kasparov's operator k-theory and applications 2. kk-theory

Kasparov’s operator K-theory and applications2. KK-theory

Georges Skandalis

Universite Paris-Diderot Paris 7Institut de Mathematiques de Jussieu

NCGOA Vanderbilt UniversityNashville - May 2008

Georges Skandalis (Univ. P7/ IMJ) Kasparov’s KK-theory - 2. KK-theory Vanderbilt U. May 6, 2008 1 / 22

Topics:

1 K -theory.

2 KK-theory.

3 Baum-Connes conjecture.

4 Lafforgue’s Banach KK -theory.


Recall: K-homology?

Definition of Ell (via Z/2Z-graded formalism)

Ell(A) (A unital C ∗-algebra) set of pairs (π,F ) where

π : A → L(H)(0) ∗-representation of A on a graded Hilbert space H.

F ∈ L(H)(1)

1− F 2 ∈ K(H) and [F , π(a)] ∈ K(H) (for a ∈ A).

Equivalence relation?

Brown-Douglas-Fillmore studied extensions of C (X ) by K (of type0 → K → B → C (X ) → 0): These extensions form a group Ext(X )recognized to be the K -homology group K 1(C (X )) = K1(X ).

BDF Ext: answer to Atiyah’s problem of finding analytic definition ofK -homology.

Many more contributions: Voiculescu, Arveson,Pimsner-Popa-Voiculescu...Georges Skandalis (Univ. P7/ IMJ) Kasparov’s KK-theory - 2. KK-theory Vanderbilt U. May 6, 2008 2 / 22

Kasparov’s KK theoryHilbert C*-modules

‘Ell’ → KK : Hilbert spaces → Hilbert C ∗-modules: Paschke, Rieffel,Kasparov.

Definition

Prehilbert B-module: right B-module E with B-valued scalar product〈 , 〉 : E × E → B such that:

1 〈x , λy〉 = λ〈x , y〉, 〈x , yb〉 = 〈x , y〉b, (∀x , y ∈ E , λ ∈ C, b ∈ B).

2 〈y , x〉 = 〈x , y〉∗, (∀x , y ∈ E ).

3 〈x , x〉 ≥ 0, (∀x ∈ E ).

Set ‖x‖ = ‖〈x , x〉‖1/2: semi-norm on E .If E is Hausdorff and complete Hilbert B-module.


Hilbert C*-modules

Remarks

1 〈λx , y〉 = λ〈x , y〉 and 〈xb, y〉 = b∗〈x , y〉.2 E prehilbert B-module. N = {x ∈ E , 〈x , x〉 = 0} submodule of E

Hausdorff completion of E (completion of E/N) Hilbert B-module.

Examples

B is a Hilbert B-module: 〈x , y〉 = x∗y (same norm as B).

Bn Hilbert B-module (n ∈ N): 〈(xi ), (yi )〉 =∑

x∗i yi .

(Ei )i∈I Hilbert B-modules. Hilbert B-module⊕

Ei scalar product

〈(xi ), (yi )〉 =∑

〈xi , yi 〉 (completion).⊕i∈N

B fundamental Hilbert B-module noted HB .


Operators; compact operators

Adjoint: not automatic

Definition

E1,E2 Hilbert B-modules. L(E1,E2): maps T : E1 → E2 with adjointT ∗ : E2 → E1 〈Tx1, x2〉 = 〈x1,T

∗x2〉 (∀x1 ∈ E1, x2 ∈ E2).Put L(E ) = L(E ,E ).

T ∈ L(E1,E2) ⇒ linear, B-linear and bounded; T ∗ unique, (T ∗)∗ = T .

With norm of bounded operators, L(E ) is a C ∗-algebra.

Definition

E1,E2 Hilbert B-modules and x1 ∈ E1 and x2 ∈ E2.

Define θx1,x2 ∈ L(E2,E1) by θx1,x2(x) = x1〈x2, x〉. Note θx1,x2 = θ∗x2,x1.

Norm closure of vector span of θx2,x1 : compact operators K(E1,E2).

K(E ) = K(E ,E ) (closed two sided) ideal in L(E ).


Hilbert C*-modules

Theorem (Kasparov)

For every Hilbert B-module E , M(K(E )) = L(E ). In particularM(K ⊗ B) = L(HB).

Kasparov’s Stabilization Theorem

For every countably generated Hilbert B-module E , HB ⊕ E is isomorphicto HB .

Adaptation of Gram-Schmidt orthonormalization method.

Graded Hilbert modulesDecomposition E = E (0) ⊕ E (1). L(E ) is then Z/2Z-graded.


Kasparov’s bifunctor

Definition

A,B C ∗-algebras.

E(A,B) triples (E , π, F ) where

∗ E is a Z/2Z-graded Hilbert B-module.∗ π : A → L(E )(0) is a ∗-homomorphism.∗ F ∈ L(E )(1) satisfies: ∀a ∈ A, π(a)(F 2 − 1) ∈ K(E )

and [π(a),F ] ∈ K(E ).

