Plan
Groupoid cocycles and derivations
Jean Renault
Université d’Orléans
November 2, 2011
1
Introduction: the scalar case
2
Conditionally negative type functions
3
Cocycles and derivations
4
Sauvageot’s construction
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
1 / 22
Plan
Groupoid cocycles and derivations
Jean Renault
Université d’Orléans
November 2, 2011
1
Introduction: the scalar case
2
Conditionally negative type functions
3
Cocycles and derivations
4
Sauvageot’s construction
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
1 / 22
Plan
Groupoid cocycles and derivations
Jean Renault
Université d’Orléans
November 2, 2011
1
Introduction: the scalar case
2
Conditionally negative type functions
3
Cocycles and derivations
4
Sauvageot’s construction
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
1 / 22
Plan
Groupoid cocycles and derivations
Jean Renault
Université d’Orléans
November 2, 2011
1
Introduction: the scalar case
2
Conditionally negative type functions
3
Cocycles and derivations
4
Sauvageot’s construction
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
1 / 22
Plan
Groupoid cocycles and derivations
Jean Renault
Université d’Orléans
November 2, 2011
1
Introduction: the scalar case
2
Conditionally negative type functions
3
Cocycles and derivations
4
Sauvageot’s construction
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
1 / 22
Introduction: the scalar case
One-parameter automorphism groups
Let H be an unbounded self-adjoint operator on a Hilbert space H. It
defines a one-parameter automorphism group of B(H) according to
αt (A) = e itH Ae −itH
whose generator is the derivation δ where
δ(A) = i[H, A] = i(HA − AH).
We are usually interested in the C*-dynamical system (A, α), where A is
a sub C*-algebra of B(H) invariant under α.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
2 / 22
Introduction: the scalar case
One-parameter automorphism groups
Let H be an unbounded self-adjoint operator on a Hilbert space H. It
defines a one-parameter automorphism group of B(H) according to
αt (A) = e itH Ae −itH
whose generator is the derivation δ where
δ(A) = i[H, A] = i(HA − AH).
We are usually interested in the C*-dynamical system (A, α), where A is
a sub C*-algebra of B(H) invariant under α.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
2 / 22
Introduction: the scalar case
One-parameter automorphism groups
Let H be an unbounded self-adjoint operator on a Hilbert space H. It
defines a one-parameter automorphism group of B(H) according to
αt (A) = e itH Ae −itH
whose generator is the derivation δ where
δ(A) = i[H, A] = i(HA − AH).
We are usually interested in the C*-dynamical system (A, α), where A is
a sub C*-algebra of B(H) invariant under α.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
2 / 22
Introduction: the scalar case
A toy model for derivations 1
Some interesting examples of dynamical systems (A, α) appear in a
diagonal form and admit a groupoid description:
Let A = C ∗ (S1 , . . . , Sn ) be the Cuntz algebra and let αt be the
automorphism such that αt (Sk ) = e it Sk .
Its groupoid description is A = C ∗ (G ), where
G = {(x, m − n, y ) : x, y ∈ X , m, n ∈ N, T n x = T m y }
where T is the one-sided shift on X = {1, . . . , n}N .
Recall that C ∗ (G ) is the C*-completion of the ∗-algebra Cc (G ) of
continuous compactly supported functions on G , where
X
f ∗ g (γ) =
f (γ 0 )g (γ 00 );
f ∗ (γ) = f (γ −1 ).
γ 0 γ 00 =γ
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
3 / 22
Introduction: the scalar case
A toy model for derivations 1
Some interesting examples of dynamical systems (A, α) appear in a
diagonal form and admit a groupoid description:
Let A = C ∗ (S1 , . . . , Sn ) be the Cuntz algebra and let αt be the
automorphism such that αt (Sk ) = e it Sk .
Its groupoid description is A = C ∗ (G ), where
G = {(x, m − n, y ) : x, y ∈ X , m, n ∈ N, T n x = T m y }
where T is the one-sided shift on X = {1, . . . , n}N .
Recall that C ∗ (G ) is the C*-completion of the ∗-algebra Cc (G ) of
continuous compactly supported functions on G , where
X
f ∗ g (γ) =
f (γ 0 )g (γ 00 );
f ∗ (γ) = f (γ −1 ).
γ 0 γ 00 =γ
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
3 / 22
Introduction: the scalar case
A toy model for derivations 1
Some interesting examples of dynamical systems (A, α) appear in a
diagonal form and admit a groupoid description:
Let A = C ∗ (S1 , . . . , Sn ) be the Cuntz algebra and let αt be the
automorphism such that αt (Sk ) = e it Sk .
Its groupoid description is A = C ∗ (G ), where
G = {(x, m − n, y ) : x, y ∈ X , m, n ∈ N, T n x = T m y }
where T is the one-sided shift on X = {1, . . . , n}N .
Recall that C ∗ (G ) is the C*-completion of the ∗-algebra Cc (G ) of
continuous compactly supported functions on G , where
X
f ∗ g (γ) =
f (γ 0 )g (γ 00 );
f ∗ (γ) = f (γ −1 ).
