Actions of Compact Quantum Groups V Free and homogeneous actions I - - PowerPoint PPT Presentation

Actions of Compact Quantum Groups V

Free and homogeneous actions I

Kenny De Commer (VUB, Brussels, Belgium)

Free actions Homogeneous actions

Outline

Free actions Homogeneous actions

Free actions Homogeneous actions

Free actions

Definition

X

α

G free if ∀x ∈ X, Gx = {g ∈ G | xg = x} = {eG}.

Free actions Homogeneous actions

A C∗-algebraic characterisation

Lemma

X

α

G free iff [(C0(X) ⊗ 1G)α(C0(X))] = C0(X) ⊗ C(G).

Proof.

α free iff Can : X × G → X × X, (x, g) → (x, xg) injective iff Can : C0(X) ⊗ C0(X) → C0(X) ⊗ C(G), F = f ⊗ h → Can(F) = (f ⊗ 1G)α(h), (x, g) → F(x, xg) surjective.

Free actions Homogeneous actions

Freeness for compact quantum group actions

Definition (Ellwood)

Let X

α

• G. Then α free if

[(C0(X) ⊗ 1G)α(C0(X))] = C0(X) ⊗ C(G).

Free actions Homogeneous actions

C∗-correspondences

Definition

C0(X)-C0(Y)-correspondence:

◮ right Hilbert C0(Y)-module Γ(E), ◮ non-degenerate ∗-representation

λ : C0(X) → L(Γ(E)).

Free actions Homogeneous actions

Interior tensor product

Definition (Interior tensor product)

Assume

◮ (Γ(E), λ) is C0(X)-C0(Y) correspondence. ◮ (Γ(F), τ) is C0(Y)-C0(Z) correspondence.

Then C0(X)-C0(Z) correspondence (Γ(E) ⊗

C0(Y) Γ(F), λ

⊗

C0(Y) id):

separation-completion of Γ(E) ⊗

alg Γ(F),

s ⊗ u, t ⊗ v = u, τ(s, t)v.

Free actions Homogeneous actions

The Galois isometry

Lemma

Let X

α

• G. Then ∃ isometry, Galois (or canonical) isometry,

Gα : L2

Y(X) ⊗ C0(Y)L2 Y(X) → L2 Y(X)⊗L2(G),

a⊗b → α(a)(b⊗1G).

Proof.

α(c)(d ⊗ 1), α(a)(b ⊗ 1) = (EY ⊗ ϕ)((d∗ ⊗ 1)α(c∗a)((b ⊗ 1)) = EY(d∗EY(c∗a)b) = c ⊗ d, a ⊗ b.

Free actions Homogeneous actions

What’s Galois got to do with it?

Theorem (Chase-Harrison-Rosenberg)

Let

◮ E ⊆ F finite field extension, ◮ G = AutE(F).

Then

◮ H = Map(G, E) Hopf algebra over E, ◮ Hopf algebraic coaction

α : F → F ⊗

E H,

α(f)(g) = αg(f), and E ⊆ F Galois if and only if the following map is bijective, F ⊗

E F → F ⊗ E H,

a ⊗ b → α(a)(b ⊗ 1).

Free actions Homogeneous actions

Unitarity of the Galois map

Theorem (DC-Yamashita; Baum-DC-Hajac)

Let X

α

• G. The following are equivalent.
• 1. The action is free.
• 2. The Galois map is unitary.
• 3. C0(X ⋊ G)

∼ =

→ K(L2

Y(X)).

Remark: Last condition: ‘saturatedness’ (Rieffel)

Free actions Homogeneous actions

Example: Action by compact quantum subgroup

Example

Let H ⊆ G compact quantum subgroup: π : C(G) ։ C(H), (π ⊗ π) ◦ ∆G = ∆H ◦ π. Then free action α = (id ⊗ π) ◦ ∆ : C(G) → C(G) ⊗ C(H).

Proof.

Exercise.

Free actions Homogeneous actions

Example: free action on smash product

Lemma

Let X

• G. Then (X ⋊

G) G free.

Proof.

[α(C0(X ⋊ G))(C0(X ⋊ G) ⊗ 1G)] ⊇ α(O(X ⋊ G))(O(X ⋊ G) ⊗ 1G) ⊇ (O(X) ⊗ 1G)∆(O(G))(O(G)O(X) ⊗ 1G) = O(X)O(G)O(X) ⊗ O(G) = O(X ⋊ G) ⊗ O(G).

Corollary (Takesaki-Takai duality)

C0(X ⋊ G ⋊ G) ∼ = B0(L2(G)) ⊗ C0(X) (since L2

X(X ⋊

G) = L2(G) ⊗ C0(X)).

Free actions Homogeneous actions

Homogeneous actions

Definition

X

α

G homogeneous (or ergodic) if α(x) = x ⊗ 1G ↔ x ∈ C1X. ⇒ C(X) unital.

Lemma

If X

α

G homogeneous, then α transitive (in the ordinary sense).

Proof.

