The Generalized E�ros-Hahn Conjecture forGroupoids
Marius Ionescu
Cornell University
Joint with Dana P. Williams, Dartmouth College
June 21, 2008
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
EH-regular dynamical systems
De�nition
A dynamical system (A,G , α) is called EH-regular if every primitive
ideal in A oα G is induced from a stability group.
Fact
In their 1967 Memoir, E�ros and Hahn conjectured that if (G ,X )was second countable locally compact transformation group with G
amenable, C0(X ) olt G is EH-regular.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
EH-regular dynamical systems
De�nition
A dynamical system (A,G , α) is called EH-regular if every primitive
ideal in A oα G is induced from a stability group.
Fact
In their 1967 Memoir, E�ros and Hahn conjectured that if (G ,X )was second countable locally compact transformation group with G
amenable, C0(X ) olt G is EH-regular.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Sauvageot-Gootman-Rosenberg Theorem
Theorem
A separable dynamical system (A,G , α) with G amenable is
EH-regular.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Our result
Theorem
Assume that G is a second countable locally compact Hausdor�
groupoid with Haar system {λu }u∈G (0) . Assume also that G is
amenable. If K ⊂ C ∗(G ) is a primitive ideal then K is induced from
an isotropy group. That is
K = IndGG(u)J
for a primitive ideal J ∈ Prim(C ∗(G (u)).
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Renault's Results
Fact
In his JOT '91 paper, Renault shows that given any representation
R of C ∗(G ), then we can form a restriction R ′ to the isotropy
group bundle and a corresponding induced representation IndR ′ ofC ∗(G ) such that ker
(IndR ′
)= kerR (if G is amenable).
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
The isotropy groups
De�nition
For u ∈ G (0) the isotropy group at u is
G (u) := Guu = {γ ∈ G : r(γ) = u = s(γ)}.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Inducing Representations
We assume that G is a second countable locally compact
groupoid with Haar system {λu }u∈G (0) .
Let H be a closed subgroupoid of G with Haar system
{αu }u∈H(0) .
Then GH(0) := s−1(H(0)) is a locally compact free and proper
right H-space.
If L is a representation of C ∗(H) the induced representation
Ind L of C ∗(G ) acts on Cc(GH(0))⊗HL via
(Ind L)(F )(ϕ⊗ h) = F · ϕ⊗ h.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Inducing Representations
We assume that G is a second countable locally compact
groupoid with Haar system {λu }u∈G (0) .
Let H be a closed subgroupoid of G with Haar system
{αu }u∈H(0) .
Then GH(0) := s−1(H(0)) is a locally compact free and proper
right H-space.
If L is a representation of C ∗(H) the induced representation
Ind L of C ∗(G ) acts on Cc(GH(0))⊗HL via
(Ind L)(F )(ϕ⊗ h) = F · ϕ⊗ h.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Inducing Representations
We assume that G is a second countable locally compact
groupoid with Haar system {λu }u∈G (0) .
Let H be a closed subgroupoid of G with Haar system
{αu }u∈H(0) .
Then GH(0) := s−1(H(0)) is a locally compact free and proper
right H-space.
If L is a representation of C ∗(H) the induced representation
Ind L of C ∗(G ) acts on Cc(GH(0))⊗HL via
(Ind L)(F )(ϕ⊗ h) = F · ϕ⊗ h.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Inducing Representations
We assume that G is a second countable locally compact
groupoid with Haar system {λu }u∈G (0) .
Let H be a closed subgroupoid of G with Haar system
{αu }u∈H(0) .
Then GH(0) := s−1(H(0)) is a locally compact free and proper
right H-space.
If L is a representation of C ∗(H) the induced representation
Ind L of C ∗(G ) acts on Cc(GH(0))⊗HL via
(Ind L)(F )(ϕ⊗ h) = F · ϕ⊗ h.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Inducing Representations
We assume that G is a second countable locally compact
groupoid with Haar system {λu }u∈G (0) .
Let H be a closed subgroupoid of G with Haar system
{αu }u∈H(0) .
Then GH(0) := s−1(H(0)) is a locally compact free and proper
right H-space.