γ 0 γ 00 =γ
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
3 / 22
Introduction: the scalar case
A toy model for derivations 1
Some interesting examples of dynamical systems (A, α) appear in a
diagonal form and admit a groupoid description:
Let A = C ∗ (S1 , . . . , Sn ) be the Cuntz algebra and let αt be the
automorphism such that αt (Sk ) = e it Sk .
Its groupoid description is A = C ∗ (G ), where
G = {(x, m − n, y ) : x, y ∈ X , m, n ∈ N, T n x = T m y }
where T is the one-sided shift on X = {1, . . . , n}N .
Recall that C ∗ (G ) is the C*-completion of the ∗-algebra Cc (G ) of
continuous compactly supported functions on G , where
X
f ∗ g (γ) =
f (γ 0 )g (γ 00 );
f ∗ (γ) = f (γ −1 ).
γ 0 γ 00 =γ
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
3 / 22
Introduction: the scalar case
A toy model for derivations 2
The above gauge automorphism group α is then given by
αt (f )(γ) = e itc(γ) f (γ)
where c : G → R is the gauge cocycle such that c(x, k, y ) = k. Here,
cocycle just means groupoid homomorphism.
The associated derivation is defined on Cc (G ) by
δ(f )(γ) = ic(γ)f (γ).
All this makes sense in the general framework of locally compact
groupoids with Haar systems: a continuous cocycle c : G → R defines an
automorphism group α of C ∗ (G ) and a derivation δ on the ∗-algebra
Cc (G ) by the above formulas.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
4 / 22
Introduction: the scalar case
A toy model for derivations 2
The above gauge automorphism group α is then given by
αt (f )(γ) = e itc(γ) f (γ)
where c : G → R is the gauge cocycle such that c(x, k, y ) = k. Here,
cocycle just means groupoid homomorphism.
The associated derivation is defined on Cc (G ) by
δ(f )(γ) = ic(γ)f (γ).
All this makes sense in the general framework of locally compact
groupoids with Haar systems: a continuous cocycle c : G → R defines an
automorphism group α of C ∗ (G ) and a derivation δ on the ∗-algebra
Cc (G ) by the above formulas.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
4 / 22
Introduction: the scalar case
A toy model for derivations 2
The above gauge automorphism group α is then given by
αt (f )(γ) = e itc(γ) f (γ)
where c : G → R is the gauge cocycle such that c(x, k, y ) = k. Here,
cocycle just means groupoid homomorphism.
The associated derivation is defined on Cc (G ) by
δ(f )(γ) = ic(γ)f (γ).
All this makes sense in the general framework of locally compact
groupoids with Haar systems: a continuous cocycle c : G → R defines an
automorphism group α of C ∗ (G ) and a derivation δ on the ∗-algebra
Cc (G ) by the above formulas.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
4 / 22
Introduction: the scalar case
A toy model for derivations 3
In particular, we see that our derivation, which is a pregenerator, is
closable. We can ask the usual questions:
When is δ bounded?
When is δ inner, i.e. of the form δ(A) = [H, A], where H belongs to the
C*-algebra C ∗ (G ) or its multiplier algebra? In our toy model, these
properties correspond exactly to cocycle properties.
Proposition
Let G , c, δ be as above. Then,
1
δ is bounded iff c is bounded;
2
δ is inner if c is a coboundary.
We say that a cocycle c : G → R is a coboundary if there exists
b : G (0) → R such that c = b ◦ r − b ◦ s. Here, we are dealing with
continuous cocycles and we insist on having b continuous.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
5 / 22
Introduction: the scalar case
A toy model for derivations 3
In particular, we see that our derivation, which is a pregenerator, is
closable. We can ask the usual questions:
When is δ bounded?
When is δ inner, i.e. of the form δ(A) = [H, A], where H belongs to the
C*-algebra C ∗ (G ) or its multiplier algebra? In our toy model, these
properties correspond exactly to cocycle properties.
Proposition
Let G , c, δ be as above. Then,
1
δ is bounded iff c is bounded;
2
δ is inner if c is a coboundary.
We say that a cocycle c : G → R is a coboundary if there exists
b : G (0) → R such that c = b ◦ r − b ◦ s. Here, we are dealing with
continuous cocycles and we insist on having b continuous.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
5 / 22
Introduction: the scalar case
A toy model for derivations 3
In particular, we see that our derivation, which is a pregenerator, is
closable. We can ask the usual questions:
When is δ bounded?
When is δ inner, i.e. of the form δ(A) = [H, A], where H belongs to the
C*-algebra C ∗ (G ) or its multiplier algebra? In our toy model, these
properties correspond exactly to cocycle properties.
Proposition
Let G , c, δ be as above. Then,
1
δ is bounded iff c is bounded;
2
δ is inner if c is a coboundary.
We say that a cocycle c : G → R is a coboundary if there exists
b : G (0) → R such that c = b ◦ r − b ◦ s. Here, we are dealing with
continuous cocycles and we insist on having b continuous.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
5 / 22
Introduction: the scalar case
A toy model for derivations 4
The converse in 2 holds in many cases. This can be shown directly, for
example if G is étale, principal and amenable or undirectly by the following
theorem.
