Plan Bounded cocycles and coboundaries - UFSCexel/boas/proceedings/Renault.pdfPlan Bounded cocycles and coboundaries Jean Renault Universit e d’Orl eans February 3, 2010 1 Measured

Plan

Bounded cocycles and coboundaries

Jean Renault

Université d’Orléans

February 3, 2010

1 Measured Groupoid Cohomology

2 Continuous Groupoid Cohomology

J. Renault (Université d’Orléans) February 3, 2010 1 / 23

Some references

F. Browder : On the iteration of transformations in noncompactminimal dynamical systems, Proc. Amer. Math. Soc., (1958),773–780.

J. Westman : Cohomology for ergodic groupoids, Trans. Amer.Math. Soc., 146 (1969), 465–471.B. Johnson : Cohomology in Banach algebras, Memoirs Amer. Math.Soc., 127 (1972).K. Schmidt: Cocycles on ergodic transformation groups, Mac Millan(Company of India, Ltd, Delhi), 1977.

J. Feldman and C.C. Moore : Ergodic equivalence relations,cohomology and von Neumann algebras I,II, Trans. Amer. Math.Soc., 234 (1977), 289–359.J. Renault : A groupoid approach to C ∗-algebras, Lecture Notes inMath., 793, Springer, 1980.J. Dugundji and A. Granas: A proof of the Ryll-Nardzewski fixed pointtheorem, J. of Math. Analysis and Applications, 97, (1983), 301–305.

Some references, cont’d

R.J. Zimmer : Ergodic theory and semisimple groups, Birkhäuser,1984.

P. de la Harpe and A. Valette : La propriété (T) de Kazhdan pour lesgroupes localement compacts, Astérisque, 175, 1989.

C. Anantharaman-Delaroche : A cohomological characterization ofamenable actions, Math. Proc. Camb. Phil. Soc., 110 (1991),491–504.

C. Anantharaman-Delaroche and J. Renault : Amenable groupoids,Enseignement Mathématique 36, 2000.

C. Anantharaman-Delaroche : Cohomology of property T groupoidsand applications, Ergod. Th. & Dynam. Sys, 25 (2005), 465–471.

J.-L. Tu : Groupoid cohomology and extensions, Trans. Amer. Math.Soc., 358, (2006), 4721–4747.

Some references, end

U. Bader, T. Gelander and N. Monod: A fixed point theorem for L1

spaces, Math arXiv :1012.1488v1, (2010).

D. Coronel, A. Navas, M. Ponce: Bounded orbits versus invariantsections for cocycles of affine isometries over a minimal dynamics,Math arXiv :1101.3523v2, (2011).

Measured Groupoid Cohomology.

Axioms for a cohomology theory

The problem is to define cohomology groups

Hn(G ,A)

where

G is a groupoid

A is a space of coefficients, called a G -module.

satisfying the usual axioms:

1 for G fixed, Hn(G , .) sends short exact sequences of G -modules tolong exact sequences of abelian groups,

2 H0(G , .) is the fixed point functor,

3 equivalent groupoids have isomorphic cohomologies.

Hn(G ,A)

where

G is a groupoid

Hn(G ,A)

where

G is a groupoid

Measured groupoid cohomology

This cohomology was introduced by Westman (1969) and developped byFeldman and Moore (1977) for countable measured equivalence relationsand by Schmidt (1977) for discrete group actions. Here G is a measuredgroupoid and A is a Borel bundle of polonais abelian groups over G (0).One assumes that A is a G -module, i.e. G acts by isomorphismsL(γ) : As(γ) → Ar(γ) and the action map G ∗ A→ A is Borel. Thecohomology groups Hn(G ,A) can be concretely realized as cocyclesmodulo coboundaries. In particular

Proposition

H1(G ,A) consists of cocycles modulo coboundaries where a Borel mapc : G → A such that c(γ) ∈ Ar(γ) is

a cocycle if c(γ1γ2) = c(γ1) + L(γ1)c(γ2) a.e.

a coboundary if there exists a Borel map b : G (0) → A such thatb(x) ∈ Ax and c(γ) = b(r(γ))− L(γ)b(s(γ)) a.e.

