Universal coefficient theorems for C*-algebras over finite...

MotivationIngredients

ExamplesMain resultProof ideas

Universal coefficient theorems for C ∗-algebrasover finite topological spaces

Rasmus Bentmann(joint work with Manuel Kohler)

April 30, 2011EU-NCG 4th Annual Meeting

Bucharest

Rasmus Bentmann UCT for C∗-algebras over finite topological spaces

Outline

1 Motivation: the non-equivariant setting

2 Ingredients in the equivariant settingC∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces

3 Examples

4 Main result

5 Proof ideas

Classification of simple nuclear purely infinite C ∗-algebras

Theorem (Rosenberg-Schochet 1987)Let A and B be separable C∗-algebras.If A belongs to the bootstrap class, then there isa short exact sequence of Z/2-graded Abelian groups

Ext1(K∗+1(A),K∗(B))� KK∗(A,B) � Hom

(K∗(A),K∗(B)

Theorem (Kirchberg-Phillips 2000)A KK-equivalence between two stable, nuclear, separable,purely infinite, simple C∗-algebras lifts to a ∗-isomorphism.

C∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces

C ∗-algebras over topological spaces

Throughout this talk, X denotes a finite T0-space.

DefinitionA C∗-algebra over X is a pair (A, ψ) consisting of a C∗-algebra Aand a continuous map ψ : Prim(A)→ X .

An open subset U ⊆ X gives a distinguished ideal A(U) of A.A ∗-homomorphism f : A→ B is X -equivariant iff(A(U)

)⊆ B(U) for all U ∈ O(X ).

A subset Y ⊆ X is locally closed if and only ifY = U \ V for V ,U ∈ O(X ) with V ⊆ U.Define distinguished subquotient A(Y ) := A(U)/A(V ).

Bivariant equivariant K -theory

Kirchberg constructed an X -equivariant version KK(X )of Kasparov’s KK -theory.The cycles fulfill an appropriate equivariance condition.

Kasparov product

KK∗(X ; A,B)⊗ KK∗(X ; B,C)→ KK∗(X ; A,C)

DefinitionLet KK(X ) be the category of separable C∗-algebras over X withKK0(X ; A,B) as morphisms, Kasparov product as composition.

Properties of the category KK(X )

The category KK(X ) isadditive,triangulated (by the suspension functor and the class oftriangles isomorphic to a mapping cone triangle);

it hasBott periodicity,six-term exact sequences for semi-split extensions,countable coproducts.

Moreover,KK(X ) is functorial in the space variable,has exterior products ⊗ : KK(X )× KK(Y )→ KK(X × Y ).

Universal property and classification

Higson’s Description in the equivariant case (Bonkat)The canonical functor C∗sep(X )→ KK(X ) is the universalsplit-exact C∗-stable (homotopy invariant) functor.

Theorem (Kirchberg 2000)A KK (X )-equivalence between two stable, nuclear, separable,purely infinite, tight C∗-algebras over X lifts toan X-equivariant ∗-isomorphism.

Here a C∗-algebra (A, ψ) over X is called tightif ψ : Prim(A)→ X is homeomorphic.

The equivariant bootstrap class

Definition (Meyer-Nest)The bootstrap class B(X ) ⊆ KK(X ) is the localising subcategoryof KK(X ) generated by the objects ixC for all x ∈ X .It is the smallest class of objects containing ixC closed under

suspensions,KK(X )-equivalence,mapping cones,countable direct sums.

Proposition (Meyer-Nest)If A ∈ KK(X ) and A(X ) is nuclear, then A ∈ B(X ) if and only ifA({x}) ∈ B for every x ∈ X .

Filtrated K -theory

For each locally closed subset Y ∈ LC(X ), define a functor

FKY : KK(X )→ AbZ/2c , FKY (A) := K∗

(A(Y )

For Y ,Z ∈ LC(X ), let NT (Y ,Z ) be the Abelian group ofnatural transformations FKY ⇒ FKZ .The category NT with object set LC(X ) and morphismsNT (Y ,Z ) is Z/2-graded and pre-additive.Filtrated K -theory is the functor

FK =(FKY

)Y∈LC(X)

: KK(X )→Mod(NT )c,

A 7→(

K∗(A(Y )

))Y∈LC(X)

Canonical generators and relations

Canonical generatorsFor Y ∈ LC(X ), U ∈ O(Y ), C = Y \ U there is an extension

A(U) � A(Y ) � A(C)

and hence a natural sequence

FKU(A)i−→ FKY (A)

r−→ FKC (A)δ−→ FKU(A).

Canonical relationsthe compositions rC

Y ◦ iYU , δU

C ◦ rCY , iY

U ◦ δUC vanish

various relations following from the naturality of the six-termsequence with respect to morphisms of extensions

Representability and universality

Representability theorem (Meyer-Nest)For every Y ∈ LC(X ) there is a separable C∗-algebra RY over Xand a natural isomorphism KK∗(X ;RY , ) ∼= FKY .