Recall the well-known result of Sakai that every (bounded) derivation of a
simple C*-algebra with unit is inner. Here is a groupoid version of Sakai’s
theorem (which is essentially Gottschalk-Hedlund’s theorem). A groupoid
G is minimal if every orbit [x] in G (0) is dense.
Theorem (R, 1980)
Let G be a minimal topological groupoid on a compact space X . For a
continuous cocycle c : G → R, the following conditions are equivalent:
1
c is a continuous coboundary;
2
there exists x ∈ X such that c(Gx ) is bounded;
3
c is bounded.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
6 / 22
Introduction: the scalar case
A toy model for derivations 4
The converse in 2 holds in many cases. This can be shown directly, for
example if G is étale, principal and amenable or undirectly by the following
theorem.
Recall the well-known result of Sakai that every (bounded) derivation of a
simple C*-algebra with unit is inner. Here is a groupoid version of Sakai’s
theorem (which is essentially Gottschalk-Hedlund’s theorem). A groupoid
G is minimal if every orbit [x] in G (0) is dense.
Theorem (R, 1980)
Let G be a minimal topological groupoid on a compact space X . For a
continuous cocycle c : G → R, the following conditions are equivalent:
1
c is a continuous coboundary;
2
there exists x ∈ X such that c(Gx ) is bounded;
3
c is bounded.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
6 / 22
Introduction: the scalar case
Our objectives
Let us summarize our study of our toy model:
A continuous cocycle c : G → R defines a derivation
δ : Cc (G ) → C ∗ (G ).
This derivation is a pregenerator.
Properties of the derivation and of the cocycle are intimately related.
There are interesting examples of derivations constructed in that
fashion.
Our goal is to extend this study to cocycles for G -Hilbert bundles. When
G is a group, this is the study of cocycles for unitary (or orthogonal)
representations and it amounts to the study of Kazhdan property (T).
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
7 / 22
Conditionally negative type functions
G -Hilbert bundles
We use Fell’s definition of a Banach bundle.
Definition
A bundle consists of topological spaces E and X and a continuous,
open and surjective map π : E → X .
A bundle π : E → X is a Hilbert bundle if each fiber Ex = π −1 (x) is a
Hilbert space and
1
2
addition, scalar multiplication and norm are continuous;
if (ui ) is a net in E such that kui k → 0 and π(ui ) → x, then ui → 0x .
Let G be a topological groupoid with G (0) = X . A G -Hilbert bundle
is a Hilbert bundle π : E → X endowed with a continuous, linear and
isometric action of G .
Linear and isometric means that for all γ ∈ G , L(γ) : Es(γ) → Er (γ) is a
linear isometry. An important example is the regular G -Hilbert bundle
L2 (G , λ) where (G , λ) is a locally compact groupoid with Haar system.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
8 / 22
Conditionally negative type functions
G -Hilbert bundles
We use Fell’s definition of a Banach bundle.
Definition
A bundle consists of topological spaces E and X and a continuous,
open and surjective map π : E → X .
A bundle π : E → X is a Hilbert bundle if each fiber Ex = π −1 (x) is a
Hilbert space and
1
2
addition, scalar multiplication and norm are continuous;
if (ui ) is a net in E such that kui k → 0 and π(ui ) → x, then ui → 0x .
Let G be a topological groupoid with G (0) = X . A G -Hilbert bundle
is a Hilbert bundle π : E → X endowed with a continuous, linear and
isometric action of G .
Linear and isometric means that for all γ ∈ G , L(γ) : Es(γ) → Er (γ) is a
linear isometry. An important example is the regular G -Hilbert bundle
L2 (G , λ) where (G , λ) is a locally compact groupoid with Haar system.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
8 / 22
Conditionally negative type functions
G -Hilbert bundles
We use Fell’s definition of a Banach bundle.
Definition
A bundle consists of topological spaces E and X and a continuous,
open and surjective map π : E → X .
A bundle π : E → X is a Hilbert bundle if each fiber Ex = π −1 (x) is a
Hilbert space and
1
2
addition, scalar multiplication and norm are continuous;
if (ui ) is a net in E such that kui k → 0 and π(ui ) → x, then ui → 0x .
Let G be a topological groupoid with G (0) = X . A G -Hilbert bundle
is a Hilbert bundle π : E → X endowed with a continuous, linear and
isometric action of G .
Linear and isometric means that for all γ ∈ G , L(γ) : Es(γ) → Er (γ) is a
linear isometry. An important example is the regular G -Hilbert bundle
L2 (G , λ) where (G , λ) is a locally compact groupoid with Haar system.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
8 / 22
Conditionally negative type functions
Positive type functions
The well-known relation between positive type functions and G -Hilbert
spaces for groups still holds for groupoids.
Definition
Let G be a groupoid. A function ϕ : G → C is said PT (of positive type)
if ∀n ∈ N∗ , ∀ζ1 , . . . , ζn ∈ C, ∀x ∈ G (0) and ∀γ1 , . . . , γn ∈ G x ,
X
ϕ(γi−1 γj )ζ i ζj ≥ 0.
i,j
We are interested here in topological groupoids and in continuous PT
functions. If e : X → E is a continuous section of a G -Hilbert bundle,
then the coefficient ϕ = (e, e) defined by
ϕ(γ) = (e ◦ r (γ)|L(γ)e ◦ s(γ))
is a continuous PT function.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
9 / 22
Conditionally negative type functions
Positive type functions
The well-known relation between positive type functions and G -Hilbert
spaces for groups still holds for groupoids.