If L is a representation of C ∗(H) the induced representation
Ind L of C ∗(G ) acts on Cc(GH(0))⊗HL via
(Ind L)(F )(ϕ⊗ h) = F · ϕ⊗ h.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Inducing irreducible representations
Theorem (M.I., Dana P. Williams)
Let G be a second countable groupoid with Haar system
{λu }u∈G (0) . Suppose that L is an irreducible representation of the
stability group G (u) at u ∈ G (0). Then IndGG(u) L is an irreducible
representation of C ∗(G ).
Theorem
There is a well-de�ned continuous map
IndGG(u) : I(C ∗(G (u)))→ I(C ∗(G ))
characterized by
ker(IndGG(u)L) = IndG
G(u)(kerL),
where, for a C ∗-algebra A, I(A) is the set of all closed two sided
ideals.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Inducing irreducible representations
Theorem (M.I., Dana P. Williams)
Let G be a second countable groupoid with Haar system
{λu }u∈G (0) . Suppose that L is an irreducible representation of the
stability group G (u) at u ∈ G (0). Then IndGG(u) L is an irreducible
representation of C ∗(G ).
Theorem
There is a well-de�ned continuous map
IndGG(u) : I(C ∗(G (u)))→ I(C ∗(G ))
characterized by
ker(IndGG(u)L) = IndG
G(u)(kerL),
where, for a C ∗-algebra A, I(A) is the set of all closed two sided
ideals.
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Main result
Theorem
Assume that G is a second countable locally compact Hausdor�
groupoid with Haar system {λu }u∈G (0) . Assume also that G is
amenable. If K ⊂ C ∗(G ) is a primitive ideal then K is induced from
an isotropy group. That is
K = IndGG(u)J
for a primitive ideal J ∈ Prim(C ∗(G (u)).
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Sketch of the Proof
Let R be an irreducible representation of C ∗(G ) and assume
that R is the integrated form of (µ,G (0) ∗ H, V̂ ).
We let Σ(0) be the space of closed subgroups of G .
Let Σ = { (γ,H) ∈ G ×Σ(0) : γ ∈ H } be the associated group
bundle.
The groupoid C ∗-algebra C ∗(Σ) is a C0(G (0))-algebra.
For η ∈ G de�ne
αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))
by
αη(F )(r(γ),H, γ
):= ω(η−1,H)−1F
(s(η), η−1 · H, η−1γη
).
The triple (C ∗(Σ),G , α) is a groupoid dynamical system
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Sketch of the Proof
Let R be an irreducible representation of C ∗(G ) and assume
that R is the integrated form of (µ,G (0) ∗ H, V̂ ).
We let Σ(0) be the space of closed subgroups of G .
Let Σ = { (γ,H) ∈ G ×Σ(0) : γ ∈ H } be the associated group
bundle.
The groupoid C ∗-algebra C ∗(Σ) is a C0(G (0))-algebra.
For η ∈ G de�ne
αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))
by
αη(F )(r(γ),H, γ
):= ω(η−1,H)−1F
(s(η), η−1 · H, η−1γη
).
The triple (C ∗(Σ),G , α) is a groupoid dynamical system
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Sketch of the Proof
Let R be an irreducible representation of C ∗(G ) and assume
that R is the integrated form of (µ,G (0) ∗ H, V̂ ).
We let Σ(0) be the space of closed subgroups of G .
Let Σ = { (γ,H) ∈ G ×Σ(0) : γ ∈ H } be the associated group
bundle.
The groupoid C ∗-algebra C ∗(Σ) is a C0(G (0))-algebra.
For η ∈ G de�ne
αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))
by
αη(F )(r(γ),H, γ
):= ω(η−1,H)−1F
(s(η), η−1 · H, η−1γη
).
The triple (C ∗(Σ),G , α) is a groupoid dynamical system
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Sketch of the Proof
Let R be an irreducible representation of C ∗(G ) and assume
that R is the integrated form of (µ,G (0) ∗ H, V̂ ).
We let Σ(0) be the space of closed subgroups of G .
Let Σ = { (γ,H) ∈ G ×Σ(0) : γ ∈ H } be the associated group
bundle.
The groupoid C ∗-algebra C ∗(Σ) is a C0(G (0))-algebra.
For η ∈ G de�ne
αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))
by
αη(F )(r(γ),H, γ
):= ω(η−1,H)−1F
(s(η), η−1 · H, η−1γη
).
