Groupoid cocycles and derivations
Jean Renault
Universit´e d’Orl´eans
November 2, 2011
1 Introduction: the scalar case
2 Conditionally negative type functions
3 Cocycles and derivations
4 Sauvageot’s construction
Groupoid cocycles and derivations
Jean Renault
Universit´e d’Orl´eans
November 2, 2011
1 Introduction: the scalar case
2 Conditionally negative type functions
3 Cocycles and derivations
4 Sauvageot’s construction
Groupoid cocycles and derivations
Jean Renault
Universit´e d’Orl´eans
November 2, 2011
1 Introduction: the scalar case
2 Conditionally negative type functions
3 Cocycles and derivations
4 Sauvageot’s construction
Groupoid cocycles and derivations
Jean Renault
Universit´e d’Orl´eans
November 2, 2011
1 Introduction: the scalar case
2 Conditionally negative type functions
3 Cocycles and derivations
4 Sauvageot’s construction
Groupoid cocycles and derivations
Jean Renault
Universit´e d’Orl´eans
November 2, 2011
1 Introduction: the scalar case
2 Conditionally negative type functions
3 Cocycles and derivations
4 Sauvageot’s construction
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) =eitHAe−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), whereA is a sub C*-algebra of B(H) invariant underα.
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) =eitHAe−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), whereA is a sub C*-algebra of B(H) invariant underα.
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) =eitHAe−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), whereA is a sub C*-algebra of B(H) invariant underα.
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 letαt be the automorphism such that αt(Sk) =eitSk.
Its groupoid description is A=C∗(G), where
G ={(x,m−n,y) :x,y∈X,m,n∈N,Tnx =Tmy}
whereT is the one-sided shift onX ={1, . . . ,n}N.
Recall that C∗(G) is the C*-completion of the∗-algebra Cc(G) of continuous compactly supported functions on G, where
f ∗g(γ) = X
γ0γ00=γ
f(γ0)g(γ00); f∗(γ) =f(γ−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 letαt be the automorphism such that αt(Sk) =eitSk.
Its groupoid description is A=C∗(G), where
G ={(x,m−n,y) :x,y∈X,m,n∈N,Tnx =Tmy}
whereT is the one-sided shift onX ={1, . . . ,n}N.
Recall that C∗(G) is the C*-completion of the∗-algebra Cc(G) of continuous compactly supported functions on G, where
f ∗g(γ) = X
γ0γ00=γ
f(γ0)g(γ00); f∗(γ) =f(γ−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 letαt be the automorphism such that αt(Sk) =eitSk.
Its groupoid description is A=C∗(G), where
G ={(x,m−n,y) :x,y∈X,m,n∈N,Tnx =Tmy}
whereT is the one-sided shift onX ={1, . . . ,n}N.
Recall that C∗(G) is the C*-completion of the∗-algebra Cc(G) of continuous compactly supported functions on G, where
f ∗g(γ) = X
γ0γ00=γ
f(γ0)g(γ00); f∗(γ) =f(γ−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 letαt be the automorphism such that αt(Sk) =eitSk.
Its groupoid description is A=C∗(G), where
G ={(x,m−n,y) :x,y∈X,m,n∈N,Tnx =Tmy}
whereT is the one-sided shift onX ={1, . . . ,n}N.
Recall that C∗(G) is the C*-completion of the∗-algebra Cc(G) of continuous compactly supported functions on G, where
f ∗g(γ) = X
γ0γ00=γ
f(γ0)g(γ00); f∗(γ) =f(γ−1).
The above gauge automorphism group α is then given by αt(f)(γ) =eitc(γ)f(γ)
wherec :G →Ris the gauge cocycle such thatc(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 →Rdefines an automorphism group α of C∗(G) and a derivation δ on the∗-algebra Cc(G) by the above formulas.
The above gauge automorphism group α is then given by αt(f)(γ) =eitc(γ)f(γ)
wherec :G →Ris the gauge cocycle such thatc(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 →Rdefines an automorphism group α of C∗(G) and a derivation δ on the∗-algebra Cc(G) by the above formulas.
The above gauge automorphism group α is then given by αt(f)(γ) =eitc(γ)f(γ)
wherec :G →Ris the gauge cocycle such thatc(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 →Rdefines an automorphism group α of C∗(G) and a derivation δ on the∗-algebra Cc(G) by the above formulas.
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, δbe as above. Then,
1 δ is bounded iff c is bounded;
2 δ is inner if c is a coboundary.
We say that a cocyclec :G →Ris a coboundary if there exists b :G(0)→R such thatc =b◦r−b◦s. Here, we are dealing with continuous cocycles and we insist on having b continuous.