Measured groupoid cohomology

This cohomology was introduced by Westman (1969) and developped byFeldman and Moore (1977) for countable measured equivalence relationsand by Schmidt (1977) for discrete group actions. Here G is a measuredgroupoid and A is a Borel bundle of polonais abelian groups over G (0).One assumes that A is a G -module, i.e. G acts by isomorphismsL(γ) : As(γ) → Ar(γ) and the action map G ∗ A→ A is Borel. Thecohomology groups Hn(G ,A) can be concretely realized as cocyclesmodulo coboundaries. In particular

Proposition

H1(G ,A) consists of cocycles modulo coboundaries where a Borel mapc : G → A such that c(γ) ∈ Ar(γ) is

a cocycle if c(γ1γ2) = c(γ1) + L(γ1)c(γ2) a.e.

a coboundary if there exists a Borel map b : G (0) → A such thatb(x) ∈ Ax and c(γ) = b(r(γ))− L(γ)b(s(γ)) a.e.

“bounded ⇒ coboundary” result I

In this framework, Feldman and Moore state the following theorem.

Theorem (Feldman-Moore, Schmidt, 1977)

Let R be an ergodic countable equivalence relation on X and letA = Rm ⊕ Zn with trivial R-action. For a Borel cocycle c : R → A, thefollowing conditions are equivalent:

1 c is a coboundary;

2 there exists Y ⊂ X of positive measure such that cY is essentiallybounded.

Feldman-Moore give an indirect proof using the fact that every derivationof a von Neumann algebra is inner.

“bounded ⇒ coboundary” result II

Moore-Schmidt have a related result:

Theorem (Moore-Schmidt, 1980)

Let G be a second countable locally compact group measure-preservingacting on a standard probability space (X , µ). Let A be a secondcountable locally compact abelian group without compact subgroups. Fora Borel cocycle c : G × X → A, the following conditions are equivalent:

2 c is bounded in the following sense: for all � > 0, there exists acompact subset K ⊂ A such that for all g ∈ G ,µ{x : c(g , x) /∈ K} < �.

When A = Rn = V , the space U(X ,V ) of Borel functions, endowed withthe convergence in measure topology is a (non locally convex) topologicalvector space. The second condition says that c : G → U(X ,V ) isbounded. This result fits in the framework of fixed point theorems.

Measurable G -Banach bundles

The above results are limited to trivial G -modules A. Let us describe nextresults about non-trivial G -modules. In what follows,

(G , λ, µ) is a measured groupoid with unit space G (0) = X

the G -module A = E is a measurable bundle of Banach spaces overX (one also says a measurable field of Banach spaces), where G actsby isomorphisms L(γ) : Es(γ) → Er(γ) and the corresponding actionmap G ∗ E → E is measurable.

Then, we say that E is a Banach G -bundle. If E is separable, one candefine its dual E ∗ as a Banach G -bundle. When the fibers Ex are Hilbertspaces, one says that E is a Hilbert G -bundle.

Triviality via amenability

Theorem (ADR 2000)

Let (G , λ, µ) be an amenable measured groupoid. Then for every separableBanach G -bundle E , every bounded cocycle c : G → E ∗ is a coboundary.

Remark

The converse is likely to be true. It is proved in AD 1991 for groupactions.

In the case of a group, the fixed point property applies to every affineaction on a compact convex set in a locally convex space. Thereshould be a generalization of this.

Triviality via amenability

Theorem (ADR 2000)

Let (G , λ, µ) be an amenable measured groupoid. Then for every separableBanach G -bundle E , every bounded cocycle c : G → E ∗ is a coboundary.

Remark

The converse is likely to be true. It is proved in AD 1991 for groupactions.

In the case of a group, the fixed point property applies to every affineaction on a compact convex set in a locally convex space. Thereshould be a generalization of this.

Some fixed point theorems

Showing that a cocycle is a coboundary amounts to showing that someaction has a fixed point. In the case of a group G , we have an affineaction of G on an affine space Z . The existence of a fixed point isguaranteed by the following theorems. We denote by E the underlyingvector space and by L the linear representation of G on E .

Some fixed point theorems, cont’d 1

1 E is locally convex, the action is continuous and distal, and thereexists a nonempty invariant compact subset.

2 (Ryll-Nardzewski) E is locally convex, the action is weakly continuousand strongly distal, and there exists a nonempty invariant weaklycompact subset.