The triple (C ∗(Σ),G , α) is a groupoid dynamical system
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Sketch of the Proof
Let R be an irreducible representation of C ∗(G ) and assume
that R is the integrated form of (µ,G (0) ∗ H, V̂ ).
We let Σ(0) be the space of closed subgroups of G .
Let Σ = { (γ,H) ∈ G ×Σ(0) : γ ∈ H } be the associated group
bundle.
The groupoid C ∗-algebra C ∗(Σ) is a C0(G (0))-algebra.
For η ∈ G de�ne
αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))
by
αη(F )(r(γ),H, γ
):= ω(η−1,H)−1F
(s(η), η−1 · H, η−1γη
).
The triple (C ∗(Σ),G , α) is a groupoid dynamical system
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Sketch of the Proof
Let R be an irreducible representation of C ∗(G ) and assume
that R is the integrated form of (µ,G (0) ∗ H, V̂ ).
We let Σ(0) be the space of closed subgroups of G .
Let Σ = { (γ,H) ∈ G ×Σ(0) : γ ∈ H } be the associated group
bundle.
The groupoid C ∗-algebra C ∗(Σ) is a C0(G (0))-algebra.
For η ∈ G de�ne
αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))
by
αη(F )(r(γ),H, γ
):= ω(η−1,H)−1F
(s(η), η−1 · H, η−1γη
).
The triple (C ∗(Σ),G , α) is a groupoid dynamical system
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Restriction to the stability groups
We de�ne a representation r of C ∗(Σ) on L2(G (0) ∗ H, µ)which we call the restriction of R to the isotropy groups of G
by
r(F )h(u) :=
ˆG(u)
F (u,G (u), γ)Vγh(u) dβG(u)(γ).
Note that
r =
ˆ ⊕G (0)
ru dµ(u).
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Restriction to the stability groups
We de�ne a representation r of C ∗(Σ) on L2(G (0) ∗ H, µ)which we call the restriction of R to the isotropy groups of G
by
r(F )h(u) :=
ˆG(u)
F (u,G (u), γ)Vγh(u) dβG(u)(γ).
Note that
r =
ˆ ⊕G (0)
ru dµ(u).
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Sketch of the Proof (cont'd)
(r , V̂ ) is a covariant representation of(C ∗(Σ),G , α
).
let L := r o V̂ be the representation of the groupoid crossed
product C ∗(Σ) oα G
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Sketch of the Proof (cont'd)
(r , V̂ ) is a covariant representation of(C ∗(Σ),G , α
).
let L := r o V̂ be the representation of the groupoid crossed
product C ∗(Σ) oα G
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Disintegration over PrimC ∗(Σ) (Renault)
the representation L above is equivalent to a representation L̂
which is built from
1 a measure ν on PrimC∗(Σ) which is quasi-invariant for the
G-action,2 a Borel Hilbert bundle K over PrimC∗(Σ),3 a Borel homomorphism L̃ : G → Iso(PrimC∗(Σ) ∗ K) such that
L̃(P, γ) =(P, L̃(P,γ),P · γ
)and
4 a representation r̃ of C∗(Σ) such that
1 er =
ˆ ⊕
PrimC∗(Σ)
erP dν(P),
with erP homogeneous with kernel P for each P, and
2 so that (er ,eL) is covariant. Thus, there is a ν-conull setU ⊂ PrimC∗(Σ) such that
eL(P, γ)erP·γ(F ) = erP`αγ(F )´er(P, γ)
for all (P, γ) ∈ G|U .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Disintegration over PrimC ∗(Σ) (Renault)
the representation L above is equivalent to a representation L̂
which is built from
1 a measure ν on PrimC∗(Σ) which is quasi-invariant for the
G-action,
2 a Borel Hilbert bundle K over PrimC∗(Σ),3 a Borel homomorphism L̃ : G → Iso(PrimC∗(Σ) ∗ K) such that
L̃(P, γ) =(P, L̃(P,γ),P · γ
)and
4 a representation r̃ of C∗(Σ) such that
1 er =
ˆ ⊕
PrimC∗(Σ)
erP dν(P),
with erP homogeneous with kernel P for each P, and
2 so that (er ,eL) is covariant. Thus, there is a ν-conull setU ⊂ PrimC∗(Σ) such that