We have FK ∼= KK∗(X ;R, ) for R =⊕

Y∈LC(X)RYand NT ∼= KK∗(X ;R,R).

Universality of filtrated K -theory (Meyer-Nest)Filtrated K -theory FK : KK(X )→Mod(NT )cis universal among the stable homological functors F withF (f ) = 0 for all f ∈ KK(X ; A,B) with FK(f ) = 0.

UCT with homological condition

Theorem (Meyer-Nest)Let A,B ∈ KK(X ). Suppose thatFK(A) ∈Mod(NT )c has a projective resolution of length 1and that A ∈ B(X ). Then there are natural short exact sequences

Ext1NT(FK(A)[j + 1],FK(B)

)� KKj(X ; A,B)

� HomNT(FK(A)[j],FK(B)

)for j ∈ Z/2.

Intermezzo: finite T0-spaces as directed graphs

Let X be a finite set. There are the following bijections:

{T0-topologies on X}

specialisation preorder

{partial orders on X}

Alexandrov topology

Hasse diagram

{transitively reduced directed acyclic graphs with vertex set X}

reachability relation

Examples

1 X = •C∗alg(X ) = plain C∗-algebrasUniversal Coefficient Theorem: Rosenberg-Schochet (1987)

2 X = • // •C∗alg(X ) = Extensions of C∗-algebrasFK = Six-term exact sequenceRørdam (1997): Classification of extensions of purely infinitesimple stable separable C∗-algebras in the bootstrap classUCT: Bonkat (2002)

3 X = • // • // •UCT: Restorff (2008)

4 X = • // • // • · · · · · · • // •UCT: Meyer-Nest (2008)

Type (A) spaces•

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Main result

Theorem (B-Kohler)Let X be finite T0-space. The following statements are equivalent:

1 X is a disjoint union of spaces of type (A).2 Let A and B be separable C∗-algebras over X.

Suppose A ∈ B(X ).Then there is a natural short exact UCT sequence

Ext1NT(FK(A)[1],FK(B)

)� KK∗(X ; A,B)

� HomNT(FK(A),FK(B)

3 Let A,B ∈ B(X ). Then FK(A) ∼= FK(B) implies A 'KK(X) B.

Classification of certain non-simple C ∗-algebras

CorollaryLet X be a disjoint union of spaces of type (A).Filtrated K-theory induces a bijection between the sets ofisomorphism classes of

tight, stable, purely infinite, separable, nuclear C∗-algebrasover X with simple subquotients in the bootstrap class

and ofcountable, exact NT -modules.

Here M is called exact if the sequenceM(U)

i−→ M(Y )r−→ M(C)

δ−→ M(U) is exactfor every Y ∈ LC(X ), U ∈ O(Y ), C = Y \ U.

Positive directionNegative direction

Proof of UCT for type (A) spaces

Reduce to the totally ordered case proven by Meyer-Nest using thefollowing observation:Let X be of type (A). Let O be the totally ordered space with thesame number of points.

Theorem (B-Kohler)There is an ungraded isomorphism Φ: NT ∗(X )→ NT ∗(O), and

Φ∗ : Modungr(NT ∗(O))

c →Modungr(NT ∗(X ))

restricts to a bijective correspondence between exact ungradedNT ∗(O)-modules and exact ungraded NT ∗(X )-modules.

Proof of UCT for type (A) spaces

This allows to carry over the arguments by Meyer-Nest.

LemmaLet M ∈Mod

(NT ∗(X )

M is projective ⇐⇒ M is exact and has free entries.M has a projective resolution of length 1⇐⇒ M is exact,⇐⇒ M = FK(A) for some A ∈ KK(X ).

Some non-type (A) spaces

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Some non-type (A) spaces

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Counterexamples

Meyer-Nest construct to non-equivalent objects with isomorphicfiltrated K -theory for the space

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Counterexamples

We copy their method to exhibit counterexamples for allspaces in the previous list.We reduce to spaces from this list:

Lemma1 Let Y ∈ LC(X ). ¬UCT (Y ) =⇒ ¬UCT (X ).2 Let f : X → Y , g : Y → X be continuous with f ◦ g = idY .¬UCT (Y ) =⇒ ¬UCT (X ).

ObservationIf X is a connected non-type (A) space, then either

some vertex of Γ(X ) has unoriented degree 3 or more;every vertex of Γ(X ) has unoriented degree 2;

Counterexamples

If there is a vertex of Γ(X ) with unoriented degree 3, “transport”counterexamples from

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Counterexamples

If every vertex of Γ(X ) has unoriented degree 2 “transport”counterexamples from

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Thank you for your attention!•

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Universal coefficient theorems for C*-algebras over finite...

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