groupoid correspondences and k-theory - uni-goettingen.de · a groupoid g is a small category in...

Groupoidcorrespondences

and (some)KK-theory

Rohit Dilip Holkar

Topologicalcorrespondences

Introduction andwarm-up

Groupoids and theiractions

Haar groupoids

C∗- algebra of a Haargroupoid

Various notions ofgroupoid morphisms

Definition

Composition oftopologicalcorrespondences

Towards a maintheorem

First theorem

Correspondencesand K-theory

KK-theory viaunbounded operators

Summary

Groupoid correspondences and K-theory

Rohit Dilip Holkar

Mathematics Institute,Georg-August-Universäty, Göttingen.

03 June 2014,K-theory and Index Theory ,

Metz.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Outline

Topological correspondences

Correspondences and K-theory

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Outline

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Plan of talk

1. Some basics about groupoids with Haar system andC ∗-correspondences.

2. Various notions of groupoid morphisms in literature.

3. Topological correspondences.

4. KK-theory via unbounded operators.

5. Constructing some unbounded odd KK-cycles usingtopological correspondences.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Groupoids

Definition

A groupoid G is a small category in which every arrow isinvertible.

We denote the base space of groupoid G by G (0) and thearrow space by G (1).Important terminology: inverse map, range map and sourcemap.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Topological groupoids

Definition

G is a topological groupoid if G (0) and G (1) are topologicalspaces , all the above maps and the composition arecontinuous.

Examples

1. A topological space X , where X (0) = X andX (1) = identity arrows = X .

2. A topological group W , where W (0) = {?} andW (1) = W .

3. Action groupoid.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Action of a groupoid

Definition: Left action

G is said to act on X if there is an open map ρ : X → G (0),called anchor map, such that,

1. ρ is an open surjection,

2. If G ×s,ρ X := {(g , x) ∈ G × X : s(g) = ρ(x)},then ∃ map m : G ×s,ρ X → X satisfying

m(g , (h, x)) = m(gh, x),

m(id(g), x) = x ,

∀g , h ∈ G (1) and x ∈ X .

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Action of a groupoid

Examples

1. A space Y acting on another space X :A map f : X → Y can be viewed as an anchor map fortrivial action y · x = x , for all x ∈ X , and all y ∈ Y .

2. A group action on a space.

3. Right (or left) multiplication of a groupoid G on itself isa right (resp. left) action of G on itself. The anchormap is the source map s (resp. range map r) in thiscase.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Haar groupoid

Definition

A groupoid G is called locally compact Haar groupoid if,

1. G (0) is locally compact Hausdorff subspace of G ,

2. Topology of G has a countable basis consisting ofrelatively compact Hausdorff subsets.

3. For every u ∈ G (0) Gu := r−1(u) is locally compactHausdorff in the relative topology inherited from G ,

4. G admits a left Haar system {αu}u∈G (0) .

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Haar groupoid: Left Haar system

A heuristic definition

A left Haar system for a locally compact groupoid G is afamily of positive Borel measures {αu}u∈G (0) , where each αuis defined on Gu, such that following conditions hold:

1. Support condition.

2. Continuity conditions for the measure.

3. Invariance under the right multiplication.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Haar groupoid

Examples

1. A second countable, locally compact, Hausdorff groupwith a Haar measure is a Haar groupoid.

2. If (H, βh) is a left Haar groupoid then (H, βh) is a rightHaar groupoid where we define∫f (x)dβh(x) :=

∫f (x−1)dβh(x−1).

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

C ∗- algebra of a Haar groupoid

Let G is locally compact Haar groupoid. For f , g ∈ Cc(G )define:

1. Convolution: f ∗ g(x) :=∫G r(x) f (y) · g(y

−1x) dλ(y).

2. Involution: f ∗(x) = f −1(x−1).

3. I -norm:

Let ||f ||I ,r = supu∈G (0)

∫Gu|f (t)| dλu(t)

||f ||I ,s = supu∈G (0)

∫Gu

|f (t)| dλu(t)

and ||f ||I := max{||f ||I ,r , ||f ||I ,s}.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

C ∗- algebra of a Haar groupoid

Theorem

Let G be a locally compact Haar groupoid. Then Cc(G ) is aseparable, normed ∗-algebra under the convolutionmultiplication and the I -norm and the involution is isometric.

