Actions of Compact Quantum Groups I Definition Kenny De Commer - - PowerPoint PPT Presentation

Actions of Compact Quantum Groups I

Definition

Kenny De Commer (VUB, Brussels, Belgium)

CQG Compact actions Non-compact actions

Course material

Material that will be treated:

◮ Actions and coactions of compact quantum groups. ◮ Actions on C∗-algebras and Hilbert modules. ◮ Crossed products. ◮ Free actions, ergodic actions, and their interrelationship.

CQG Compact actions Non-compact actions

Outline Lecture I

Compact quantum groups Actions of compact quantum groups on compact quantum spaces Actions on non-compact quantum spaces

CQG Compact actions Non-compact actions

Compact quantum groups

Definition (Woronowicz)

Compact quantum group (CQG) G: ◮ unital C∗-algebra C(G), ◮ unital ∗-homomorphism, comultiplication ∆ : C(G) → C(G) ⊗ C(G) s.t. ◮ coassociativity: (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆, ◮ cancellation: [(C(G) ⊗ 1G)∆(C(G))] = [∆(C(G))(1G ⊗ C(G))] = C(G) ⊗ C(G). Here: [S] = closed linear span of S (in some Banach space).

CQG Compact actions Non-compact actions

Classical CQG

Lemma

X, Y compact Hausdorff: C(X) ⊗ C(Y ) ∼ = C(X × Y ), (a ⊗ b)(x, y) = a(x)b(y).

Example

G compact Hausdorff group ⇒ CQG (C(G), ∆), ∆ : C(G) → C(G) ⊗ C(G), f → (∆(f) : (g, h) → f(gh)) . Conversely: CQG G with C(G) commutative ⇓ G = Spec(C(G)) compact Hausdorff group.

CQG Compact actions Non-compact actions

C(G)-corepresentations

Definition

Unitary C(G)-corepresentation:

◮ finite dimensional Hilbert space H, ◮ U ∈ B(H) ⊗ C(G)

s.t.

◮ U unitary, ◮ (id ⊗ ∆)(U) = U12U13, where U12 = U ⊗ 1 etc.

U ∈ B(H) ⊗ C(G)

• δ : H → H ⊗C(G),

ξ → U(ξ ⊗ 1G) s.t. . . . ?

CQG Compact actions Non-compact actions

G-representations

Definition

G compact quantum group. (Continuous finite dimensional unitary left) G-representation π: ◮ finite dimensional Hilbert space Hπ, ◮ linear map δπ : Hπ → Hπ ⊗C(G) s.t. ◮ right comodule: (id ⊗ ∆) ◦ δπ = (δπ ⊗ id) ◦ δπ, ◮ isometric: δπ(ξ)∗δπ(η) = ξ, η1C(G), ◮ density: [δπ(H)(1 ⊗ C(G))] = H ⊗ C(G). ◮ Here Hπ ∼ = B(C, Hπ), so (ξ ⊗ a)∗(η ⊗ b) = ξ∗η ⊗ a∗b ∼ = ξ, ηa∗b. ◮ Density condition automatically satisfied. ◮ C(G)-corepresentations ↔ G-representations.

CQG Compact actions Non-compact actions

Classical representations

Example

Let G compact Hausdorff group. Then G-representations as compact quantum group

• G-representations as compact group

by δπ : Hπ → Hπ ⊗C(G) ∼ = C(G, Hπ)

• π : G × Hπ → Hπ,

(g, ξ) → π(g)ξ = δπ(ξ)(g).

CQG Compact actions Non-compact actions

The canonical Hopf ∗-algebra

Theorem (Woronowicz)

Let O(G) = {(ξ∗ ⊗ id)δπ(η) | π G-representation, ξ, η ∈ Hπ}. Then

◮ (O(G), ∆) Hopf ∗-algebra, (O(G), ∆, ǫ, S), ◮ O(G) dense in C(G), ◮ (O(G), ∆) unique dense Hopf ∗-algebra, ◮ δπ : H → H ⊗O(G) is O(G)-comodule:

◮ (id ⊗ ∆) ◦ δπ = (δπ ⊗ id) ◦ δπ, ◮ (idH ⊗ ǫ)δπ = idH.

