Semidirect Product Fell Bundles L. Hall*, S. Kaliszewski, J. Quigg, - - PowerPoint PPT Presentation

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Semidirect Product Fell Bundles

• L. Hall*, S. Kaliszewski, J. Quigg, D. Williams

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Groupoid

A set G and a subset G(2) ⊂ G × G together with maps

i.

Composition: G(2) → G, denoted (γ, η) → γ · η

ii.

Inversion: G → G, denoted γ → γ−1

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Groupoid

A set G and a subset G(2) ⊂ G × G together with maps

i.

Composition: G(2) → G, denoted (γ, η) → γ · η

ii.

Inversion: G → G, denoted γ → γ−1 Associative composition Involutive inversion Cancellative products γ−1 · γ · η = η and γ · η · η−1 = γ

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Groupoid

A set G and a subset G(2) ⊂ G × G together with maps

i.

Composition: G(2) → G, denoted (γ, η) → γ · η

ii.

Inversion: G → G, denoted γ → γ−1 Associative composition Involutive inversion Cancellative products γ−1 · γ · η = η and γ · η · η−1 = γ G(0)

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Groupoid

A set G and a subset G(2) ⊂ G × G together with maps

i.

Composition: G(2) → G, denoted (γ, η) → γ · η

ii.

Inversion: G → G, denoted γ → γ−1 Associative composition Involutive inversion Cancellative products γ−1 · γ · η = η and γ · η · η−1 = γ G(0) Locally compact groupoids, with Haar systems.

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Examples

Groups

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Examples

Groups T

• pological Spaces

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Examples

Groups T

• pological Spaces

Group Bundles

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Examples

Groups T

• pological Spaces

Group Bundles Equivalence Relations R ⊂ S × S

associative composition (t, s) · (s, r) = (t, r) involutive inversion (t, s)−1 = (s, t) cancellative products (t, s) · (s, r) · (r, s) = (t, s) · (s, s) = (t, s).

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Examples

Groups T

• pological Spaces

Group Bundles Equivalence Relations R ⊂ S × S

associative composition (t, s) · (s, r) = (t, r) involutive inversion (t, s)−1 = (s, t) cancellative products (t, s) · (s, r) · (r, s) = (t, s) · (s, s) = (t, s).

Group Transformation Groupoids Group Γ acting on a space T. Define Γ ⋊ T by

Composition (h, y) · (g, x) = (hg, x) whenever y = gx Inverse (g, x)−1 = (g−1, gx)

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Haar Systems

Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = {λu}u∈G(0) of radon measures on G such that

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Haar Systems

Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = {λu}u∈G(0) of radon measures on G such that

1

supp λu = r−1(u)

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Haar Systems

Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = {λu}u∈G(0) of radon measures on G such that

1

supp λu = r−1(u)

2

For γ ∈ G, γ · λs(γ) = λr(γ)

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Haar Systems

Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = {λu}u∈G(0) of radon measures on G such that

1

supp λu = r−1(u)

2

For γ ∈ G, γ · λs(γ) = λr(γ)

3

Continuous assembly: For f ∈ Cc(G), the map u →

• G f dλu

is continuous.

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Haar Systems

Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = {λu}u∈G(0) of radon measures on G such that

1

supp λu = r−1(u)

2

For γ ∈ G, γ · λs(γ) = λr(γ)

3

Continuous assembly: For f ∈ Cc(G), the map u →

• G f dλu

is continuous. Define a convolution product on Cc(G): f ∗ g(η) =

• G f(γ)g(γ−1 · η)dλr(η)(γ)

Involution: f ∗ (γ) = f(γ−1)

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Haar Systems

Definition A Haar system[Wil19] on a locally compact groupoid G is a family λ = {λu}u∈G(0) of radon measures on G such that

1

supp λu = r−1(u)

2

For γ ∈ G, γ · λs(γ) = λr(γ)

3

Continuous assembly: For f ∈ Cc(G), the map u →

• G f dλu

is continuous. Define a convolution product on Cc(G): f ∗ g(η) =

• G f(γ)g(γ−1 · η)dλr(η)(γ)

Involution: f ∗ (γ) = f(γ−1) Complete to get a C∗-algebra!