Definition

The C ∗-enveloping algebra C ∗(G ) of Cc(G ) above is definedas the C ∗-algebra of G .

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Various notions of groupoid morphisms

Definition: Usual morphism

A usual morphism from G to H is a functor between thegroupoids, when they are considered as small categories.

Definition: Hilsum-Skandalis morphism

A Hilsum-Skandalis morphism from G to H is given by atopological space X with a left G -action and a rightH-action, such that

1. the actions of G and H commute,

2. the right H-action is free and proper,

3. the anchor map X → G (0) for the left action induces ahomeomorphism X/H ∼= G (0).

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition: Usual morphism

A usual morphism from G to H is a functor between thegroupoids, when they are considered as small categories.

Definition: Hilsum-Skandalis morphism

A Hilsum-Skandalis morphism from G to H is given by atopological space X with a left G -action and a rightH-action, such that

1. the actions of G and H commute,

2. the right H-action is free and proper,

3. the anchor map X → G (0) for the left action induces ahomeomorphism X/H ∼= G (0).

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition: Morita equivalence (Muhly-Renault-Williams)

A Morita equivalence between two topological groupoids Gand H is give by a space X with G y X , X x H andfollowing conditions are satisfied,

1. the actions are principal,

2. the actions commute i.e. g(xh) = (gx)h for all g ∈ G ,x ∈ X and h ∈ H,

3. the anchor map for the right action induces a bijectionbetween G/X to H(0),

4. the anchor map for the left action induces a bijectionbetween X\H and G (0).

A M.E. from G to H produces strong M.E. from C ∗(G ) toC ∗(H).

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition: Algebraic morphism (Buneci and Stachura ;2005)

An algebraic morphism from G to H is a right action of Gon H that commutes with the right action of H on itself byright multiplication.

Use of algebraic morphism (Buneci and Stachura

An algebraic morphism from groupoid G to groupoid Hinduces a ∗-homomorphism C ∗(G )→M(C ∗(H)).

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition: Generalized algebraic morphism

A generalized algebraic morphism from G to H is atopological space X and actions G y X and X x H suchthat following conditions hold,

1. Actions of G and H on X commute i.e. g(xh) = (gx)hfor all a ∈ G , x ∈ X and h ∈ H,

2. The right action X x H is principal.

Examples

1. A Morita equivalence.

2. An algebraic morphism.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition: Generalized algebraic morphism

A generalized algebraic morphism from G to H is atopological space X and actions G y X and X x H suchthat following conditions hold,

1. Actions of G and H on X commute i.e. g(xh) = (gx)hfor all a ∈ G , x ∈ X and h ∈ H,

2. The right action X x H is principal.

Examples

1. A Morita equivalence.

2. An algebraic morphism.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

(Left) Haar space

Left H-Haar space

For a (left) Haar groupoid (H, β) a left H-Haar space is atuple (X , {λh}h∈H(0)) where X x H, ρ : X → H(0) is theanchor map and {λh}h∈H(0) is a family of Borel measuresthat satisfy,

1. supp(λh) = ρ−1(h) := X h, ∀ h ∈ H(0),

2. H-invariance:∫Xr(h)

f (xh)dλr(h)(x) =

∫Xs(y)

f (x̃)dλs(y)(x̃)

For all f ∈ Cc(X )3. Continuity : The map Λ : Cc(X )→ Cc(G (0)) given by

Λ(f )(h) =∫X f (x) dλr(h)(x) is continuous surjection.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Topological correspondence

Definition

A correspondence from groupoid (G , λ) to groupoid (H, µ)is a triple (X , α,∆) where,

1. X is a G -H bispace and action of H is proper.

2. α is a H-invariant family of measures on X .