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, δbe as above. Then,
1 δ is bounded iff c is bounded;
2 δ is inner if c is a coboundary.
We say that a cocyclec :G →Ris a coboundary if there exists b :G(0)→R such thatc =b◦r−b◦s. Here, we are dealing with continuous cocycles and we insist on having b continuous.
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, δbe as above. Then,
1 δ is bounded iff c is bounded;
2 δ is inner if c is a coboundary.
We say that a cocyclec :G →Ris a coboundary if there exists b :G(0)→R such thatc =b◦r−b◦s. Here, we are dealing with continuous cocycles and we insist on having b continuous.
The converse in 2 holds in many cases. This can be shown directly, for example ifG is ´etale, 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] inG(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.
The converse in 2 holds in many cases. This can be shown directly, for example ifG is ´etale, 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] inG(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.
Let us summarize our study of our toy model:
A continuous cocyclec :G →Rdefines 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 forG-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).
We use Fell’s definition of a Banach bundle.
Definition
A bundle consists of topological spacesE and X and a continuous, open and surjective map π:E →X.
A bundleπ :E →X is a Hilbert bundle if each fiberEx =π−1(x) is a Hilbert space and
1 addition, scalar multiplication and norm are continuous;
2 if (ui) is a net inE such thatkuik →0 andπ(ui)→x, thenui →0x. Let G be a topological groupoid withG(0)=X. AG-Hilbert bundle is a Hilbert bundleπ :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 regularG-Hilbert bundle L2(G, λ) where (G, λ) is a locally compact groupoid with Haar system.
We use Fell’s definition of a Banach bundle.
Definition
A bundle consists of topological spacesE and X and a continuous, open and surjective map π:E →X.
A bundleπ :E →X is a Hilbert bundle if each fiberEx =π−1(x) is a Hilbert space and
1 addition, scalar multiplication and norm are continuous;
2 if (ui) is a net inE such thatkuik →0 andπ(ui)→x, thenui →0x. Let G be a topological groupoid withG(0)=X. AG-Hilbert bundle is a Hilbert bundleπ :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 regularG-Hilbert bundle L2(G, λ) where (G, λ) is a locally compact groupoid with Haar system.
We use Fell’s definition of a Banach bundle.
Definition
A bundle consists of topological spacesE and X and a continuous, open and surjective map π:E →X.
A bundleπ :E →X is a Hilbert bundle if each fiberEx =π−1(x) is a Hilbert space and
1 addition, scalar multiplication and norm are continuous;
2 if (ui) is a net inE such thatkuik →0 andπ(ui)→x, thenui →0x. Let G be a topological groupoid withG(0)=X. AG-Hilbert bundle is a Hilbert bundleπ :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 regularG-Hilbert bundle L2(G, λ) where (G, λ) is a locally compact groupoid with Haar system.
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 functionϕ:G →Cis said PT (of positive type) if ∀n∈N∗,∀ζ1, . . . , ζn∈C,∀x∈G(0) and∀γ1, . . . , γn∈Gx,
X
i,j
ϕ(γi−1γj)ζiζj ≥0.
We are interested here in topological groupoids and in continuous PT functions. If e :X →E is a continuous section of aG-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.
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 functionϕ:G →Cis said PT (of positive type) if ∀n∈N∗,∀ζ1, . . . , ζn∈C,∀x∈G(0) and∀γ1, . . . , γn∈Gx,
X
i,j
ϕ(γi−1γj)ζiζj ≥0.
We are interested here in topological groupoids and in continuous PT functions. If e :X →E is a continuous section of aG-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.
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 functionϕ:G →Cis said PT (of positive type) if ∀n∈N∗,∀ζ1, . . . , ζn∈C,∀x∈G(0) and∀γ1, . . . , γn∈Gx,
X
i,j
ϕ(γi−1γj)ζiζj ≥0.
We are interested here in topological groupoids and in continuous PT functions. If e :X →E is a continuous section of aG-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.
The converse holds. More precisely:
Theorem (J.-L. Tu, 1999)
Let (G, λ) be a locally compact groupoid with Haar system. Let ϕ:G →Cbe 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 allγ∈G ,ϕ(γ) = (e◦r(γ)|L(γ)e◦s(γ));
for all x∈G(0),{L(γ)e◦s(γ), γ∈Gx}is total in Ex.
2 If(E0,e0) is another pair satisfying the same properties, there exists a G -equivariant isometric continuous bundle map u:E →E0 such that e0 =u◦e.
The converse holds. More precisely:
Theorem (J.-L. Tu, 1999)
Let (G, λ) be a locally compact groupoid with Haar system. Let ϕ:G →Cbe 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 allγ∈G ,ϕ(γ) = (e◦r(γ)|L(γ)e◦s(γ));
for all x∈G(0),{L(γ)e◦s(γ), γ∈Gx}is total in Ex.