Definition
Let G be a groupoid. A function ϕ : G → C is said PT (of positive type)
if ∀n ∈ N∗ , ∀ζ1 , . . . , ζn ∈ C, ∀x ∈ G (0) and ∀γ1 , . . . , γn ∈ G x ,
X
ϕ(γi−1 γj )ζ i ζj ≥ 0.
i,j
We are interested here in topological groupoids and in continuous PT
functions. If e : X → E is a continuous section of a G -Hilbert bundle,
then the coefficient ϕ = (e, e) defined by
ϕ(γ) = (e ◦ r (γ)|L(γ)e ◦ s(γ))
is a continuous PT function.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
9 / 22
Conditionally negative type functions
Positive type functions
The well-known relation between positive type functions and G -Hilbert
spaces for groups still holds for groupoids.
Definition
Let G be a groupoid. A function ϕ : G → C is said PT (of positive type)
if ∀n ∈ N∗ , ∀ζ1 , . . . , ζn ∈ C, ∀x ∈ G (0) and ∀γ1 , . . . , γn ∈ G x ,
X
ϕ(γi−1 γj )ζ i ζj ≥ 0.
i,j
We are interested here in topological groupoids and in continuous PT
functions. If e : X → E is a continuous section of a G -Hilbert bundle,
then the coefficient ϕ = (e, e) defined by
ϕ(γ) = (e ◦ r (γ)|L(γ)e ◦ s(γ))
is a continuous PT function.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
9 / 22
Conditionally negative type functions
GNS representation 1
The converse holds. More precisely:
Theorem (J.-L. Tu, 1999)
Let (G , λ) be a locally compact groupoid with Haar system. Let
ϕ : G → C be a continuous function of positive type. Then
1 There exists a pair (E , e) consisting of a continuous G -Hilbert bundle
E and a continuous section e such that
for all γ ∈ G , ϕ(γ) = (e ◦ r (γ)|L(γ)e ◦ s(γ));
for all x ∈ G (0) , {L(γ)e ◦ s(γ), γ ∈ G x } is total in Ex .
2
If (E 0 , e 0 ) is another pair satisfying the same properties, there exists a
G -equivariant isometric continuous bundle map u : E → E 0 such that
e 0 = u ◦ e.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
10 / 22
Conditionally negative type functions
GNS representation 1
The converse holds. More precisely:
Theorem (J.-L. Tu, 1999)
Let (G , λ) be a locally compact groupoid with Haar system. Let
ϕ : G → C be a continuous function of positive type. Then
1 There exists a pair (E , e) consisting of a continuous G -Hilbert bundle
E and a continuous section e such that
for all γ ∈ G , ϕ(γ) = (e ◦ r (γ)|L(γ)e ◦ s(γ));
for all x ∈ G (0) , {L(γ)e ◦ s(γ), γ ∈ G x } is total in Ex .
2
If (E 0 , e 0 ) is another pair satisfying the same properties, there exists a
G -equivariant isometric continuous bundle map u : E → E 0 such that
e 0 = u ◦ e.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
10 / 22
Conditionally negative type functions
G -affine Hilbert bundles
Definition
An affine Hilbert bundle is a bundle π : E → X endowed with a
~ → E of a Hilbert bundle π : E
~ → X which is
continuous action E ∗ E
fiberwise free and transitive.
Let G be a topological groupoid with G (0) = X . A G -affine Hilbert
bundle is an affine Hilbert bundle π : E → X endowed with a
continuous, affine and isometric action of G .
Affine and isometric means that for all γ ∈ G , a(γ) : Es(γ) → Er (γ) is an
affine isometry. Thus, we have
a(γ)(e + ~u ) = a(γ)e + L(γ)~u .
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
11 / 22
Conditionally negative type functions
G -affine Hilbert bundles
Definition
An affine Hilbert bundle is a bundle π : E → X endowed with a
~ → E of a Hilbert bundle π : E
~ → X which is
continuous action E ∗ E
fiberwise free and transitive.
Let G be a topological groupoid with G (0) = X . A G -affine Hilbert
bundle is an affine Hilbert bundle π : E → X endowed with a
continuous, affine and isometric action of G .
Affine and isometric means that for all γ ∈ G , a(γ) : Es(γ) → Er (γ) is an
affine isometry. Thus, we have
a(γ)(e + ~u ) = a(γ)e + L(γ)~u .
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
11 / 22
Conditionally negative type functions
Cocycles and affine bundles
The relation between cocycles and affine bundles is well-known: let E be a
G -affine Hilbert bundle. A continuous section e : X → E identifies E with
~ (as topological spaces) and defines a continuous section c : G → r ∗ E
~
E
∗
~
(where r E is the pull-back along r : G → X ) according to:
c(γ) = a(γ)e ◦ s(γ) − e ◦ r (γ).
c satisfies the cocycle property:
c(γγ 0 ) = c(γ) + L(γ)c(γ 0 ).
Another choice of section gives a cohomologous cocycle.