3. ∆ : G n X → R+ is a continuous function such that foreach u ∈ H(0)∫∫

F (g , x) dλ(g) dαu(x) =∫∫F (g−1, gx)∆(g−1, gx) dλ(g)dαr(u)(x)

In short a topological correspondences from G to H is aspace with a family of measures that is G -invariant andH-quasi invariant.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Topological correspondence

Definition

A correspondence from groupoid (G , λ) to groupoid (H, µ)is a triple (X , α,∆) where,

1. X is a G -H bispace and action of H is proper.

2. α is a H-invariant family of measures on X .

3. ∆ : G n X → R+ is a continuous function such that foreach u ∈ H(0)∫∫

F (g , x) dλ(g) dαu(x) =∫∫F (g−1, gx)∆(g−1, gx) dλ(g)dαr(u)(x)

In short a topological correspondences from G to H is aspace with a family of measures that is G -invariant andH-quasi invariant.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

CompositionLet (X , α,∆1) : (G1, λ1)→ (G2, λ2) and(Y , β,∆2) : (G2, λ2)→ (G3, λ3) be correspondences,their composition consists of

A space with left G1 and right G3 action,

Space

Ω := (X ×G (0) Y )/G2

Family of measure on this spaceDoes it exit...?Yes!

Adjoining function for the left action.

Adjoining function

∆12(γ, [x , y ]) = ∆1(γ, x)

This makes sense as ∆1 is invariant under G2 action.

Skip composition details

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Space

Ω := (X ×G (0) Y )/G2

Adjoining function

∆12(γ, [x , y ]) = ∆1(γ, x)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Space

Ω := (X ×G (0) Y )/G2

Adjoining function

∆12(γ, [x , y ]) = ∆1(γ, x)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Space

Ω := (X ×G (0) Y )/G2

Adjoining function

∆12(γ, [x , y ]) = ∆1(γ, x)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Space

Ω := (X ×G (0) Y )/G2

Adjoining function

∆12(γ, [x , y ]) = ∆1(γ, x)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Space

Ω := (X ×G (0) Y )/G2

Adjoining function

∆12(γ, [x , y ]) = ∆1(γ, x)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Space

Ω := (X ×G (0) Y )/G2

Adjoining function

∆12(γ, [x , y ]) = ∆1(γ, x)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Space

Ω := (X ×G (0) Y )/G2

Adjoining function

∆12(γ, [x , y ]) = ∆1(γ, x)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Space

Ω := (X ×G (0) Y )/G2

Adjoining function

∆12(γ, [x , y ]) = ∆1(γ, x)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Composition

Family of measures

1. (X ×G (0) Y ) := Z carries a family G3-invariant family ofmeasures, which is α×G (0) β.

2. A G3-equivariant continuous positive function b on Zthat is a 0-cocycle b in G3-equivariant continuous R+cohomology of groupoid Z ×Ω Z . It is special function.

3. π : (X ×G (0) Y )→ Ω be the projection map.

λ(f )[x , y ] =

∫f (xγ, γ−1y) dλ2(γ)

is a system of measures along π.

4. Take measure µ on Ω such that b(α×G (0) β) = µ ◦ λ

The function b has to satisfy some technical computationconditions.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Composition

Family of measures

λ(f )[x , y ] =

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Composition

Family of measures

λ(f )[x , y ] =

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Composition

Family of measures

λ(f )[x , y ] =

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Composition of topological correspondences

Definition (Composition)

For correspondences (X , α,∆1) : (G1, λ1)→ (G2, λ2) and(Y , β,∆2) : (G2, λ2)→ (G3, λ3) a composed correspondence(Ω, µ,∆1,2) : (G1, λ1)→ (G3, λ3) is defined by:

1. Space Ω := (X ×G

(0)3

Y )/G2,

2. Family of measure µ = {µu}u∈G (0)3

is a family of

measure that lifts to bα× β on Z for someb′ ∈ HH0G3(Z ∗ Z ,R

∗+) satisfying condition that

d0(b) = ∆.

3. Adjoining function ∆1,2 is the one given by equation.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Bicategory of topological correspondences

We use definition of bicaegory introduced by Bénabou.