2 If(E0,e0) is another pair satisfying the same properties, there exists a G -equivariant isometric continuous bundle map u:E →E0 such that e0 =u◦e.
Definition
An affine Hilbert bundle is a bundle π:E →X endowed with a continuous actionE ∗E~ →E of a Hilbert bundleπ:E~ →X which is fiberwise free and transitive.
Let G be a topological groupoid withG(0)=X. AG-affine Hilbert bundle is an affine Hilbert bundleπ :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.
Definition
An affine Hilbert bundle is a bundle π:E →X endowed with a continuous actionE ∗E~ →E of a Hilbert bundleπ:E~ →X which is fiberwise free and transitive.
Let G be a topological groupoid withG(0)=X. AG-affine Hilbert bundle is an affine Hilbert bundleπ :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.
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 E~ (as topological spaces) and defines a continuous sectionc :G →r∗~E (where r∗E~ is the pull-back alongr :G →X) according to:
c(γ) =a(γ)e◦s(γ)−e◦r(γ).
c satisfies the cocycle property:
c(γγ0) =c(γ) +L(γ)c(γ0).
Another choice of section gives a cohomologous cocycle.
Conversely, a continuous cocyclec :G →r∗~E endowE =~E with a G-affine action according to: a(γ)e =c(γ) +L(γ)e. TheG affine Hilbert bundle defined by c will be denoted byE(c).
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 E~ (as topological spaces) and defines a continuous sectionc :G →r∗~E (where r∗E~ is the pull-back alongr :G →X) according to:
c(γ) =a(γ)e◦s(γ)−e◦r(γ).
c satisfies the cocycle property:
c(γγ0) =c(γ) +L(γ)c(γ0).
Another choice of section gives a cohomologous cocycle.
Conversely, a continuous cocyclec :G →r∗~E endowE =~E with a G-affine action according to: a(γ)e =c(γ) +L(γ)e. TheG affine Hilbert bundle defined by c will be denoted by E(c).
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 E~ (as topological spaces) and defines a continuous sectionc :G →r∗~E (where r∗E~ is the pull-back alongr :G →X) according to:
c(γ) =a(γ)e◦s(γ)−e◦r(γ).
c satisfies the cocycle property:
c(γγ0) =c(γ) +L(γ)c(γ0).
Another choice of section gives a cohomologous cocycle.
Conversely, a continuous cocyclec :G →r∗~E endowE =~E with a G-affine action according to: a(γ)e =c(γ) +L(γ)e. TheG affine Hilbert bundle defined by c will be denoted by E(c).
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 functionψ: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∈Rsuch that Pn 1ζi = 0,
∀x ∈G(0), ∀γ1, . . . , γn ∈Gx, X
i,j
ψ(γi−1γj)ζiζj ≤0.
Again, we are interested in topological groupoids and in continuous CNT functions.
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 →Rbe 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 allγ∈G ,ψ(γ) =kc(γ)k2;
for all x∈G(0),{c(γ), γ∈Gx}is total in Ex.
2 If(E0,c0) is another pair satisfying the same properties, there exists a G -equivariant isometric continuous bundle map u:E →E0 such that c0=u◦c.
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 →Rbe 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 allγ∈G ,ψ(γ) =kc(γ)k2;
for all x∈G(0),{c(γ), γ∈Gx}is total in Ex.
2 If(E0,c0) is another pair satisfying the same properties, there exists a G -equivariant isometric continuous bundle map u:E →E0 such that c0=u◦c.
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.
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.
Let (G, λ) be a locally compact groupoid with Haar system. We construct from aG-Hilbert bundleE aC∗(G)-bimoduleC∗(G,r∗E). LetCc(G,r∗E) be the space of compactly supported continuous sections ofr∗E. We define the left and right actions: for f,g ∈Cc(G) andξ∈Cc(G,r∗E),
fξ(γ) = Z
f(γ0)L(γ0)ξ(γ0−1γ)dλr(γ)(γ0)
ξg(γ) = Z
ξ(γγ0)g(γ0−1)dλs(γ)(γ0)
and the left and right inner products: for ξ, η ∈Cc(G),
l < ξ, η >(γ) = Z
(ξ(γ0)|L(γ)η(γ−1γ0))r(γ)dλr(γ)(γ0)
< ξ, η >r (γ) = Z
(ξ(γ0−1)|η(γ0−1γ))s(γ0)dλr(γ)(γ0)
Let (G, λ) be a locally compact groupoid with Haar system. We construct from aG-Hilbert bundleE aC∗(G)-bimoduleC∗(G,r∗E). LetCc(G,r∗E) be the space of compactly supported continuous sections ofr∗E. We define the left and right actions: for f,g ∈Cc(G) andξ∈Cc(G,r∗E),
fξ(γ) = Z
f(γ0)L(γ0)ξ(γ0−1γ)dλr(γ)(γ0)
ξg(γ) = Z
ξ(γγ0)g(γ0−1)dλs(γ)(γ0)
and the left and right inner products: for ξ, η ∈Cc(G),
l < ξ, η >(γ) = Z
(ξ(γ0)|L(γ)η(γ−1γ0))r(γ)dλr(γ)(γ0)
< ξ, η >r (γ) = Z
(ξ(γ0−1)|η(γ0−1γ))s(γ0)dλr(γ)(γ0)
We define C∗(G,r∗E) as the completion ofCc(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. Defineδ :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+fδ(g);
2 < δ(f), δ(g)>r= l < δ(f∗), δ(g∗)>.