~ endow E = E
~ with a
Conversely, a continuous cocycle c : G → r ∗ E
G -affine action according to: a(γ)e = c(γ) + L(γ)e. The G affine Hilbert
bundle defined by c will be denoted by E (c).
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
12 / 22
Conditionally negative type functions
Cocycles and affine bundles
The relation between cocycles and affine bundles is well-known: let E be a
G -affine Hilbert bundle. A continuous section e : X → E identifies E with
~ (as topological spaces) and defines a continuous section c : G → r ∗ E
~
E
∗
~
(where r E is the pull-back along r : G → X ) according to:
c(γ) = a(γ)e ◦ s(γ) − e ◦ r (γ).
c satisfies the cocycle property:
c(γγ 0 ) = c(γ) + L(γ)c(γ 0 ).
Another choice of section gives a cohomologous cocycle.
~ endow E = E
~ with a
Conversely, a continuous cocycle c : G → r ∗ E
G -affine action according to: a(γ)e = c(γ) + L(γ)e. The G affine Hilbert
bundle defined by c will be denoted by E (c).
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
12 / 22
Conditionally negative type functions
Cocycles and affine bundles
The relation between cocycles and affine bundles is well-known: let E be a
G -affine Hilbert bundle. A continuous section e : X → E identifies E with
~ (as topological spaces) and defines a continuous section c : G → r ∗ E
~
E
∗
~
(where r E is the pull-back along r : G → X ) according to:
c(γ) = a(γ)e ◦ s(γ) − e ◦ r (γ).
c satisfies the cocycle property:
c(γγ 0 ) = c(γ) + L(γ)c(γ 0 ).
Another choice of section gives a cohomologous cocycle.
~ endow E = E
~ with a
Conversely, a continuous cocycle c : G → r ∗ E
G -affine action according to: a(γ)e = c(γ) + L(γ)e. The G affine Hilbert
bundle defined by c will be denoted by E (c).
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
12 / 22
Conditionally negative type functions
Conditionally negative type functions
The well-known relation between conditionally negative type functions and
G -affine real Hilbert spaces for groups still holds for groupoids.
Definition
Let G be a groupoid. A function ψ : G → R is said CNT (conditionally of
negative type) if
1
∀x ∈ G (0) , ψ(x) = 0,
2
∀γ ∈ G , ψ(γ −1 ) = ψ(γ),
3
∀n ∈ N∗ , ∀ζ1 , . . . , ζn ∈ R such that
∀x ∈ G (0) ,
Pn
∀γ1 , . . . , γn ∈ G x ,
1 ζi
= 0,
X
ψ(γi−1 γj )ζi ζj ≤ 0.
i,j
Again, we are interested in topological groupoids and in continuous CNT
functions.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
13 / 22
Conditionally negative type functions
GNS representation 2
Just as for continuous PT functions, we have a universal construction for
continuous CNT functions.
Theorem (J.-L. Tu, 1999)
Let (G , λ) be a locally compact groupoid with Haar system. Let
ψ : G → R be a continuous function conditionally of negative type. Then
1 There exists a pair (E , c) consisting of a continuous G -Hilbert bundle
E and a continuous cocycle c : G → r ∗ E such that
for all γ ∈ G , ψ(γ) = kc(γ)k2 ;
for all x ∈ G (0) , {c(γ), γ ∈ G x } is total in Ex .
2
If (E 0 , c 0 ) is another pair satisfying the same properties, there exists a
G -equivariant isometric continuous bundle map u : E → E 0 such that
c 0 = u ◦ c.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
14 / 22
Conditionally negative type functions
GNS representation 2
Just as for continuous PT functions, we have a universal construction for
continuous CNT functions.
Theorem (J.-L. Tu, 1999)
Let (G , λ) be a locally compact groupoid with Haar system. Let
ψ : G → R be a continuous function conditionally of negative type. Then
1 There exists a pair (E , c) consisting of a continuous G -Hilbert bundle
E and a continuous cocycle c : G → r ∗ E such that
for all γ ∈ G , ψ(γ) = kc(γ)k2 ;
for all x ∈ G (0) , {c(γ), γ ∈ G x } is total in Ex .
2
If (E 0 , c 0 ) is another pair satisfying the same properties, there exists a
G -equivariant isometric continuous bundle map u : E → E 0 such that
c 0 = u ◦ c.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
14 / 22
Conditionally negative type functions
Bounded cocycles are coboundaries
The following theorem extends the groupoid version of the
Gottschalk-Hedlund’s theorem given in the introduction. It is inspired a
preprint of D. Coronel, A. Navas, M. Ponce [Bounded orbits versus
invariant sections for cocycles of affine isometries over a minimal
dynamics, Math arXiv :1101.3523v2, (2011)] which considers the case of
a semi-group action and a constant Hilbert bundle.
