Actions of Compact Quantum Groups II Examples, spectral components - - PowerPoint PPT Presentation

Actions of Compact Quantum Groups II

Examples, spectral components and Podle´ s algebra

Kenny De Commer (VUB, Brussels, Belgium)

Examples Coinvariants Isotypical components Algebraic actions

Outline

Examples of compact quantum group actions The C∗-algebra of coinvariants for an action Isotypical components Algebraic actions

Examples Coinvariants Isotypical components Algebraic actions

Recall: 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) → C0(X) ⊗ C(G) s.t. ◮ coaction property, (α ⊗ idG) ◦ α = (idX ⊗ ∆) ◦ α, ◮ density: [α(C0(X))(1X ⊗ C(G))] = C0(X) ⊗ C(G).

Examples Coinvariants Isotypical components Algebraic actions

Actions from representations

Example

Let G CQG, π G-representation. Then G B(Hπ), Adπ : B(Hπ) → B(Hπ) ⊗ C(G), ξη∗ → δπ(ξ)δπ(η)∗ = Uπ(ξη∗ ⊗ 1)U ∗

π.

For G classical Hausdorff group, π representation: (Adπ)g(x) = π(g)xπ(g)∗, x ∈ B(Hπ).

Examples Coinvariants Isotypical components Algebraic actions

Universal C∗-envelopes

Definition

Let O(X) ∗-algebra. Then O(X) admits universal C∗-envelope if

xu = sup{λ(x) | λ non-degenerate ∗-representation O(X) → B(Hλ)} < ∞.

Then C0(Xu) ∼ =

• Im
• O(X) →
• λ

B(Hλ)

• .

Remark:

◮ O(X) → C0(Xu) not necessarily injective... ◮ C0(Xu) could be zero!

Examples

◮ Examples: Cu(G), On, C(SN−1 +

), . . .

◮ Non-example: C[x], x∗ = x.

Examples Coinvariants Isotypical components Algebraic actions

C∗-algebraic actions from algebraic actions

Lemma

Let O(X) ∗-algebra with Hopf ∗-algebraic coaction α : O(X) → O(X) ⊗

alg O(G).

Assume O(X) admits a universal C∗-envelope. Then α extends to coaction αu : C0(Xu) → C0(Xu) ⊗ C(Gu).

Hopf ∗-algebraic coaction: ◮ (α ⊗ id) ◦ α = (id ⊗ ∆) ◦ α, ◮ (idX ⊗ ǫ)α = idX.

Examples Coinvariants Isotypical components Algebraic actions

Proof

◮ Existence αu as ∗-homomorphism: clear by universality. ◮ αu coaction:

◮ Coaction property: clear by continuity. ◮ Density: ◮ Write

α(a) = a(0) ⊗ a(1), (id ⊗ ∆)α(a) = a(0) ⊗ a(1) ⊗ a(2), . . .

◮ Then

α(a(0))(1 ⊗ S(a(1))) = a(0) ⊗ a(1)S(a(2)) = a(0) ⊗ ǫ(a(1))1 = a ⊗ 1.

◮ Hence α(O(X))(1X ⊗ O(G)) = O(X) ⊗ alg O(G). ◮ Hence [αu(C0(Xu))(1 ⊗ C(Gu))] = C0(Xu) ⊗ C(Gu).

Examples Coinvariants Isotypical components Algebraic actions

Actions from representations II

Definition

Let H finite dimensional Hilbert space. Cuntz C∗-algebra O(H),

◮ H ⊆ O(H) linearly, ◮ ξ∗η = ξ, η, ◮ i ξiξ∗ i = 1 for {ξi} o.n. basis.

Example

On = O(Cn).

Example

Let G CQG, π G-representation. Then action G O(Hπ), απ : O(Hπ) → O(Hπ) ⊗ C(G), ξ → δπ(ξ).

Examples Coinvariants Isotypical components Algebraic actions

Liberated and free

Definition (Wang)

Universal C∗-algebra C(O+

N),

C∗(uij | 1 ≤ i, j ≤ N, u∗

ij = uij and U = (uij)i,j unitary)

is CQG by ∆(uij) =

• k

uik ⊗ ukj.

Example

SN−1

+

O+

N by

α(Vi) =

• j

Vj ⊗ uji.