We define C∗(G,r∗E) as the completion ofCc(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. Defineδ :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+fδ(g);
2 < δ(f), δ(g)>r= l < δ(f∗), δ(g∗)>.
What can be said about this derivation in the C*-algebraic framework?
In the scalar case, i.e. E =X ×Cand c :G →R, the bimodule isC∗(G).
We have seen that the derivation δ defined byc 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.
What can be said about this derivation in the C*-algebraic framework?
In the scalar case, i.e. E =X ×Cand c :G →R, the bimodule isC∗(G).
We have seen that the derivation δ defined byc 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.
What can be said about this derivation in the C*-algebraic framework?
In the scalar case, i.e. E =X ×Cand c :G →R, the bimodule isC∗(G).
We have seen that the derivation δ defined byc 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.
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
L(α, β) = 1
2[α∗∆(β) + ∆(α∗)β−∆(α∗β)].
One can prove that L is completely positive in the sense that
∀n∈N,∀α1, . . . , αn∈ A,[L(αi, αj)]∈Mn(A)+.
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
L(α, β) = 1
2[α∗∆(β) + ∆(α∗)β−∆(α∗β)].
One can prove that L is completely positive in the sense that
∀n∈N,∀α1, . . . , αn∈ A,[L(αi, αj)]∈Mn(A)+.
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
L(α, β) = 1
2[α∗∆(β) + ∆(α∗)β−∆(α∗β)].
One can prove that L is completely positive in the sense that
∀n∈N,∀α1, . . . , αn∈ A,[L(αi, αj)]∈Mn(A)+.
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
L(α, β) = 1
2[α∗∆(β) + ∆(α∗)β−∆(α∗β)].
One can prove that L is completely positive in the sense that
∀n∈N,∀α1, . . . , αn∈ A,[L(αi, αj)]∈Mn(A)+.
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 letL be its associated Dirichlet form. Then
1 There exists an (A,A)-C*-bimoduleE and a derivationδ :A → E such that
for allα, β∈ A,L(α, β) =< δ(α), δ(β)>A;
the range ofδgeneratesE as a right Banach A-module.
2 If(E0, δ0) is another pair satisfying the same properties, there exists a C*-bimodule isomorphism u :E → E0 such that δ0 =u◦δ.
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 letL be its associated Dirichlet form. Then
1 There exists an (A,A)-C*-bimoduleE and a derivationδ :A → E such that
for allα, β∈ A,L(α, β) =< δ(α), δ(β)>A;
the range ofδgeneratesE as a right Banach A-module.
2 If(E0, δ0) is another pair satisfying the same properties, there exists a C*-bimodule isomorphism u :E → E0 such that δ0 =u◦δ.
Let (G, λ) be a locally compact groupoid with Haar system.
Proposition
Given a continuous CNT function ψ:G →R
1 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).
2 Its generator is−∆with the dense sub-∗algebraA=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 ofψ described in GNS representation 2.
Let (G, λ) be a locally compact groupoid with Haar system.
Proposition
Given a continuous CNT function ψ:G →R
1 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).
2 Its generator is−∆with the dense sub-∗algebraA=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 ofψ described in GNS representation 2.
Let (G, λ) be a locally compact groupoid with Haar system.
Proposition
Given a continuous CNT function ψ:G →R
1 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).
2 Its generator is−∆with the dense sub-∗algebraA=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 ofψ described in GNS representation 2.
One can apply Sauvageot and Cipriani-Sauvageot results to conclude that the derivation δc associated with a cocyclec :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 thatδ is a ∗-derivation.
One can apply Sauvageot and Cipriani-Sauvageot results to conclude that the derivation δc associated with a cocyclec :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 thatδ is a ∗-derivation.