Theorem (R, 2011)
Let G be a minimal topological groupoid on a compact space X . Let
c : G → r ∗ E be a continuous cocycle, where E is a G -Hilbert bundle and
let E (c) be the associated G -affine bundle. Then the following conditions
are equivalent:
1
c is a continuous coboundary;
2
E (c) admits an equivariant continuous section;
3
E (c) admits a bounded orbit;
4
kck is bounded.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
15 / 22
Conditionally negative type functions
Bounded cocycles are coboundaries
The following theorem extends the groupoid version of the
Gottschalk-Hedlund’s theorem given in the introduction. It is inspired a
preprint of D. Coronel, A. Navas, M. Ponce [Bounded orbits versus
invariant sections for cocycles of affine isometries over a minimal
dynamics, Math arXiv :1101.3523v2, (2011)] which considers the case of
a semi-group action and a constant Hilbert bundle.
Theorem (R, 2011)
Let G be a minimal topological groupoid on a compact space X . Let
c : G → r ∗ E be a continuous cocycle, where E is a G -Hilbert bundle and
let E (c) be the associated G -affine bundle. Then the following conditions
are equivalent:
1
c is a continuous coboundary;
2
E (c) admits an equivariant continuous section;
3
E (c) admits a bounded orbit;
4
kck is bounded.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
15 / 22
Cocycles and derivations
Construction of the C*-bimodule
Let (G , λ) be a locally compact groupoid with Haar system. We construct
from a G -Hilbert bundle E a C ∗ (G )-bimodule C ∗ (G , r ∗ E ). Let Cc (G , r ∗ E )
be the space of compactly supported continuous sections of r ∗ E . We
define the left and right actions: for f , g ∈ Cc (G ) and ξ ∈ Cc (G , r ∗ E ),
Z
−1
f ξ(γ) = f (γ 0 )L(γ 0 )ξ(γ 0 γ)dλr (γ) (γ 0 )
Z
ξg (γ) =
ξ(γγ 0 )g (γ 0
−1
)dλs(γ) (γ 0 )
and the left and right inner products: for ξ, η ∈ Cc (G ),
Z
(ξ(γ 0 )|L(γ)η(γ −1 γ 0 ))r (γ) dλr (γ) (γ 0 )
l < ξ, η > (γ) =
Z
< ξ, η >r (γ) =
J. Renault (Université d’Orléans)
(ξ(γ 0
−1
)|η(γ 0
−1
γ))s(γ 0 ) dλr (γ) (γ 0 )
Cocycles and derivations — Settat 11
November 2, 2011
16 / 22
Cocycles and derivations
Construction of the C*-bimodule
Let (G , λ) be a locally compact groupoid with Haar system. We construct
from a G -Hilbert bundle E a C ∗ (G )-bimodule C ∗ (G , r ∗ E ). Let Cc (G , r ∗ E )
be the space of compactly supported continuous sections of r ∗ E . We
define the left and right actions: for f , g ∈ Cc (G ) and ξ ∈ Cc (G , r ∗ E ),
Z
−1
f ξ(γ) = f (γ 0 )L(γ 0 )ξ(γ 0 γ)dλr (γ) (γ 0 )
Z
ξg (γ) =
ξ(γγ 0 )g (γ 0
−1
)dλs(γ) (γ 0 )
and the left and right inner products: for ξ, η ∈ Cc (G ),
Z
(ξ(γ 0 )|L(γ)η(γ −1 γ 0 ))r (γ) dλr (γ) (γ 0 )
l < ξ, η > (γ) =
Z
< ξ, η >r (γ) =
J. Renault (Université d’Orléans)
(ξ(γ 0
−1
)|η(γ 0
−1
γ))s(γ 0 ) dλr (γ) (γ 0 )
Cocycles and derivations — Settat 11
November 2, 2011
16 / 22
Cocycles and derivations
Construction of the derivation
We define C ∗ (G , r ∗ E ) as the completion of Cc (G , r ∗ E ) for the norm
kξk = k < ξ, ξ >r k1/2 .
Proposition
C ∗ (G , r ∗ E ) is a (C ∗ (G ), C ∗ (G ))-bimodule and the inner products have a
continuous extension.
Let c : G → r ∗ E be a continuous cocycle. Define δ : Cc (G ) → Cc (G , r ∗ E )
by
δ(f )(γ) = ic(γ)f (γ).
Proposition
1
δ is a derivation: δ(f ∗ g ) = δ(f )g + f δ(g );
2
< δ(f ), δ(g ) >r =
J. Renault (Université d’Orléans)
l
< δ(f ∗ ), δ(g ∗ ) >.
Cocycles and derivations — Settat 11
November 2, 2011
17 / 22
Cocycles and derivations
Construction of the derivation
We define C ∗ (G , r ∗ E ) as the completion of Cc (G , r ∗ E ) for the norm
kξk = k < ξ, ξ >r k1/2 .
Proposition
C ∗ (G , r ∗ E ) is a (C ∗ (G ), C ∗ (G ))-bimodule and the inner products have a
continuous extension.
Let c : G → r ∗ E be a continuous cocycle. Define δ : Cc (G ) → Cc (G , r ∗ E )
by
δ(f )(γ) = ic(γ)f (γ).
Proposition
1
δ is a derivation: δ(f ∗ g ) = δ(f )g + f δ(g );
2
< δ(f ), δ(g ) >r =
J. Renault (Université d’Orléans)
l
< δ(f ∗ ), δ(g ∗ ) >.
Cocycles and derivations — Settat 11
November 2, 2011
17 / 22
Cocycles and derivations
Some questions
What can be said about this derivation in the C*-algebraic framework?