C(X)G = C(X/G).

Free actions Homogeneous actions

Homogeneity and reduced and universal actions

Lemma

If X

α

G homogeneous, then αu and αred homogeneous.

Proof.

Y = X/G = Xu/Gu = Xred/Gred.

Free actions Homogeneous actions

Associated von Neumann algebra

Definition

Let X

α

G homogeneous. Then invariant state ϕX on C(X), ∀a ∈ C0(X), ϕX(a)1X = (id ⊗ ϕ)α(a).

Lemma

Let X

α

G homogeneous.

◮ ϕX is invariant:

∀a ∈ C(X), (ϕX ⊗ idG)α(a) = ϕX(a)1G.

◮ With L∞(X) = C(Xred)′′ ⊆ B(L2(X, ϕX)), normal

coassociative ∗-homomorphism αvN : L∞(X) → L∞(X)⊗L∞(G).

Free actions Homogeneous actions

Actions of quotient type

Definition (Podle´ s)

Let X

α

• G. One calls α of quotient type if ∃ H ⊆ G and

θ : C(X) ∼ = C(H\G) = {f ∈ C(G) | (π ⊗ id)∆(f) = 1H ⊗ f} such that (θ ⊗ id) ◦ α = ∆ ◦ θ. Remarks:

◮ Quotient type ⇒ Homogeneous. ◮ G, X classical: Homogeneous ⇒ Quotient type. ◮ In general: Homogeneous Quotient type.

Free actions Homogeneous actions

Example: Standard Podle´ s sphere

Definition

For q ∈ [−1, 1] \ {0}: C∗-algebra C(SUq(2)) as C∗(a, b | U = u11 u12 u21 u22

• =

a −qb∗ b a∗

• unitary)

is CQG for ‘matrix comultiplication’ ∆(uij) = ui1 ⊗ u1j + ui2 ⊗ u2j.

Lemma

S1 ⊆ SUq(2) by a −qb∗ b a∗

• →

z ¯ z

• .

Definition

Standard Podle´ s sphere S2

q = S1\SUq(2).

Free actions Homogeneous actions

Concrete representation standard Podle´ s sphere

Lemma

C(S2

q) generated by X = ab, Z = qb∗b and Y = b∗a∗.

Lemma

C(S2

q) universal C∗-algebra generated by X, Y, Z s.t. ◮

◮ X∗ = Y , ◮ Z∗ = Z,

◮

◮ XZ = q2ZX, ◮ Y Z = q−2ZY ,

◮

◮ Y X = q−1Z − q−2Z2, ◮ XY = q

Z − q2 Z2.

Remark: For q = 1, |X|2 + (Z − 1

2)2 = 1 4.

Free actions Homogeneous actions

Embeddable actions

Definition (Podle´ s)

Let X

α

• G. One calls α embeddable if ∃ faithful ∗-homomorphism

θ : C(X) ֒ → C(G) such that (θ ⊗ id) ◦ α = ∆ ◦ θ. Remark:

◮ Embeddable ⇒ Homogeneous. ◮ Quotient type ⇒ Embeddable. ◮ Embeddable Quotient type (e.g. non-standard Podle´ s spheres) ◮ Homogeneous Embeddable.

Free actions Homogeneous actions

Non-embeddable actions

Example

Let π irreducible left G-representation. Adπ : B(Hπ) → B(Hπ) ⊗ C(G), ξη∗ → δπ(ξ)δπ(η)∗ homogeneous, but not embeddable for dim(Hπ) ≥ 2

Free actions Homogeneous actions

Homogeneous actions with classical point

Lemma

X

α

G homogeneous. TFAE:

• 1. αred is embeddable.
• 2. C(Xu) has a character.

Free actions Homogeneous actions

Proof

• 1. ⇒ 2. θ : C(Xred) → C(Gred) equivariant, so

θalg : OG(X) → O(G) ⇒ θu : C(Xu) → C(Gu). Then ǫ ◦ θu character.

• 2. ⇒ 1.

◮ If χ character, then equivariant ∗-homomorphism

θu : C(Xu) → C(Gu), a → (χ ⊗ id) ◦ αu.

◮ Hence equivariant ∗-homomorphism

θalg : OG(X) → O(G).

◮ Then

ϕ(θalg(a)∗θalg(a)) = χ(EY(a∗a)). But, by homogeneity, EY values in C1X, so EY(a∗a) = χ(EY(a∗a))1X.

◮ Hence θr : C(Xred) ֒

→ C(Gred) since EY faithful on C(Xred).

Free actions Homogeneous actions

Boca’s theorems

Theorem (Boca)

If X

α G homogeneous, then all C(X)π finite dimensional.

In fact, Boca gives concrete estimate in terms of ‘quantum multiplicity’. Combined with Takesaki-Takai duality:

Theorem (Boca)

X

α

G homogeneous ⇒ ∃ set I and Hilbert spaces Hi, C0(X ⋊ G) ∼ = ⊕i∈IB0(Hi)