3 G is amenable, E is locally convex, the action is continuous, andthere exists a nonempty invariant compact subset.

4 E is a uniformly convex Banach space, the action is isometric, andthere exists a nonempty invariant bounded subset.

5 (Bader-Gelander-Monod) E is an L-embedded Banach space, theaction is isometric, and there exists a nonempty invariant boundedsubset.

Recall that a Banach space E is said to be uniformly convex if for all� > 0, there exists δ > 0 such that for all x , y ∈ E ,

‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ > �⇒ ‖x + y2‖ < 1− δ.

The circumcenter lemma says that every nonempty bounded subset A of auniformly convex Banach space has a unique center. i.e. there exists aunique ball B of minimal radius containing A. The center of this ball B iscalled the center (or circumcenter) of A and denoted by cA; the minimalradius is called the circumradius and is denoted by rA.

It is well known that Hilbert spaces and Lp(X ) (for 1 < p

Every nonempty bounded set A of a metric space X has a circumradius rA,defined as the infimum of the radii of the balls containing A. Its Chebyshevcenter CA is defined as the set of centers of balls of radius rA containingA. The key point of the B-G-M theorem is that CA is nonempty andweakly compact when X = E is an L-embedded Banach space.

A Banach space E is called L-embedded if E ∗∗ = E ⊕ F with‖x + y‖ = ‖x‖+ ‖y‖. A basic example is L1(X ), or more generally thepredual of a von Neumann algebra.

Thus, when E is L-embedded and the representation L is isometric,bounded cocycles c : G → E are coboundaries.

Triviality for isometric representations

In her study of property T for groupoids, C. Anantharaman generalizesfrom groups to measured groupoids above result about bounded cocycles:

Theorem (AD 2005)

Let (G , λ, µ) be an ergodic measured groupoid. Let H be a measurableG -Hilbert bundle. For a Borel cocycle c : G → H, the following conditionsare equivalent:

2 there exists a Borel subset Y ⊂ G (0) of positive measure such that cYis bounded.

Her proof is based on the lemma of the center: Let ξ(y) be the center ofc(G yY ) ⊂ Hy . Then y 7→ ξ(y) is Borel and satisfies

c(γ) + L(γ)ξ ◦ s(γ) = ξ ◦ r(γ)

for all γ ∈ GYY .J. Renault (Université d’Orléans) February 3, 2010 15 / 23

Triviality for isometric representations

In her study of property T for groupoids, C. Anantharaman generalizesfrom groups to measured groupoids above result about bounded cocycles:

Theorem (AD 2005)

Let (G , λ, µ) be an ergodic measured groupoid. Let H be a measurableG -Hilbert bundle. For a Borel cocycle c : G → H, the following conditionsare equivalent:

2 there exists a Borel subset Y ⊂ G (0) of positive measure such that cYis bounded.

Her proof is based on the lemma of the center: Let ξ(y) be the center ofc(G yY ) ⊂ Hy . Then y 7→ ξ(y) is Borel and satisfies

c(γ) + L(γ)ξ ◦ s(γ) = ξ ◦ r(γ)

for all γ ∈ GYY .J. Renault (Université d’Orléans) February 3, 2010 15 / 23

Continuous Groupoid Cohomology

Continuous groupoid cohomology

Are there analogous results about continuous cocycles on topologicalgroupoids?

The first problem now is to define the continuous cohomology groupsHn(G ,A) where G is a topological groupoid and A is a G -sheaf of abeliangroups. As said earlier, the requirements are

1 for G fixed, Hn(G , .) sends short exact sequences of G -sheaves tolong exact sequences of abelian groups,

Continuous groupoid cohomology

Are there analogous results about continuous cocycles on topologicalgroupoids?

The first problem now is to define the continuous cohomology groupsHn(G ,A) where G is a topological groupoid and A is a G -sheaf of abeliangroups. As said earlier, the requirements are

1 for G fixed, Hn(G , .) sends short exact sequences of G -sheaves tolong exact sequences of abelian groups,

Continuous groupoid cohomology, cont’d

Following Grothendieck (1955) who considered the case of a discretegroup acting on a topological space, these cohomology groups weredefined by Haefliger (1979) and Kumjian (1988) for étale groupoids. Thegeneral case is done by Tu (2006), who observes that this fits into theframework of sheaf cohomology for simplicial spaces (where G defines thesimplicial space G• = (G

(n))) and Čech cohomology.