In the scalar case, i.e. E = X × C and c : G → R, the bimodule is C ∗ (G ).
We have seen that the derivation δ defined by c is a pregenerator of a
one-parameter automorphism group (αt )t∈R of C ∗ (G ). In particular, it is
closable.
In the Hilbert bundle case, we can relate the derivation δ to a
one-parameter semi-group of contractions of C ∗ (G ) as a particular case of
Sauvageot’s construction.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
18 / 22
Cocycles and derivations
Some questions
What can be said about this derivation in the C*-algebraic framework?
In the scalar case, i.e. E = X × C and c : G → R, the bimodule is C ∗ (G ).
We have seen that the derivation δ defined by c is a pregenerator of a
one-parameter automorphism group (αt )t∈R of C ∗ (G ). In particular, it is
closable.
In the Hilbert bundle case, we can relate the derivation δ to a
one-parameter semi-group of contractions of C ∗ (G ) as a particular case of
Sauvageot’s construction.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
18 / 22
Cocycles and derivations
Some questions
What can be said about this derivation in the C*-algebraic framework?
In the scalar case, i.e. E = X × C and c : G → R, the bimodule is C ∗ (G ).
We have seen that the derivation δ defined by c is a pregenerator of a
one-parameter automorphism group (αt )t∈R of C ∗ (G ). In particular, it is
closable.
In the Hilbert bundle case, we can relate the derivation δ to a
one-parameter semi-group of contractions of C ∗ (G ) as a particular case of
Sauvageot’s construction.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
18 / 22
Sauvageot’s construction
Non-commutative Dirichlet forms
In the theory of non-commutative Dirichlet forms, one finds the following
construction (J.-L. Sauvageot, 1989).
Let (Tt )t≥0 be a strongly continuous semi-group of completely positive
contractions of a C*-algebra A. We denote by −∆ its generator (it is a
dissipation). We assume that A is a dense sub ∗-algebra in its domain.
The associated Dirichlet form is the sesquilinear map L : A × A → A
defined by
1
L(α, β) = [α∗ ∆(β) + ∆(α∗ )β − ∆(α∗ β)].
2
One can prove that L is completely positive in the sense that
∀n ∈ N, ∀α1 , . . . , αn ∈ A, [L(αi , αj )] ∈ Mn (A)+ .
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
19 / 22
Sauvageot’s construction
Non-commutative Dirichlet forms
In the theory of non-commutative Dirichlet forms, one finds the following
construction (J.-L. Sauvageot, 1989).
Let (Tt )t≥0 be a strongly continuous semi-group of completely positive
contractions of a C*-algebra A. We denote by −∆ its generator (it is a
dissipation). We assume that A is a dense sub ∗-algebra in its domain.
The associated Dirichlet form is the sesquilinear map L : A × A → A
defined by
1
L(α, β) = [α∗ ∆(β) + ∆(α∗ )β − ∆(α∗ β)].
2
One can prove that L is completely positive in the sense that
∀n ∈ N, ∀α1 , . . . , αn ∈ A, [L(αi , αj )] ∈ Mn (A)+ .
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
19 / 22
Sauvageot’s construction
Non-commutative Dirichlet forms
In the theory of non-commutative Dirichlet forms, one finds the following
construction (J.-L. Sauvageot, 1989).
Let (Tt )t≥0 be a strongly continuous semi-group of completely positive
contractions of a C*-algebra A. We denote by −∆ its generator (it is a
dissipation). We assume that A is a dense sub ∗-algebra in its domain.
The associated Dirichlet form is the sesquilinear map L : A × A → A
defined by
1
L(α, β) = [α∗ ∆(β) + ∆(α∗ )β − ∆(α∗ β)].
2
One can prove that L is completely positive in the sense that
∀n ∈ N, ∀α1 , . . . , αn ∈ A, [L(αi , αj )] ∈ Mn (A)+ .
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
19 / 22
Sauvageot’s construction
Non-commutative Dirichlet forms
In the theory of non-commutative Dirichlet forms, one finds the following
construction (J.-L. Sauvageot, 1989).
Let (Tt )t≥0 be a strongly continuous semi-group of completely positive
contractions of a C*-algebra A. We denote by −∆ its generator (it is a
dissipation). We assume that A is a dense sub ∗-algebra in its domain.
The associated Dirichlet form is the sesquilinear map L : A × A → A
defined by
1
L(α, β) = [α∗ ∆(β) + ∆(α∗ )β − ∆(α∗ β)].
2
One can prove that L is completely positive in the sense that
∀n ∈ N, ∀α1 , . . . , αn ∈ A, [L(αi , αj )] ∈ Mn (A)+ .
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
19 / 22
Sauvageot’s construction
GNS construction 3
The positivity of the non-commutative Dirichlet form gives the following
GNS representation.
Theorem (J.-L. Sauvageot, 1989)
Let (Tt )t≥0 be a semi-group of CP contractions of a C*-algebra A as
above and let L be its associated Dirichlet form. Then
1 There exists an (A, A)-C*-bimodule E and a derivation δ : A → E
such that
for all α, β ∈ A, L(α, β) =< δ(α), δ(β) >A ;
the range of δ generates E as a right Banach A-module.
