On Baum Connes conjecture Homology KK G Kasparov product KK G - PowerPoint PPT Presentation

On Baum Connes conjecture Ryszard Nest Non- commutative topology C*-algebras On Baum Connes conjecture Homology KK G Kasparov product KK G -category Assembly Ryszard Nest Baum-Connes conjecture Categorical reformulation University of Copenhagen Example: Γ = Z 16th June 2010

On Baum Basic generalisation of a locally compact Hausdorff space is a Connes conjecture C*-algebra. The idea is to look at the functor Ryszard Nest X � C 0 ( X ) Non- commutative topology and replace C 0 ( X ) by a non-commutative C*-algebra. C*-algebras Homology A C*-algebra is a norm closed subalgebra of B ( H ) (bounded KK G Kasparov product operators on a Hilbert space H ) closed under taking adjoints KK G -category a → a ∗ . First examples Assembly Baum-Connes 1 M n ( C ); H = C n , conjecture Categorical reformulation 2 C 0 ( X ); h = L 2 ( X , µ ) where µ is any positive Radon Example: Γ = Z measure nonvanishing on any open subset of X and f ∈ C 0 ( X ) acts by multiplication L 2 ( X ) ∋ ξ → f ξ ∈ L 2 ( X ) . In fact, any abelian C*-algebra is of this form. 3 K ( H ) the algebra of all compact operators on H .

On Baum Connes conjecture Ryszard Nest The basic norm identity is Non- commutative topology || a ∗ a || = || a || 2 . C*-algebras Homology KK G Kasparov product C*-algebras form a category, with KK G -category Assembly Mor C ∗ ( A , B ) = { φ : A → B | φ is a *-homomorphism } . Baum-Connes conjecture Categorical reformulation The basic C*-identity implies a sensible notion of positivity, and Example: Γ = Z in particular, every *-homomorphism is automatically continuous. What distinguishes a C*-algebra from complex numbers is the fact that the unit ball is not round.

On Baum Connes conjecture We can always add some extra structure, f. ex. a G - action Ryszard Nest Non- α : G → Aut ( A ) commutative topology C*-algebras by *-automorphisms, where G is a (second countable) locally Homology KK G compact group and α is a pointwise continuous Kasparov product KK G -category homomorphism. In this case Assembly Baum-Connes Mor G conjecture C ∗ ( A , B ) Categorical reformulation Example: Γ = Z consists of *-homomorphisms preserving group action.

On Baum Connes conjecture We can always add some extra structure, f. ex. a G - action Ryszard Nest Non- α : G → Aut ( A ) commutative topology C*-algebras by *-automorphisms, where G is a (second countable) locally Homology KK G compact group and α is a pointwise continuous Kasparov product KK G -category homomorphism. In this case Assembly Baum-Connes Mor G conjecture C ∗ ( A , B ) Categorical reformulation Example: Γ = Z consists of *-homomorphisms preserving group action. Topology The category of Abelian G-C*-algebras coincides with the category of pointed compact Hausdorff G-spaces.

On Baum Connes conjecture Ryszard Nest Definition A non-commutative homology theory is a functor on a category Non- commutative of (separable) C ∗ -algebras (with extra structure) that is topology C*-algebras Homology KK G Kasparov product KK G -category Assembly Baum-Connes conjecture Categorical reformulation Example: Γ = Z

On Baum Connes conjecture Ryszard Nest Definition A non-commutative homology theory is a functor on a category Non- commutative of (separable) C ∗ -algebras (with extra structure) that is topology C*-algebras • C ∗ -stable (Morita invariant) Homology KK G Kasparov product KK G -category Assembly Baum-Connes conjecture Categorical reformulation Example: Γ = Z

On Baum Connes conjecture Ryszard Nest Definition A non-commutative homology theory is a functor on a category Non- commutative of (separable) C ∗ -algebras (with extra structure) that is topology C*-algebras • C ∗ -stable (Morita invariant) Homology KK G Kasparov product • split-exact KK G -category Assembly Baum-Connes conjecture Categorical reformulation Example: Γ = Z

On Baum Connes conjecture Ryszard Nest Definition A non-commutative homology theory is a functor on a category Non- commutative of (separable) C ∗ -algebras (with extra structure) that is topology C*-algebras • C ∗ -stable (Morita invariant) Homology KK G Kasparov product • split-exact KK G -category • homotopy invariant Assembly Baum-Connes conjecture Categorical reformulation Example: Γ = Z

On Baum Connes conjecture Ryszard Nest Definition A non-commutative homology theory is a functor on a category Non- commutative of (separable) C ∗ -algebras (with extra structure) that is topology C*-algebras • C ∗ -stable (Morita invariant) Homology KK G Kasparov product • split-exact KK G -category • homotopy invariant Assembly Baum-Connes conjecture • has Puppe exact sequence for mapping cones Categorical reformulation Example: Γ = Z

