The Generalized Effros-Hahn Conjecture for Groupoids

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.

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.

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.

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.

Sauvageot-Gootman-Rosenberg Theorem

Theorem

A separable dynamical system (A,G , α) with G amenable is

EH-regular.

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)).

Renault's Results

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).

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)}.

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.

{αu }u∈H(0) .

right H-space.

(Ind L)(F )(ϕ⊗ h) = F · ϕ⊗ h.

{αu }u∈H(0) .

right H-space.

(Ind L)(F )(ϕ⊗ h) = F · ϕ⊗ h.

{αu }u∈H(0) .

right H-space.

(Ind L)(F )(ϕ⊗ h) = F · ϕ⊗ h.

{αu }u∈H(0) .

right H-space.

(Ind L)(F )(ϕ⊗ h) = F · ϕ⊗ h.

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.

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.

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)).

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))

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

Sketch of the Proof

bundle.

αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))

):= ω(η−1,H)−1F

(s(η), η−1 · H, η−1γη

Sketch of the Proof

bundle.

αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))

):= ω(η−1,H)−1F

(s(η), η−1 · H, η−1γη

Sketch of the Proof

bundle.

αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))

):= ω(η−1,H)−1F

(s(η), η−1 · H, η−1γη

Sketch of the Proof

bundle.

αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))

):= ω(η−1,H)−1F

(s(η), η−1 · H, η−1γη

Sketch of the Proof

bundle.

αη : C ∗(Σ)(s(η))→ C ∗(Σ)(r(η))

):= ω(η−1,H)−1F

(s(η), η−1 · H, η−1γη

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

r(F )h(u) :=

ˆG(u)

F (u,G (u), γ)Vγh(u) dβG(u)(γ).

Note that

ˆ ⊕G (0)

ru dµ(u).

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

r(F )h(u) :=

ˆG(u)

F (u,G (u), γ)Vγh(u) dβG(u)(γ).

Note that

ˆ ⊕G (0)

ru dµ(u).

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

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

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

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 .

which is built from

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

1 er =

ˆ ⊕

PrimC∗(Σ)

erP dν(P),

which is built from

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

1 er =

ˆ ⊕

PrimC∗(Σ)

erP dν(P),

which is built from

L̃(P, γ) =(P, L̃(P,γ),P · γ

1 er =

ˆ ⊕

PrimC∗(Σ)

erP dν(P),

which is built from

L̃(P, γ) =(P, L̃(P,γ),P · γ

1 er =

ˆ ⊕

PrimC∗(Σ)

erP dν(P),

which is built from

L̃(P, γ) =(P, L̃(P,γ),P · γ

1 er =

ˆ ⊕

PrimC∗(Σ)

erP dν(P),

which is built from

L̃(P, γ) =(P, L̃(P,γ),P · γ

1 er =

ˆ ⊕

PrimC∗(Σ)

erP dν(P),

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 .

Prim(C ∗(Σ)).

indr̃ :=

ˆ ⊕PrimC∗(Σ)

ideal of C ∗(G ).

Prim(C ∗(Σ)).

indr̃ :=

ˆ ⊕PrimC∗(Σ)

ideal of C ∗(G ).

Prim(C ∗(Σ)).

indr̃ :=

ˆ ⊕PrimC∗(Σ)

ideal of C ∗(G ).

Prim(C ∗(Σ)).

indr̃ :=

ˆ ⊕PrimC∗(Σ)

ideal of C ∗(G ).

Prim(C ∗(Σ)).

indr̃ :=

ˆ ⊕PrimC∗(Σ)

ideal of C ∗(G ).

The Generalized Effros-Hahn Conjecture for Groupoids

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