D(A,B): degenerate (E , π, F ): π(a)(F 2 − 1) = 0 and [π(a),F ] = 0.

Addition: (E , π, F ) + (E ′, π′,F ′) = (E ⊕ E ′, π ⊕ π′,F ⊕ F ′).

Homotopy in E(A,B): element of E(A,B[0, 1]).

KK (A,B) set of homotopy classes of elements of E(A,B).


The KK -groups

Theorem

KK (A,B) is an abelian group

Degenerate elements are homotopic to 0.The opposite of (E , π, F ) is (−E , π,−F ), where −E is E with oppositegrading.

Theorem

KK (A,B) homotopy invariant bifunctor: covariant in B, contravariant inA.

Theorem

For every C ∗-algebra B, KK (C,B) = K0(B).


The KK-product (Kasparov)

Definition

1A ∈ KK (A,A) class of (A, iA, 0) where A(1) = 0 andiA : A → K(A) ⊂ L(A) (i(a)b = ab, a, b ∈ A).

Theorem1 There is a well defined bilinear coupling (Kasparov product)

KK (A,D)× KK (D,B) → KK (A,B) noted (x , y) 7→ x ⊗D y.

2 Covariant in B, contravariant in A;

3 f : D → E ∗-homomorphism, x ∈ KK (A,D) and y ∈ KK (E ,B)f∗(x)⊗E y = x ⊗D f ∗(y).

4 ∀x ∈ KK (A,B), x ⊗B 1B = 1A ⊗ x = x.

5 Associative: ∀x ∈ KK (A,D), y ∈ KK (D,E ), z ∈ KK (E ,B) we have:(x ⊗D y)⊗E z = x ⊗D (y ⊗E z).


Extending KK-product

Definition

τD : KK (A,B) → KK (A⊗ D,B ⊗ D):(E , π, F ) 7→ (E ⊗ D, π ⊗ iD ,F ⊗ 1).

A1,A2,B1,B2,D C ∗-algebras,x ∈ KK (A1,B1 ⊗ D), y ∈ KK (D ⊗ A2,B2).

Put x ⊗D y = τA2(x)⊗B1⊗D⊗A2 τB1(y).

Theorem1 Same properties as above.

2 Product over C commutative.


Abstract periodicity, abstract duality

Theorem (abstract periodicity)

α ∈ KK (D,E ), β ∈ KK (E ,D) such that α⊗E β = 1D and β ⊗D α = 1E .We say D and E are K-equivalent.• ⊗D α : KK (A,B ⊗ D) → KK (A,B ⊗ E ) andβ ⊗D • : KK (D ⊗ A,B) → KK (E ⊗ A,B) isomorphismsinverses • ⊗E β and α⊗E •.

K -equivalent C ∗-algebras cannot be distinguished by KK -theory.

Theorem (abstract duality)

ρ ∈ KK (D ⊗ E , C), σ ∈ KK (C,E ⊗ D) such that σ ⊗D ρ = 1E andσ ⊗E ρ = 1D . We say D and E are K-dual.σ ⊗D • : KK (D ⊗ A,B) → KK (A,E ⊗ B) andσ ⊗E • : KK (A⊗ E ,B) → KK (A,B ⊗ D) isomorphismsinverses • ⊗E ρ and • ⊗D ρ).


Bott periodicity

β ∈ KK (C,C0(R2)) class of (E , π, F ) where

E (0) = E (1) = C0(R2);

π : C → L(E )

F (ξ, η) = (P∗η, Pξ) where P : C0(R2) → C0(R2) givenPξ(s, t) = (1 + s2 + t2)−1/2(s + it)ξ(s, t), (s, t) ∈ R2.

1− F 2 ∈ K(E ) (multiplication by (1 + s2 + t2)−1 ∈ C0(R2)).

α ∈ KK ((C0(R2), C) class of (H, π, D) where

H(0) = H(1) = L2(R2);

π action by multiplication;

D(ξ, η) = (Tη, T ∗ξ) where T Fourier transform of P (T ξ = P ξ).

Theorem (Bott periodicity)

KK (A,B(R2)) ' KK (A,B) and KK (A(R2),B) ' KK (A,B).


Other consequences of KK-product

Theorem (Bott periodicity)

m, n ≥ 0 we have:

if m + n is even, KK (A(Rm),B(Rn)) ' KK (A,B);

if m + n is odd,KK (A(Rm),B(Rn)) ' KK (A,B(R)) ' KK (A(R),B) := KK 1(A,B).

Theorem (Thom isomorphism)

X locally compact space and let E (total space) complex vector bundleover X . C0(X ) and C0(E ) are K-equivalent.

Proposition (Stability)

For every C ∗-algebra A, A and A⊗K are K-equivalent.

Generalization: Morita equivalence.


Equivariant KK-theory

Actions of G locally compact group on A and B by same letter α.