CQG Compact actions Non-compact actions

Notation (Sweedler-Heynemann notation)

h ∈ O(G): ∆(h) = h(1) ⊗h(2), (∆⊗ι)∆(h) = ∆(2)(h) = h(1) ⊗h(2) ⊗h(3), ...

Example

Let h ∈ O(G). Then ∆(h(1))(1 ⊗ S(h(2))) = h(1) ⊗ h(2)S(h(3)) = h(1) ⊗ ǫ(h(2))1 = h ⊗ 1. Hence (Linear span) ∆(O(G))(1 ⊗ O(G)) = O(G) ⊗

alg O(G).

CQG Compact actions Non-compact actions

Universal C∗-algebra

Lemma

G CQG.

◮ Universal C∗-envelope C(Gu) of O(G) exists. ◮ CQG Gu by

∆u : C(Gu) → C(Gu) ⊗ C(Gu).

Definition

Gu universal CQG (associated to G).

CQG Compact actions Non-compact actions

Right actions of compact quantum groups on C∗-algebras

Definition (Podle´ s)

Right action X G:

◮ Compact quantum group G, ◮ C∗-algebra C(X) (with X ‘compact quantum space’), ◮ Unital ∗-homomorphism, right coaction

α : C(X) → C(X) ⊗ C(G) s.t.

◮ coaction property:

(α ⊗ idG) ◦ α = (idX ⊗ ∆) ◦ α,

◮ density (Podle´

s condition): [α(C(X))(1X ⊗ C(G))] = C(X) ⊗ C(G).

CQG Compact actions Non-compact actions

Right translations

Example

Let G compact quantum group. Then G

∆

G by ∆ : C(G) → C(G) ⊗ C(G).

CQG Compact actions Non-compact actions

Half-classical case

Lemma (All C(G) commutative)

◮ G compact Hausdorff group, ◮ C∗-algebra C(X), ◮ G α

C(X) continuous action:

◮ (g, a) → αg(a) continuous, ◮ each αg ∗-automorphism, ◮ αgh = αg ◦ αh, ◮ αe = idX, for e ∈ G identity element.

⇒ X G, α : C(X) → C(X) ⊗ C(G) ∼ = C(G, C(X)), a → (α(a) : g → αg(a)) .

CQG Compact actions Non-compact actions

Proof, Part I

◮ Forgetting group structure:

◮ Using partitions of unity on G: ◮ C(X) ⊗ C(G) ∼ =

→ C(G, C(X)) by a ⊗ f → (g → f(g)a).

◮ C(X) ⊗ C(G) ⊗ C(G) ∼

= C(G × G, C(X)), etc.

◮ continuous G

α

C(X) by unital ∗-endomorphisms ⇔ α : C(X) → C(G, C(X)) unital ∗-homomorphism.

◮ ((id ⊗ ∆)α)(a)(g, h) = ((α ⊗ id)α)(a)(g, h) ⇔ αgh(a) = αg(αh(a)).

Conclusion: one-to-one correspondence between

◮ α with coaction property, and ◮ actions of a group on a C∗-algebra by endomorphisms.

To do: Density ⇔ αe = idC(X) for e unit G.

CQG Compact actions Non-compact actions

Proof, Part II

◮ ∗-homomorphism

• α : C(X) ⊗ C(G) → C(X) ⊗ C(G),

a ⊗ f → α(a)(1 ⊗ f).

◮ Density ⇔

α surjective.

◮ On level of C(G, C(X)) ∼

= C(X) ⊗ C(G): ∀F ∈ C(G, C(X)),

• α(F)(g) = αg(F(g)).

◮ Assume αe = idC(X). Then

α has inverse β,

• β(F)(g) = αg−1(F(g)).

Hence range α dense.

◮ If αe = idC(X) ⇒ αe non-trivial idempotent ∗-endomorphism. ◮ Put C(Xe) = αe(C(X)) = C(X). ◮ ∀g ∈ G: αg(C(X)) = αe(αg(C(X))) ⊆ C(Xe). ◮ ⇒ If a /

∈ C(Xe), then g → a not in range α.

