universal coefficient theorems for c*-algebras over finite...
TRANSCRIPT
![Page 1: Universal coefficient theorems for C*-algebras over finite ...purice/conferences/2011/EUNCG4/Talks/Bentmann.pdf · C∗-algebras over topological spaces Bivariant equivariant K-theory](https://reader033.vdocuments.mx/reader033/viewer/2022042418/5f34c31ecd69ac5a0a116c19/html5/thumbnails/1.jpg)
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
![Page 2: Universal coefficient theorems for C*-algebras over finite ...purice/conferences/2011/EUNCG4/Talks/Bentmann.pdf · C∗-algebras over topological spaces Bivariant equivariant K-theory](https://reader033.vdocuments.mx/reader033/viewer/2022042418/5f34c31ecd69ac5a0a116c19/html5/thumbnails/2.jpg)
MotivationIngredients
ExamplesMain resultProof ideas
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
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
![Page 3: Universal coefficient theorems for C*-algebras over finite ...purice/conferences/2011/EUNCG4/Talks/Bentmann.pdf · C∗-algebras over topological spaces Bivariant equivariant K-theory](https://reader033.vdocuments.mx/reader033/viewer/2022042418/5f34c31ecd69ac5a0a116c19/html5/thumbnails/3.jpg)
MotivationIngredients
ExamplesMain resultProof 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.
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
![Page 4: Universal coefficient theorems for C*-algebras over finite ...purice/conferences/2011/EUNCG4/Talks/Bentmann.pdf · C∗-algebras over topological spaces Bivariant equivariant K-theory](https://reader033.vdocuments.mx/reader033/viewer/2022042418/5f34c31ecd69ac5a0a116c19/html5/thumbnails/4.jpg)
MotivationIngredients
ExamplesMain resultProof ideas
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 ).
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
![Page 5: Universal coefficient theorems for C*-algebras over finite ...purice/conferences/2011/EUNCG4/Talks/Bentmann.pdf · C∗-algebras over topological spaces Bivariant equivariant K-theory](https://reader033.vdocuments.mx/reader033/viewer/2022042418/5f34c31ecd69ac5a0a116c19/html5/thumbnails/5.jpg)
MotivationIngredients
ExamplesMain resultProof ideas
C∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces
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.
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
C∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces
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 ).
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
![Page 7: Universal coefficient theorems for C*-algebras over finite ...purice/conferences/2011/EUNCG4/Talks/Bentmann.pdf · C∗-algebras over topological spaces Bivariant equivariant K-theory](https://reader033.vdocuments.mx/reader033/viewer/2022042418/5f34c31ecd69ac5a0a116c19/html5/thumbnails/7.jpg)
MotivationIngredients
ExamplesMain resultProof ideas
C∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces
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.
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
C∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces
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 .
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
C∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces
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)
.
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
![Page 10: Universal coefficient theorems for C*-algebras over finite ...purice/conferences/2011/EUNCG4/Talks/Bentmann.pdf · C∗-algebras over topological spaces Bivariant equivariant K-theory](https://reader033.vdocuments.mx/reader033/viewer/2022042418/5f34c31ecd69ac5a0a116c19/html5/thumbnails/10.jpg)
MotivationIngredients
ExamplesMain resultProof ideas
C∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces
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
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
C∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces
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.
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
C∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces
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.
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
C∗-algebras over topological spacesBivariant equivariant K -theoryFiltrated K -theoryFinite T0-spaces
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
II
Hasse diagram
{transitively reduced directed acyclic graphs with vertex set X}
reachability relation
II
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
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)
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
Type (A) spaces•
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Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
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.
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
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.
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
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 ))
c
restricts to a bijective correspondence between exact ungradedNT ∗(O)-modules and exact ungraded NT ∗(X )-modules.
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
Positive directionNegative direction
Proof of UCT for type (A) spaces
This allows to carry over the arguments by Meyer-Nest.
LemmaLet M ∈Mod
(NT ∗(X )
)c.
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 ).
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
Positive directionNegative direction
Some non-type (A) spaces
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Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
Positive directionNegative direction
Some non-type (A) spaces
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Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
Positive directionNegative direction
Counterexamples
Meyer-Nest construct to non-equivalent objects with isomorphicfiltrated K -theory for the space
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Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
Positive directionNegative direction
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;
Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
Positive directionNegative direction
Counterexamples
If there is a vertex of Γ(X ) with unoriented degree 3, “transport”counterexamples from
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Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
Positive directionNegative direction
Counterexamples
If every vertex of Γ(X ) has unoriented degree 2 “transport”counterexamples from
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Rasmus Bentmann UCT for C∗-algebras over finite topological spaces
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MotivationIngredients
ExamplesMain resultProof ideas
Positive directionNegative direction
Thank you for your attention!•
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Rasmus Bentmann UCT for C∗-algebras over finite topological spaces