On Baum Connes conjecture Ryszard Nest Definition A non-commutative homology theory is a functor on a category Non- commutative of (separable) C ∗ -algebras (with extra structure) that is topology C*-algebras • C ∗ -stable (Morita invariant) Homology KK G Kasparov product • split-exact KK G -category • homotopy invariant Assembly Baum-Connes conjecture • has Puppe exact sequence for mapping cones Categorical reformulation Example: Γ = Z Example K-theory is a non-commutative homology theory for C ∗ -algebras. It maps separable C ∗ -algebras to the category Ab Z / 2 of Z / 2-graded countable Abelian groups. c

Example On Baum Connes conjecture KK G is a (bivariant) non-commutative homology theory Ryszard Nest for C ∗ -algebras with a G -action. Non- commutative Cycles in KK G ( A , B ) topology C*-algebras Homology KK G • H B is a right Hilbert B -module; Kasparov product KK G -category • ϕ : A → B ( H B ) is a ∗ -representation; Assembly • F ∈ B ( H B ) ; Baum-Connes conjecture Categorical • ϕ ( a )( F 2 − 1 ) , ϕ ( a )( F − F ∗ ) , and [ ϕ ( a ) , F ] are compact reformulation Example: Γ = Z for all a ∈ A ; • in the even case, γ is a Z / 2-grading on H B ; • H B carries a representation U of G which implements action of G and commutes with F up to compacts. A cycle is trivial, if all the "compacts" above vanish, and two cycles are equivalent, if they are homotopic after adding trivial cycles.

� � � � On Baum Some properties of KK G Connes conjecture Ryszard Nest 1 The classes in KK G 1 ( A , B ) are given by semisplit extensions: 0 → B ⊗ K → E → A → 0 Non- commutative topology 2 Kasparov product C*-algebras KK G i ( A , B ) × KK G j ( B , C ) → KK G i + j ( A , C ) Homology KK G 3 Excision. Given a semisplit short exact sequence Kasparov product KK G -category 0 → I → A → A / I → 0, there exists an associated six Assembly term exact sequence Baum-Connes conjecture Categorical reformulation � KK G � KK G KK G 0 ( A / I , B ) 0 ( A , B ) 0 ( I , B ) Example: Γ = Z ◦ ◦ KK G KK G KK G 1 ( A , B ) 1 ( A , B ) 1 ( A / I , B ) and similarly in the second variable. 4 For G compact group • KK G ∗ ( C , A ) = K G ∗ ( A ) = K ∗ ( A ⋊ G ) - equivariant K-theory • KK G ∗ ( C , C ) = R G - the representation ring of G .

On Baum Connes Suppose that G = Z . Then conjecture The cycles are given as follows Ryszard Nest • An even representation of Z on a Hilbert space Non- H = H + ⊕ H − (hence a pair of unitary operators commutative topology U + ⊕ U − ), C*-algebras Homology • A Fredholm operator F : H + → H − which intertwines U + KK G Kasparov product with U − modulo compacts. KK G -category Assembly Then the class of ( U , F ) gives Baum-Connes conjecture Categorical reformulation Index ( F ) = dim ker F − dim coker F ∈ Z . Example: Γ = Z Theorem (BC for Z ) KK Z 0 ( C , C ) ∋ F → Index ( F ) ∈ Z is an isomorphism.

� � � � On Baum Connes The Kasparov product conjecture Ryszard Nest KK G ∗ ( C , B ) × KK G 1 ( B , C ) → KK G ∗ + 1 ( C , C ) Non- commutative has an explicit description as follows. topology C*-algebras Homology KK G Given class [ D ] ∈ KK G 1 ( B , C ) , represent it by a semisplit extension Kasparov product KK G -category 0 → C ⊗ K → E → B → 0 . Assembly Baum-Connes conjecture Then the pairing Categorical ∩ [ D ] : K G ∗ ( B ) → K G reformulation ∗ + 1 ( C ) Example: Γ = Z coincides with the boundary map δ in the six-term exact sequence � K G � K G K G 0 ( C ) 0 ( E ) 0 ( B ) δ ◦ δ ◦ K G KK G K G 1 ( B ) 1 ( E ) 1 ( C )

On Baum Connes conjecture The universality of Kasparov theory Ryszard Nest Non- commutative Theorem (Joachim Cuntz and Nigel Higson) topology C*-algebras Bivariant KK -theory is the universal C ∗ -stable, split-exact Homology KK G functor on the category of separable C ∗ -algebras. Kasparov product KK G -category That is, a functor from the category of separable C ∗ -algebras Assembly to some additive category factors through KK if and only if it is Baum-Connes conjecture Categorical C ∗ -stable and split-exact, and this factorisation is unique if it reformulation Example: Γ = Z exists.