The definition agrees with the Haefliger/Kumjian definition when G isétale. It agrees with Moore’s Borel cohomology when G is a locallycompact group and A is a Polish G -module.

Continuous cocycles

We are only interested here in H1(G ,A), for which we have the convenientdescription.

Proposition (Tu)

Let G be a topological groupoid and let A be a continuous G-module.Denote by A• the associated sheaf over G•. Then H1(G•,A•) is the groupof isomorphism classes of G-equivariant locally trivial A-principal bundlesover G (0).

The relation with continuous cocycles is as follows.

A continuous cocycle c : G → A defines the A-principal bundleZ (c) = A, where G acts on the left by γz = L(γ)z + c(γ).

c is a continuous coboundary if and only if Z (c) is trivial.

Replacing G by an equivalent groupoid, each G -equivariant locallytrivial A-principal bundle can be realized by a continuous cocycle.

Continuous cocycles

Proposition (Tu)

Continuous cocycles

Proposition (Tu)

The Gottschalk-Hedlund theorem

For specific classes of topological groupoids and of coefficients, H1(G ,A)is known; for example, if G is proper, H1(G ,R) = 0.

The only “bounded ⇒ coboundary” result I know for continuouscohomology is the Gottschalk-Hedlund theorem, which appeared in their1955’s book. It is quite similar to above Feldman-Moore’s result. Here is aversion adapted from my thesis.

Theorem (R, 1980)

Let G be a topological groupoid on a locally compact Hausdorff space Xand let A be a topological abelian group endowed with trivial G -action.Assume that G is minimal and that A has no compact subgroups. For acontinuous cocycle c : G → A, the following conditions are equivalent:

1 c is a continuous coboundary;

2 for each compact subset Y ⊂ X , c(GYY ) has compact closure.

Theorem (R, 1980)

The proof

The classical proof, due to F. Browder, relies on the existence of anonempty minimal closed invariant subset F0 of Z (c) = X × A. Then, it iseasy to show that F0 is the graph of a continuous function b : X → A.

My proof is different. One first establishes the following continuitycondition: for each neighborhood V of 0 in A and each x ∈ X , there existsa neighborhood U of x in X such c(GUU ) ⊂ V . One deduces that c notonly factors through the associated principal groupoid R ⊂ X × X but alsoextends continuously to X × X .

It seems difficult to generalize either proofs to the case of a non-trivialG -module A.

The proof

Continuous G -Banach bundles

The formulation of our problem in the continuous setting isstraightforward. We have

a topological groupoid G with unit space G (0) = X ;

a continuous bundle of Banach spaces over X (one also says acontinuous field of Banach spaces), where G acts by isomorphismsL(γ) : Es(γ) → Er(γ) and the corresponding action map G ∗ E → E iscontinuous.

Then, we say that E is a (continuous) G -Banach bundle. If E is separable,one can define its dual E ∗ as a G -Banach bundle. When the fibers Ex areHilbert spaces, one says that E is a G -Hilbert bundle.

Bounded continuous cocycles

I expect that the above theorems [ADR 2000] (case when G is amenableand E ∗ is a dual G -Banach bundle) and [AD 2005] (case of an isometricG -Hilbert bundle) have a continuous version.

For example, if the continuous cocycle c : G → H is bounded, the centerξ(x) of c(G x) ⊂ Hx depends continuously on x (so far I have looked onlyat the case of a constant G -bundle H). As before, this expresses c as acontinuous coboundary.

Bounded continuous cocycles

I expect that the above theorems [ADR 2000] (case when G is amenableand E ∗ is a dual G -Banach bundle) and [AD 2005] (case of an isometricG -Hilbert bundle) have a continuous version.

For example, if the continuous cocycle c : G → H is bounded, the centerξ(x) of c(G x) ⊂ Hx depends continuously on x (so far I have looked onlyat the case of a constant G -bundle H). As before, this expresses c as acontinuous coboundary.

Bounded continuous cocycles, cont’d

Added on March 5, 2011:

Coronel, Navas and Ponce have definite answers when G arises from aminimal group (or semigroup) action on a compact space and H is aconstant Hilbert bundle. They show that the above function ξ iscontinuous if H is finite-dimensional but that this may fail in the infinitedimensional case; nevertheless, they prove that a bounded continuouscocycle is a continuous coboundary also in the infinite dimensional case.

Measured Groupoid Cohomology.Continuous Groupoid Cohomology