2
If (E 0 , δ 0 ) is another pair satisfying the same properties, there exists a
C*-bimodule isomorphism u : E → E 0 such that δ 0 = u ◦ δ.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
20 / 22
Sauvageot’s construction
GNS construction 3
The positivity of the non-commutative Dirichlet form gives the following
GNS representation.
Theorem (J.-L. Sauvageot, 1989)
Let (Tt )t≥0 be a semi-group of CP contractions of a C*-algebra A as
above and let L be its associated Dirichlet form. Then
1 There exists an (A, A)-C*-bimodule E and a derivation δ : A → E
such that
for all α, β ∈ A, L(α, β) =< δ(α), δ(β) >A ;
the range of δ generates E as a right Banach A-module.
2
If (E 0 , δ 0 ) is another pair satisfying the same properties, there exists a
C*-bimodule isomorphism u : E → E 0 such that δ 0 = u ◦ δ.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
20 / 22
Sauvageot’s construction
CNT functions and Sauvageot’s construction
Let (G , λ) be a locally compact groupoid with Haar system.
Proposition
Given a continuous CNT function ψ : G → R
1
2
Pointwise multiplication by e −tψ , where t ≥ 0 defines a completely
positive contraction Tt : C ∗ (G ) → C ∗ (G ) and (Tt )t≥0 is a strongly
continuous semi-group of completely positive contractions of C ∗ (G ).
Its generator is −∆ with the dense sub-∗algebra A = Cc (G ) as
essential domain and with ∆f = ψf .
Proposition
Up to isomorphism, the Sauvageot pair (E, δ) associated with ∆ is
(C ∗ (G , r ∗ E ), δc ), where (E , c) is the representation of ψ described in GNS
representation 2.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
21 / 22
Sauvageot’s construction
CNT functions and Sauvageot’s construction
Let (G , λ) be a locally compact groupoid with Haar system.
Proposition
Given a continuous CNT function ψ : G → R
1
2
Pointwise multiplication by e −tψ , where t ≥ 0 defines a completely
positive contraction Tt : C ∗ (G ) → C ∗ (G ) and (Tt )t≥0 is a strongly
continuous semi-group of completely positive contractions of C ∗ (G ).
Its generator is −∆ with the dense sub-∗algebra A = Cc (G ) as
essential domain and with ∆f = ψf .
Proposition
Up to isomorphism, the Sauvageot pair (E, δ) associated with ∆ is
(C ∗ (G , r ∗ E ), δc ), where (E , c) is the representation of ψ described in GNS
representation 2.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
21 / 22
Sauvageot’s construction
CNT functions and Sauvageot’s construction
Let (G , λ) be a locally compact groupoid with Haar system.
Proposition
Given a continuous CNT function ψ : G → R
1
2
Pointwise multiplication by e −tψ , where t ≥ 0 defines a completely
positive contraction Tt : C ∗ (G ) → C ∗ (G ) and (Tt )t≥0 is a strongly
continuous semi-group of completely positive contractions of C ∗ (G ).
Its generator is −∆ with the dense sub-∗algebra A = Cc (G ) as
essential domain and with ∆f = ψf .
Proposition
Up to isomorphism, the Sauvageot pair (E, δ) associated with ∆ is
(C ∗ (G , r ∗ E ), δc ), where (E , c) is the representation of ψ described in GNS
representation 2.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
21 / 22
Sauvageot’s construction
Consequences
One can apply Sauvageot and Cipriani-Sauvageot results to conclude that
the derivation δc associated with a cocycle c : G → r ∗ E is closable. It
would be desirable to have a direct proof. Just as in the scalar case, δc is
bounded iff c is bounded and it is inner if c is a continuous boundary.
Remark. The theory of unbounded derivations with values in
(M, M)-Hilbert modules, where M is a von Neumann algebra (usually
equipped with a finite trace is well developed. This is not the case for
unbounded derivations with values in (A, A)-C*-modules. The above
groupoid example may be useful to develop this theory. In our example,
there are both a left and a right A-valued inner products and we have the
relation
L(α, β) =< δ(α), δ(β) >A = A < δ(α∗ ), δ(β ∗ ) >
which seems to express that δ is a ∗-derivation.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
22 / 22
Sauvageot’s construction
Consequences
One can apply Sauvageot and Cipriani-Sauvageot results to conclude that
the derivation δc associated with a cocycle c : G → r ∗ E is closable. It
would be desirable to have a direct proof. Just as in the scalar case, δc is
bounded iff c is bounded and it is inner if c is a continuous boundary.
Remark. The theory of unbounded derivations with values in
(M, M)-Hilbert modules, where M is a von Neumann algebra (usually
equipped with a finite trace is well developed. This is not the case for
unbounded derivations with values in (A, A)-C*-modules. The above
groupoid example may be useful to develop this theory. In our example,
there are both a left and a right A-valued inner products and we have the
relation
L(α, β) =< δ(α), δ(β) >A = A < δ(α∗ ), δ(β ∗ ) >
which seems to express that δ is a ∗-derivation.
J. Renault (Université d’Orléans)
Cocycles and derivations — Settat 11
November 2, 2011
22 / 22