eL(P, γ)erP·γ(F ) = erP`αγ(F )´er(P, γ)
for all (P, γ) ∈ G|U .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Disintegration over PrimC ∗(Σ) (Renault)
the representation L above is equivalent to a representation L̂
which is built from
1 a measure ν on PrimC∗(Σ) which is quasi-invariant for the
G-action,2 a Borel Hilbert bundle K over PrimC∗(Σ),
3 a Borel homomorphism L̃ : G → Iso(PrimC∗(Σ) ∗ K) such that
L̃(P, γ) =(P, L̃(P,γ),P · γ
)and
4 a representation r̃ of C∗(Σ) such that
1 er =
ˆ ⊕
PrimC∗(Σ)
erP dν(P),
with erP homogeneous with kernel P for each P, and
2 so that (er ,eL) is covariant. Thus, there is a ν-conull setU ⊂ PrimC∗(Σ) such that
eL(P, γ)erP·γ(F ) = erP`αγ(F )´er(P, γ)
for all (P, γ) ∈ G|U .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Disintegration over PrimC ∗(Σ) (Renault)
the representation L above is equivalent to a representation L̂
which is built from
1 a measure ν on PrimC∗(Σ) which is quasi-invariant for the
G-action,2 a Borel Hilbert bundle K over PrimC∗(Σ),3 a Borel homomorphism L̃ : G → Iso(PrimC∗(Σ) ∗ K) such that
L̃(P, γ) =(P, L̃(P,γ),P · γ
)and
4 a representation r̃ of C∗(Σ) such that
1 er =
ˆ ⊕
PrimC∗(Σ)
erP dν(P),
with erP homogeneous with kernel P for each P, and
2 so that (er ,eL) is covariant. Thus, there is a ν-conull setU ⊂ PrimC∗(Σ) such that
eL(P, γ)erP·γ(F ) = erP`αγ(F )´er(P, γ)
for all (P, γ) ∈ G|U .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Disintegration over PrimC ∗(Σ) (Renault)
the representation L above is equivalent to a representation L̂
which is built from
1 a measure ν on PrimC∗(Σ) which is quasi-invariant for the
G-action,2 a Borel Hilbert bundle K over PrimC∗(Σ),3 a Borel homomorphism L̃ : G → Iso(PrimC∗(Σ) ∗ K) such that
L̃(P, γ) =(P, L̃(P,γ),P · γ
)and
4 a representation r̃ of C∗(Σ) such that
1 er =
ˆ ⊕
PrimC∗(Σ)
erP dν(P),
with erP homogeneous with kernel P for each P, and
2 so that (er ,eL) is covariant. Thus, there is a ν-conull setU ⊂ PrimC∗(Σ) such that
eL(P, γ)erP·γ(F ) = erP`αγ(F )´er(P, γ)
for all (P, γ) ∈ G|U .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Disintegration over PrimC ∗(Σ) (Renault)
the representation L above is equivalent to a representation L̂
which is built from
1 a measure ν on PrimC∗(Σ) which is quasi-invariant for the
G-action,2 a Borel Hilbert bundle K over PrimC∗(Σ),3 a Borel homomorphism L̃ : G → Iso(PrimC∗(Σ) ∗ K) such that
L̃(P, γ) =(P, L̃(P,γ),P · γ
)and
4 a representation r̃ of C∗(Σ) such that
1 er =
ˆ ⊕
PrimC∗(Σ)
erP dν(P),
with erP homogeneous with kernel P for each P, and
2 so that (er ,eL) is covariant. Thus, there is a ν-conull setU ⊂ PrimC∗(Σ) such that
eL(P, γ)erP·γ(F ) = erP`αγ(F )´er(P, γ)
for all (P, γ) ∈ G|U .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
Disintegration over PrimC ∗(Σ) (Renault)
the representation L above is equivalent to a representation L̂
which is built from
1 a measure ν on PrimC∗(Σ) which is quasi-invariant for the
G-action,2 a Borel Hilbert bundle K over PrimC∗(Σ),3 a Borel homomorphism L̃ : G → Iso(PrimC∗(Σ) ∗ K) such that
L̃(P, γ) =(P, L̃(P,γ),P · γ
)and
4 a representation r̃ of C∗(Σ) such that
1 er =
ˆ ⊕
PrimC∗(Σ)
erP dν(P),
with erP homogeneous with kernel P for each P, and
2 so that (er ,eL) is covariant. Thus, there is a ν-conull setU ⊂ PrimC∗(Σ) such that
eL(P, γ)erP·γ(F ) = erP`αγ(F )´er(P, γ)
for all (P, γ) ∈ G|U .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
The Induced Representation
r and r̃ are equivalent representations of C ∗(Σ).