Object: a groupoid (G , α) goes to its C ∗-algebraC ∗(G , α)

1-arrow: a correspondences (X , λ) from (G , α) to(H, β) goes to a C ∗-correspondence H(X , λ)from C ∗(G , α) to C ∗(H, β).

2-arrow: A G -H invariant measure preserving functionbetween the spaces that give thecorrespondence.

Proposition (R.D.H.)

Locally compact Haar groupoids with above given data forma bicategory.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

C ∗-correspondences

Definition

A C ∗-correspondence from a C ∗-algebra A to C ∗-algebra Bis a B Hilbert module H with an action of A such and theactions commute.

Definition

A C ∗-correspondence H from A to B is proper if A acts onH by compact operators.

C ∗-correspondence form a bicategory.

Objects: C ∗-algebras

1-arrows: C ∗-correspondences

2--arrows: Equivariant isomorphisms of Hilbert modules

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Towards a main theorem

Let (X , λ,∆) be correspondence from (G , α) to (H, β).Cc(X ) can be made into a Cc(G )-Cc(H) bimodule and ithas a Cc(H)-valued bilinear form for which the formulae aredefined as follows:If φ ∈ Cc(G ) and ψ ∈ Cc(H) and f ∈ Cc(X ) then,Left action: (φ · f )(x) :=∫

G r(x) φ(γ′−1x)f (x) ∆1/2(γ′, γ′−1x) dα(γ′)

Right action: (f · ψ)(x) :=∫Hs(x) f (xγ

−1)φ(γ) dβ(t)

Inner product: 〈f , g〉(γ) :=∫X r(h) f (x)g(xγ) dλ(x)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

G r(x) φ(γ′−1x)f (x) ∆1/2(γ′, γ′−1x) dα(γ′)

−1)φ(γ) dβ(t)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

G r(x) φ(γ′−1x)f (x) ∆1/2(γ′, γ′−1x) dα(γ′)

−1)φ(γ) dβ(t)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

First theorem

Theorem (R.D.H.)

Let (G , α) and (H, β) be locally compact Haar groupoidsand (X , λ,∆) be a measured correspondence. Then Cc(X )can be naturally completed into a C ∗-correspondence fromC ∗(G ) to C ∗(H).

This construction works for reduced C ∗-algebras, too.

Theorem (J. Renault & R.D.H.)

A topological correspondence going to a C ∗-correspondenceis a morphism of bicategories.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

First theorem

Theorem (R.D.H.)

Let (G , α) and (H, β) be locally compact Haar groupoidsand (X , λ,∆) be a measured correspondence. Then Cc(X )can be naturally completed into a C ∗-correspondence fromC ∗(G ) to C ∗(H).

This construction works for reduced C ∗-algebras, too.

Theorem (J. Renault & R.D.H.)

A topological correspondence going to a C ∗-correspondenceis a morphism of bicategories.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

First theorem

Examples

1. A space map f : X → Y is a measured correspondenceas YXX .

2. A group homomorphism G → H is a measuredcorrespondence GHH .

3. A groupoid action G y X is a measuredcorrespondence GXX .

4. If GXH is a Morita equivalence then X can be madeinto H-Haar space.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

One (important!) more exampleI (G , λ, σ) be a measured groupoid

(H, µ, τ) be a measured subgroupoid of G and(K , ν) be a subgroupoid of G

I δG and δH denote modular functions of measures σ ◦ λand τ ◦ µ.

Lemma

If data above is given then ( δGδH ,G , λ−1) is a correspondence

from (H, µ) to (K , ν). to corollary

What does the lemma tell?I If H = G and (KXPoint, λ) is a correspondence, then

composition of correspondence in the lemma and X isprocess of induction a representation.

I If K = G and (GYPoint, λ′) is a correspondence then the

composition of correspondence in lemma and Y isprocess of restriction of representation.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Lemma

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Lemma

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Lemma

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Lemma

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Lemma

Outline

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

R-equivariant C ∗-correspondence

Let A be a C ∗-algebra and R be a group.

