Elliott’s intertwining argument.

Classiﬁcation Results The idea behind the proof

First prove that, when A and B are both elementary ﬁelds, the map

Γ: EX (A, B) → Hom(K0(A), K0(B))

is an isomorphism. Use the approximation result to write

A = lim An, and B = lim Bk

where An, Bk are elementary ﬁelds. Classiﬁcation Results The idea behind the proof

First prove that, when A and B are both elementary ﬁelds, the map

Γ: EX (A, B) → Hom(K0(A), K0(B))

is an isomorphism. Use the approximation result to write

A = lim An, and B = lim Bk

where An, Bk are elementary ﬁelds. Use a continuity property of K0(·), together with Kirchberg’s theorem and Elliott’s intertwining argument. Thank you for listening!