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Groupoid Actions

Groupoid actions[Wil19]: For a space T, begin with moment map ρ : T → G(0).

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Groupoid Actions

Groupoid actions[Wil19]: For a space T, begin with moment map ρ : T → G(0). Fibre product G ∗ T = {(γ, t) : ρ(t) = s(γ)}. G ∗ T T G G(0)

proj proj ρ s

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Groupoid Actions

Groupoid actions[Wil19]: For a space T, begin with moment map ρ : T → G(0). Fibre product G ∗ T = {(γ, t) : ρ(t) = s(γ)}. G ∗ T T G G(0)

proj proj ρ s

The action: map G ∗ T → T, styled (γ, t) → γt with ρ(t) · t = t For (η, γ) ∈ G(2), (γ, t) ∈ G ∗ T, η · γ t = ηγt.

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Groupoid Bundle Actions

Let p : T → B be a bundle (p is continuous, open, surjective).

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Groupoid Bundle Actions

Let p : T → B be a bundle (p is continuous, open, surjective). G acts on the bundle provided G acts on T, B, and p intertwines the actions: γp(t) = p(γt).

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Groupoid Bundle Actions

Let p : T → B be a bundle (p is continuous, open, surjective). G acts on the bundle provided G acts on T, B, and p intertwines the actions: γp(t) = p(γt). Subbundle isomorphism: G, T, B as above. For x ∈ G T|Bs(x) T|Br(x) Bs(x) Br(x)

t→xt p| p| b→xb

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Actions by Isomorphisms

G acting on H a LCH groupoid. Definition G acts by isomorphisms provided

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Actions by Isomorphisms

G acting on H a LCH groupoid. Definition G acts by isomorphisms provided fibres over G(0) are subgroupoids of H;

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Actions by Isomorphisms

G acting on H a LCH groupoid. Definition G acts by isomorphisms provided fibres over G(0) are subgroupoids of H; for x ∈ G, the map h → xh is an isomorphism of the fibres.

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Actions by Isomorphisms

G acting on H a LCH groupoid. Definition G acts by isomorphisms provided fibres over G(0) are subgroupoids of H; for x ∈ G, the map h → xh is an isomorphism of the fibres. ♥ ♥ H ♠ ♣ ♠ ♣

• G(0)

ρ

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Actions on H

Definition (Semidirect Product Groupoid) Denote by H ≀ G the fibre product H ∗ G = {(h, x) : ρ(h) = r(x)} with operations (h, x) · (k, y) = (h · xk, x · y) (h, x)−1 = (x−1h−1, x−1)

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Actions on H

Definition (Semidirect Product Groupoid) Denote by H ≀ G the fibre product H ∗ G = {(h, x) : ρ(h) = r(x)} with operations (h, x) · (k, y) = (h · xk, x · y) (h, x)−1 = (x−1h−1, x−1) ♥ ♥ ♠ ♣ ♠ ♣

• h

h·xk xk k x·y y x

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Fell Bundles

Definition A Fell Bundle over H is a Banach bundle p : A → H which linearly reflects the structure of H. A(u) a C∗-algebra A(η) an A(r(η)) − A(s(η)) imprimitivity bimodule

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Fell Bundles

Definition A Fell Bundle over H is a Banach bundle p : A → H which linearly reflects the structure of H. A(u) a C∗-algebra A(η) an A(r(η)) − A(s(η)) imprimitivity bimodule It is profitable (though abusive) to think of a Fell bundle as a functor into H.