Definition1 Equivariant Hilbert B-module E : (pointwise) cont. action α of G :

αg (ξb) = αg (ξ)αg (b), αg (〈ξ, η〉) = 〈αg (ξ), αg (η)〉,∂(αg (ξ)) = ∂(ξ) (b ∈ B, ξ, η ∈ E and g ∈ G ).

2 equivariant bimodule: π : A → L(E ): αg (π(a)ξ) = π(αg (a))αg (ξ)(a ∈ A, g ∈ G , ξ ∈ E ).

3 αg (F )ξ = αg (Fαg−1(ξ)).

4 F said to be G-continuous: g 7→ αg (F ) norm continuous.

5 EG (A,B)

• (E , π, F ) ∈ E(A,B);• E equivariant bimodule;• F G -continuous and αg (F )− F ∈ K(E ) (g ∈ G ).

6 KKG (A,B) group of homotopy classes of elements of EG (A,B).


Constructions1 KK-product extends to equivariant case.2 Restriction. r : G → H continuous group homomorphism. Restriction

r∗ : KKH(A,B) → KKG (A,B).3 Induction If H is a closed subgroup of G , Kasparov constructs an

induction homomorphism indGH maps KKH(A,B) to

KKG (C (G ×H A),C (G ×H B)).

• Both restriction and induction have a good behaviour with respect toKasparov product.

4 Descent morphism jG A,B-bimodule (E , π) we associate anA oα G , B oα G -bimodule (E oα G , πα).

Theorem

1 If (E , π, F ) ∈ EG (A,B) then (E oα G , πα,F ⊗B 1) ∈ E(A oα G ,B oα G ).

2 Gives group homomorphism jG : KKG (A,B) → KK (A oα G ,B oα G ).

3 Kasparov product: jG (x ⊗B y) = jG (x)⊗BoαG jG (y) and jG (1A) = 1AoαG .


Generalizations

Groupoids: Le Gall.

Hopf algebras: Baaj-Sk


Index Theory

Ideal setting for Atiyah-Singer index theorems and generalizations.

Atiyah-Singer index theoremElliptic pseudodifferential operator P of order 0 on manifold M defines anelement [P] ∈ KK (C (M, C) (starting point of Ell)

Index of P i.e. homomorphism from K0(C (M)) to Z = K0(C) it defines iscomputed in terms of [σP ].

Correspondence between [P] ↔ [σP ] isomorphism (Poincare duality)KK (C (M), C) ↔ KK (C,C0(T

∗M)).

Topological index also seen as an element of KK .

Theorem

Topological index=Analytical index. Equality of KK-elements.


Generalizations

Many more examples of natural KK -elements related with index problems:

The Atiyah-Singer index theorem for families: Families of manifoldsindexed by a locally compact space Y : fibration M → Y with fibersmanifolds. Pseudodifferential family: KK (C (M),C (Y )).

Longitudinal index theorem for foliations (Connes+Sk)

Atiyah’s index theorem for covering spaces

Mishchenko and Fomenko index theorem

Signature operators on Lipschitz manifolds (Teleman, Hilsum)

Transversally elliptic operators for foliations (Hilsum-Sk)


C*-algebraic extensions

Many C ∗-algebras are formed out of simpler blocks through extensions.

Important to give invariants that classify different possible extensions of apair of given C ∗-algebras.

Already mentioned important work of Brown-Douglas-Fillmore extensions0 → K → B → C (X ) → 0.


Kasparov’s Bivariant Ext

Extensions of A by B ⊗K.

Trivial: if split.

Consider extensions: 0 → B ⊗Kij−→Dj

pj−→A → 0 (j = 0, 1)

Sum given by set of matrices

(x1 yz x2

)with y , z ∈ B ⊗K and

xi ∈ Di satisfying p1(x1) = p2(x2).

Unitary equivalence:

0 → B ⊗K → D1 → A → 0Ad(u) ↓ h ↓ ‖

0 → B ⊗K → D2 → A → 0

Equivalence of extensions σ1 ∼ σ2 if there exist trivial extensionsτ1, τ2 with σ1 + τ1 and σ2 + τ2 unitarily equivalent.Extensions form a semigroup Ext(A,B).


Results

Theorem (Kasparov)

Extension 0 → B ⊗K → D → A is invertible in Ext(A,B) iff it admits acompletely positive splitting (generalized ‘Stinespring theorem’).

Theorem (Kasparov)

The group of invrtible elements Ext(A,B)−1 isKK 1(A,B) := KK (A,B(R)).

In particular Ext−1 is homotopy invariant and satisfies all niceperiodicities...

Finally, Kasparov shows that in each class of Ext(A,B) there is a (unique)element which absorbs all trivial extensions (generalized ‘Voiculescutheorem’ - asumption of nuclearity).


More properties of KK

Exact sequences in both variables. Assumption of nuclearity (or cplifting).

Roseberg-Schochet: computation in terms of K -theory. UniversalCoefficient Formula. (Good cases).

Connes-Higson: E (A,B) exact in all cases. In a way: group of formaldifferences of Ext(A,B).


kasparov's operator k-theory and applications 2. kk-theory

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