Definition

The C ∗-algebra A is called a R-algebra id there is acontinuous homomorphism R → Aut(A).

Let A and B be RC ∗-algebras.

Definition

A C ∗-correspondence E from A to B is called R-equivariantif R-acts on E by unitary operators and

I t(eb) = (te)(tb)

I 〈re1, re2〉 = r〈e1, e2〉I (ra)e = a(re)

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

K-theory via unbounded operators

Definition

Let B be a C ∗-algebra and E be a C ∗-B-module. A denselydefined closed operator D : Dom(D)→ E is called regular ifD∗ is densly defined in E and 1 + DD∗ has dense range.

I Such D is automatically B-linear and Dom(D) is aB-submodule o E .

I There are two operators related to this operators:Resolvent: r(D) := (1 + D∗D)−1/2

Bounded transform: b(D) := D(1 + D∗D)−1/2

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Let R be a group.

Definition (Unbounded equivariant KK-cycle,(Baaj-Julg,1983))

A R-equivariant, odd unbounded bimodule from a R-algebraA to R-algebra B is tuple (E ,D) where E is an equivariantA-B bimodule together with an unbounded regular operatorD on E such that:

1. [D, a] ∈ End(E ) for all a in a dense subalgebra of A,2. a · r(D) ∈ KB(E ), for all a in a dense subalgebra of A.3. The map r 7→ rDr−1 is strictly continuous map

R → End(E ).

to Proof

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Let G be a groupoid and R be group.

Some definitions

A 0-cocycle: is a continuous function on G (0)/G .Set of R-valued 0-cocycles is denoted byZ0(G ;R).

A 1-cocycle: is a continuous homomorphism from G to R.Set of R-valued 1-cocycles is denoted byZ1(G ;R).

For us– a cocycle would always mean a positive 1-cocycle

Definition

A cocycle c ∈ Z1(G ;R) is regular if ker(c) = H admits aHaar system and exact if it is regular and the map

r × s :G ′ → G (0) × Rγ 7→(r(γ), c(γ))

is a quotient map onto its image.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Some definitions

Definition

r × s :G ′ → G (0) × Rγ 7→(r(γ), c(γ))

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Some definitions

Definition

r × s :G ′ → G (0) × Rγ 7→(r(γ), c(γ))

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Some definitions

Definition

r × s :G ′ → G (0) × Rγ 7→(r(γ), c(γ))

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Lemma (Renault, 1980)

Let (G , λ) be a groupoid and c be a cocycle on G for eacht ∈ R cocycle c gives an automorphism Ut of ∗-algebraCc(G ) by formula, Ut(f )(γ) = e

it c(γ)f (γ). Thisautomorphism extends to an automorphism of C ∗(G ).

Proposition (Mesland, 2011)

If (G , λ), (H, α), c , Ut be as above then Ut extends to oneparameter group of unitaries in C ∗(H) (resp. C ∗r (H)).Further more H(G ) is a R-equivariant C ∗(H) (resp. C ∗r (H))a module.

Corollary

Let (G , λ, τ), (H, µ, τ) (K , ν) be as in the example then abovementioned R action makes the C ∗-correspondence related to( δGδH ,G , λ

−1) into a R-equivariant correspondence fromC ∗(H) to C ∗(K ).

Here is the Hilbert module

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Corollary

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Corollary

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

The unbounded operator

Let G , c and K be as above. Then the operator

D : Cc(G )→ Cc(G )f (γ) 7→ c(γ)f (γ)

is a Cc(K )-linear derivation of Cc(G ) considered as abimodule over itself. Moreover, it extends to a self adjointregular operator on the C ∗(K )-Hilbert module H(G ) (similarfor C ∗(K )).

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Getting some KK-cycles– the other theorem

Theorem

Let (G , λ) be a second countable locally compact Hausdorffgroupoid, c be a real 1-exact cocycle on G and (H, α) be anopen subgroupoid of G such that H(0) = G (0). Then for aquasi-invariant measure σ and τ on G and H respectivelythe operator D in last proposition, makes the R-equivariantcorrespondence (H(G ),D) into an odd R-equivariantunbounded bimodule from C ∗(H) to C ∗(K ).Similar statement holds for (Hr (G ),Dr ) from C ∗r (H) toC ∗r (K ).