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Fell Bundle Algebras

Given a Fell bundle p : A → H, mimic the construction of C∗(H):

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Fell Bundle Algebras

Given a Fell bundle p : A → H, mimic the construction of C∗(H): A0 = Cc(H, A) compactly supported sections of p (so p ◦ f = idH)

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Fell Bundle Algebras

Given a Fell bundle p : A → H, mimic the construction of C∗(H): A0 = Cc(H, A) compactly supported sections of p (so p ◦ f = idH) Multiplication: vector valued integral f ∗ g(η) =

• H

f(γ)g(γ−1η)dλr(η)(γ)

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Fell Bundle Algebras

Given a Fell bundle p : A → H, mimic the construction of C∗(H): A0 = Cc(H, A) compactly supported sections of p (so p ◦ f = idH) Multiplication: vector valued integral f ∗ g(η) =

• H

f(γ)g(γ−1η)dλr(η)(γ) Involution: f ∗(η) = f(η−1)∗.

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Fell Bundle Algebras

Given a Fell bundle p : A → H, mimic the construction of C∗(H): A0 = Cc(H, A) compactly supported sections of p (so p ◦ f = idH) Multiplication: vector valued integral f ∗ g(η) =

• H

f(γ)g(γ−1η)dλr(η)(γ) Involution: f ∗(η) = f(η−1)∗. Complete to get a C∗-algebra!

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Fell Bundle Actions

Let’s make G act on a Fell bundle. Essentially, G acts by isomorphisms on H and any x ∈ G has a → x · a : As(x) → Ar(x) an isomorphism of Fell bundles.

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Fell Bundle Actions

Let’s make G act on a Fell bundle. Essentially, G acts by isomorphisms on H and any x ∈ G has a → x · a : As(x) → Ar(x) an isomorphism of Fell bundles. A♥ A♥ A♠ A♣ A♠ A♣

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Semidirect Product Fell Bundle

Definition (Semidirect Product Fell Bundle [KMQW10]) Denote by A ≀ G the fibre product A ∗ G = {(a, x) : ρ(p(a)) = s(x)} with operations (a, x) · (b, y) = (a · xb, x · y) (a, x)−1 = (x−1a−1, x−1)

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Dynamical System?!

Lemma Let p : A → H be a Fell bundle, and suppose p admits an action

• f G by isomorphisms. Then there is a groupoid dynamical

system (C∗(A), α, G). C∗(A) is a C0(G(0))-algebra with fibre C∗(A)(u) = C∗(Au) x ∈ G acts isomorphically on each G(0) fibre, by αx(f)(η) = xf(x−1η).

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Main Event

Theorem C∗(A ≀ G) ∼ = C∗(A) ⋊α G.

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Main Event

Theorem C∗(A ≀ G) ∼ = C∗(A) ⋊α G. Strategy: View C∗(A) ⋊α G as C∗(G, K)[MW08]

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Main Event

Theorem C∗(A ≀ G) ∼ = C∗(A) ⋊α G. Strategy: View C∗(A) ⋊α G as C∗(G, K)[MW08] Map f ∈ Γc(A ≀ G) to Γc(K) densely

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Main Event

Theorem C∗(A ≀ G) ∼ = C∗(A) ⋊α G. Strategy: View C∗(A) ⋊α G as C∗(G, K)[MW08] Map f ∈ Γc(A ≀ G) to Γc(K) densely Given a representation L of C∗(A ≀ G), produce a representation L of C∗(G, K) which L factors through. C∗(A ≀ G) B(H) C∗(G, K)

Φ L L

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References

[KMQW10] S. Kaliszewski, P . S. Muhly, J. Quigg, and D. P . Williams, Coactions and Fell bundles, New York J. Math. 16 (2010), 315–359. [MW08] P . S. Muhly and D. P . Williams, Equivalence and disintegration theorems for Fell bundles and their C∗-algebras, Dissertationes Mathematicae 456 (2008), 1–57. [Wil19] D. P . Williams, A tool kit for groupoid C∗-algegras, Mathematical Surveys and Monographs, vol. 241, American Mathematical Society, Providence, RI, 2019.

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