The measure ν on PrimC ∗(Σ) in the ideal center
decomposition is ergodic with respect to the action of G on
Prim(C ∗(Σ)).
We de�ne the induced representation
indr̃ :=
ˆ ⊕PrimC∗(Σ)
indGG(σ(P))r̃Pdν(P).
The kernel of the representation indr̃ is an induced primitive
ideal of C ∗(G ).
indr̃ is equivalent with Renault's induced representation indr .
Renault's results imply that indr is weakly contained in R and,
if G is amenable, R is also weakly contained in indr .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
The Induced Representation
r and r̃ are equivalent representations of C ∗(Σ).
The measure ν on PrimC ∗(Σ) in the ideal center
decomposition is ergodic with respect to the action of G on
Prim(C ∗(Σ)).
We de�ne the induced representation
indr̃ :=
ˆ ⊕PrimC∗(Σ)
indGG(σ(P))r̃Pdν(P).
The kernel of the representation indr̃ is an induced primitive
ideal of C ∗(G ).
indr̃ is equivalent with Renault's induced representation indr .
Renault's results imply that indr is weakly contained in R and,
if G is amenable, R is also weakly contained in indr .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
The Induced Representation
r and r̃ are equivalent representations of C ∗(Σ).
The measure ν on PrimC ∗(Σ) in the ideal center
decomposition is ergodic with respect to the action of G on
Prim(C ∗(Σ)).
We de�ne the induced representation
indr̃ :=
ˆ ⊕PrimC∗(Σ)
indGG(σ(P))r̃Pdν(P).
The kernel of the representation indr̃ is an induced primitive
ideal of C ∗(G ).
indr̃ is equivalent with Renault's induced representation indr .
Renault's results imply that indr is weakly contained in R and,
if G is amenable, R is also weakly contained in indr .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
The Induced Representation
r and r̃ are equivalent representations of C ∗(Σ).
The measure ν on PrimC ∗(Σ) in the ideal center
decomposition is ergodic with respect to the action of G on
Prim(C ∗(Σ)).
We de�ne the induced representation
indr̃ :=
ˆ ⊕PrimC∗(Σ)
indGG(σ(P))r̃Pdν(P).
The kernel of the representation indr̃ is an induced primitive
ideal of C ∗(G ).
indr̃ is equivalent with Renault's induced representation indr .
Renault's results imply that indr is weakly contained in R and,
if G is amenable, R is also weakly contained in indr .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
The Induced Representation
r and r̃ are equivalent representations of C ∗(Σ).
The measure ν on PrimC ∗(Σ) in the ideal center
decomposition is ergodic with respect to the action of G on
Prim(C ∗(Σ)).
We de�ne the induced representation
indr̃ :=
ˆ ⊕PrimC∗(Σ)
indGG(σ(P))r̃Pdν(P).
The kernel of the representation indr̃ is an induced primitive
ideal of C ∗(G ).
indr̃ is equivalent with Renault's induced representation indr .
Renault's results imply that indr is weakly contained in R and,
if G is amenable, R is also weakly contained in indr .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids
The Induced Representation
r and r̃ are equivalent representations of C ∗(Σ).
The measure ν on PrimC ∗(Σ) in the ideal center
decomposition is ergodic with respect to the action of G on
Prim(C ∗(Σ)).
We de�ne the induced representation
indr̃ :=
ˆ ⊕PrimC∗(Σ)
indGG(σ(P))r̃Pdν(P).
The kernel of the representation indr̃ is an induced primitive
ideal of C ∗(G ).
indr̃ is equivalent with Renault's induced representation indr .
Renault's results imply that indr is weakly contained in R and,
if G is amenable, R is also weakly contained in indr .
Marius Ionescu Cornell University The Generalized E�ros-Hahn Conjecture for Groupoids