Skip Proof Sketch of proof: H ⊆ G is open hence we canrealise Cc(H) ⊆ Cc(G ) by extending functions by zerooutside their domain. A similar statement holds for reducedC ∗-algebras.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Proof:

to definition

I Given φ ∈ Cc(H) ⊆ Cc(G ) we use to see that[D, φ]g = D(f ∗ g) for all g ∈ Cc(G ). Hence using thesame proposition for each φ the commutator [D, φ] isbounded.

I It is cleat that The map r 7→ rDr−1 is strictlycontinuous map R → End(E ).

I Only thing to be proven is that φ(1 + DD∗)−1 hasC ∗(K )-compact resolvant.

This operator acts on a g ∈ Cc(G ) as

φ(1 + DD∗)−1 ◦ g(ω) =∫Gf (γ)(1 + c2(γ−1ω))−1 ∆(γ, γ−1ω) dαr(ω)(γ)

∆ = δH/δG is the adjoining function.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Proof:

to definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Proof:

to definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Proof:

to definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Proof:

to definition

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Proof continued:

I Take k(γ, [ω]) := φ(γ)(1 + c2(ω))−1 ∆(γ, ω)Define Kn = rG (supp(φ)× R) ∩ c̄−1([−n, n]) ⊆ G/KDue to exactness of c Kns are compact.

I Take functions enC (G/K , [0, 1]) such that f = 1 on Knand zero outside Kn+1. Define

kn(γ, [ω]) := en[ω]k(γ, [ω])

Each kn is compact.

I Finally we show that {kn}n is a Cauchy sequence inI -norm.

It can be seen easily that this sequence converges toφ(γ)(1 + c2(ω))−1 ∆(γ, ω).

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Proof continued:

kn(γ, [ω]) := en[ω]k(γ, [ω])

Each kn is compact.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Proof continued:

kn(γ, [ω]) := en[ω]k(γ, [ω])

Each kn is compact.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Proof continued:

kn(γ, [ω]) := en[ω]k(γ, [ω])

Each kn is compact.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Proof continued:

kn(γ, [ω]) := en[ω]k(γ, [ω])

Each kn is compact.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Corollary: Mesland’s construction of odd KK-cycles,2011

Let G be a locally compact Hausdorf groupoid andc : G (0) → R be an exact cocycle. The operator D as abovemakes the C ∗-correspondence H(G ) from C ∗(G ) toC ∗(ker(c) into an odd R-equivariant unbounded bimodule.A similar statement holds for reduced C ∗-algebras.

Example

Non-commutative torus

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

Some remarks

Getting cocycles: quasi-invariant measures, correspondingmodular functions.

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

1. Topological correspondences from G to H is a G -Hbispace with a H invariant and G quasi invariant familyof measures indexed by H(0).

2. A topological correspondence between groupoids inducea C ∗-algebraic correspondence between the C ∗-algebrasof the groupoids.

3. For a groupoid G and its two appropriate subgroupoidsH and K a cocycle make the correspondence

C∗(H)H(G )C∗(K) into a R-equivariant correspondence.4. Using the same cocycle we obtained an odd element of

R-equivariant unbounded Kasparov theory of the pair(C ∗(H),C ∗(K )).

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

3. For a groupoid G and its two appropriate subgroupoidsH and K a cocycle make the correspondence

C∗(H)H(G )C∗(K) into a R-equivariant correspondence.4. Using the same cocycle we obtained an odd element of

R-equivariant unbounded Kasparov theory of the pair(C ∗(H),C ∗(K )).

and (some)KK-theory

Rohit Dilip Holkar

Haar groupoids

Definition

First theorem

Summary

3. For a groupoid G and its two appropriate subgroupoidsH and K a cocycle make th

groupoid correspondences and k-theory - uni-goettingen.de · a groupoid g is a small category in...

Documents