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

semidirect product fell bundles
SMART_READER_LITE
LIVE PREVIEW

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

References Semidirect Product Fell Bundles L. Hall*, S. Kaliszewski, J. Quigg, D. Williams *H*KQW Semidirect Products 1 / 15 References Groupoid A set G and a subset G ( 2 ) G G together with maps Composition: G ( 2 ) G , denoted (


slide-1
SLIDE 1

References

Semidirect Product Fell Bundles

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

*H*KQW Semidirect Products 1 / 15

slide-2
SLIDE 2

References

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

*H*KQW Semidirect Products 2 / 15

slide-3
SLIDE 3

References

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

*H*KQW Semidirect Products 2 / 15

slide-4
SLIDE 4

References

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)

*H*KQW Semidirect Products 2 / 15

slide-5
SLIDE 5

References

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.

*H*KQW Semidirect Products 2 / 15

slide-6
SLIDE 6

References

Examples

Groups

*H*KQW Semidirect Products 3 / 15

slide-7
SLIDE 7

References

Examples

Groups T

  • pological Spaces

*H*KQW Semidirect Products 3 / 15

slide-8
SLIDE 8

References

Examples

Groups T

  • pological Spaces

Group Bundles

*H*KQW Semidirect Products 3 / 15

slide-9
SLIDE 9

References

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).

*H*KQW Semidirect Products 3 / 15

slide-10
SLIDE 10

References

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)

*H*KQW Semidirect Products 3 / 15

slide-11
SLIDE 11

References

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

*H*KQW Semidirect Products 4 / 15

slide-12
SLIDE 12

References

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)

*H*KQW Semidirect Products 4 / 15

slide-13
SLIDE 13

References

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)

*H*KQW Semidirect Products 4 / 15

slide-14
SLIDE 14

References

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.

*H*KQW Semidirect Products 4 / 15

slide-15
SLIDE 15

References

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)

*H*KQW Semidirect Products 4 / 15

slide-16
SLIDE 16

References

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!

*H*KQW Semidirect Products 4 / 15

slide-17
SLIDE 17

References

Groupoid Actions

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

*H*KQW Semidirect Products 5 / 15

slide-18
SLIDE 18

References

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

*H*KQW Semidirect Products 5 / 15

slide-19
SLIDE 19

References

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.

*H*KQW Semidirect Products 5 / 15

slide-20
SLIDE 20

References

Groupoid Bundle Actions

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

*H*KQW Semidirect Products 6 / 15

slide-21
SLIDE 21

References

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).

*H*KQW Semidirect Products 6 / 15

slide-22
SLIDE 22

References

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

*H*KQW Semidirect Products 6 / 15

slide-23
SLIDE 23

References

Actions by Isomorphisms

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

*H*KQW Semidirect Products 7 / 15

slide-24
SLIDE 24

References

Actions by Isomorphisms

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

*H*KQW Semidirect Products 7 / 15

slide-25
SLIDE 25

References

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*KQW Semidirect Products 7 / 15

slide-26
SLIDE 26

References

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

*H*KQW Semidirect Products 7 / 15

slide-27
SLIDE 27

References

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*KQW Semidirect Products 8 / 15

slide-28
SLIDE 28

References

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

*H*KQW Semidirect Products 8 / 15

slide-29
SLIDE 29

References

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

*H*KQW Semidirect Products 9 / 15

slide-30
SLIDE 30

References

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.

*H*KQW Semidirect Products 9 / 15

slide-31
SLIDE 31

References

Fell Bundle Algebras

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

*H*KQW Semidirect Products 10 / 15

slide-32
SLIDE 32

References

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)

*H*KQW Semidirect Products 10 / 15

slide-33
SLIDE 33

References

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)

*H*KQW Semidirect Products 10 / 15

slide-34
SLIDE 34

References

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)∗.

*H*KQW Semidirect Products 10 / 15

slide-35
SLIDE 35

References

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!

*H*KQW Semidirect Products 10 / 15

slide-36
SLIDE 36

References

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.

*H*KQW Semidirect Products 11 / 15

slide-37
SLIDE 37

References

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♣

  • *H*KQW

Semidirect Products 11 / 15

slide-38
SLIDE 38

References

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)

*H*KQW Semidirect Products 12 / 15

slide-39
SLIDE 39

References

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).

*H*KQW Semidirect Products 13 / 15

slide-40
SLIDE 40

References

Main Event

Theorem C∗(A ≀ G) ∼ = C∗(A) ⋊α G.

*H*KQW Semidirect Products 14 / 15

slide-41
SLIDE 41

References

Main Event

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

*H*KQW Semidirect Products 14 / 15

slide-42
SLIDE 42

References

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

*H*KQW Semidirect Products 14 / 15

slide-43
SLIDE 43

References

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

*H*KQW Semidirect Products 14 / 15

slide-44
SLIDE 44

References

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.

*H*KQW Semidirect Products 15 / 15