CQG Compact actions Non-compact actions

Classical

Example (All C(G) and C(X) commutative)

G compact Hausdorff group, X compact Hausdorff space, X G continuous ⇒ G C(X), αg(f)(x) = f(x · g).

Example

Consider sphere SN−1 = {z = (z1, . . . , zN) ∈ RN |

• i

z2

i = 1}.

Then SN−1 O(N) by (z, g) → zg.

CQG Compact actions Non-compact actions

Example: Half-classical I

Example

Cuntz algebras, On = C∗(V1, . . . , Vn | V ∗

i Vj = δij,

• i

ViV ∗

i = 1).

Then U(n) On by αu(Vi) =

• j

ujiVj. In particular, S1 On by αz(Vi) = zVi.

CQG Compact actions Non-compact actions

Example: Half-classical II

Example (Banica)

Free spheres, C(SN−1

+

) =< V1, . . . , VN | Vi = V ∗

i ,

• i

V 2

i = 1}.

Then O(N) C(SN−1

+

) by αg(Vi) =

• j

gjiVj.

CQG Compact actions Non-compact actions

Left actions of compact quantum groups on C∗-algebras

Definition (Podle´ s)

Left action G H:

◮ Compact quantum group G, ◮ C∗-algebra C(X), ◮ Unital ∗-homomorphism, left coaction

α : C(X) → C(G) ⊗ C(X) s.t.

◮ coaction property:

(idG ⊗ α) ◦ α = (∆ ⊗ idX) ◦ α,

◮ density:

[(C(G) ⊗ 1X)α(C(X))] = C(X) ⊗ C(G).

CQG Compact actions Non-compact actions

From left to right

Definition

Let G CQG. Then Gop CQG by C(Gop) = C(G), ∆Gop = ∆op

G = ς ◦ ∆,

where ς : C(G) ⊗ C(G) → C(G) ⊗ C(G), g ⊗ h → h ⊗ g.

Lemma

G

α

X ↔ X

αop

Gop.

CQG Compact actions Non-compact actions

Non-unital C∗-algebras

Definition (Multiplier C∗-algebras)

C0(X) non-unital C∗-algebra (‘locally compact quantum space’). Multiplier C∗-algebra M(C0(X)) = Cb(X):

◮ Concrete: For C0(X) ⊆ B(H) with [C0(X) H] = H: Cb(X) = {T ∈ B(H) | ∀a ∈ C0(X), Ta, aT ∈ C0(X)}. ◮ Abstract: Cb(X) collection maps T : C0(X) → C0(X) s.t. ∃T ∗, ∀a, b ∈ C0(X), a∗(Tb) = (T ∗b)∗a.

If T ∈ Cb(X): T(ab) = T(a)b, and C0(X) ⊆ Cb(X).

Example

If X locally compact Hausdorff space, M(C0(X)) = Cb(X).

CQG Compact actions Non-compact actions

Morphisms between locally compact quantum spaces

Definition

∗-homomorphism π : C0(Y) → M(C0(X)) non-degenerate:

[π(C0(Y))C0(X)] = C0(X).

Example

Let X, Y locally compact Hausdorff spaces.

◮ Non-degenerate maps C0(Y ) → Cb(X) ⇔ continuous maps X → Y . ◮ Non-degenerate maps C0(Y ) → C0(X) ⇔ continuous proper maps X → Y . ◮ Degenerate map C0(Y ) → Cb(X): points of X to infinity.

Lemma

π : C0(Y) → Cb(X) non-degenerate ⇒ ∃!π : Cb(Y) → Cb(X).

CQG Compact actions Non-compact actions

Actions on locally compact quantum spaces

Definition

Right action X G: ◮ Compact quantum group G, ◮ C∗-algebra C0(X), ◮ non-degenerate ∗-homomorphism, right coaction α : C0(X) → Cb(X × G) s.t. ◮ coaction property: (α ⊗ idG) ◦ α = (idX ⊗ ∆) ◦ α, ◮ density: [α(C0(X))(1X ⊗ C(G))] = C0(X) ⊗ C(G). In particular... α : C0(X) → C0(X) ⊗ C(G) (proper action).