Examples Coinvariants Isotypical components Algebraic actions

Half-classical revisited

Definition (Wang)

Universal C∗-algebra C(Sym+

n ) = C(O+(n))/ < uij − u2 ij >

is CQG by ∆(uij) =

• k

uik ⊗ ukj.

Example

Let Xn = {1, 2, . . . , n}. Then Xn Sym+

n ,

α : C(Xn) → C(Xn) ⊗ C(Sym+

n ),

δi →

• j

δj ⊗ uji.

Examples Coinvariants Isotypical components Algebraic actions

Actions of discrete group duals

Example (C∗-algebraic bundles and Γ-graded C∗-algebras)

◮ Γ discrete group. ◮ Banach spaces Ag with ‘associative’ contractive multiplication

Ag × Ah → Agh.

◮ ∗ : Ag → Ag−1 antilinear, ‘involutive’, isometric. ◮ b∗b = b2 for b ∈ Ag, ◮ b∗b ≥ 0 in (the C∗-algebra) Ae for b ∈ Ag.

Then Γ A = universal C∗-envelope ⊕gAg, α : A → A ⊗ C∗(Γ), ag → ag ⊗ λg.

Examples Coinvariants Isotypical components Algebraic actions

The C∗-algebra of coinvariants

Definition

Let X

α

• G. Space of orbits: Y = X/G, given by C∗-algebra

C0(Y) = {a ∈ C0(X) | α(a) = a ⊗ 1G}.

Examples

◮ G α

C0(X) C0(Y) = C0(X)G = {a ∈ C0(X) | αg(a) = a for all g ∈ G}.

◮ X α

G: C0(X)G = {G-constant continuous functions on X} ∼ = {continuous functions on X/G}.

Examples Coinvariants Isotypical components Algebraic actions

Intertwiners and fixed point subalgebras

Definition (Space of intertwiners)

Let π1 and π2 G-representations. Then Mor(π1, π2) = {T : H1 → H2 | δ2◦T = (T⊗id)◦δ1} ⊆ B(H1, H2).

Lemma

◮ T ∈ Mor(π1, π2), T ′ ∈ Mor(π2, π3) ⇒ T ′ ◦ T ∈ Mor(π1, π3). ◮ T ∈ Mor(π1, π2) ⇒ T ∗ ∈ Mor(π2, π1).

Example

B(Hπ)απ = Mor(π, π), απ(ξη∗) = δπ(ξ)δπ(η)∗ = Uπ(ξη∗ ⊗ 1)U ∗

π.

Examples Coinvariants Isotypical components Algebraic actions

Commuting actions

Definition

Let X

α

G and H

β

• X. Commutation of α and β:

(β ⊗ idG)α = (idG ⊗ α)β.

Example

Assume X

α

G and H

β

X commute. Then H X/G, β|C(X/G) : C(X/G) → C(H) ⊗ C(X/G).

Examples Coinvariants Isotypical components Algebraic actions

Invariant functionals on CQG

Theorem (Woronowicz)

G compact quantum group. ∃! state ϕ on C(G), Haar state, s.t. (id ⊗ ϕ)∆(f) = (ϕ ⊗ id)∆(f) = ϕ(f)1G, ∀f ∈ C(G). For G compact Hausdorff group, µ Haar (probability) measure, ϕ(f) =

• G

f(g)dµ(g).

Lemma

ϕ faithful on O(G): ∀h ∈ O(G), ϕ(h∗h) = 0 ⇒ h = 0.

In fact: if h ∈ O(G) positive in C(G) and ϕ(h) = 0, then h = 0.

Examples Coinvariants Isotypical components Algebraic actions

Conditional expectation

Lemma (Integration over fibers)

X

α

G, Y = X/G. Conditional expectation onto C0(Y) EY : C0(X) → C0(X), a → (id ⊗ ϕ)α(a),

◮ range C0(Y), ◮ idempotent, ◮ completely positive, ◮ bimodular:

EY(bac) = bEY(a)c, a ∈ C0(X), b, c ∈ C0(Y),

◮ non-degenerate: [C0(X)C0(Y)] = C0(X).

Non-degeneracy: ‘Every point of X is in an orbit (point of Y)’.

Examples Coinvariants Isotypical components Algebraic actions

Examples

Examples

◮ G α

C0(X) EY(a) =

• G

αg(a)dµ(g).

◮ X α

G: integration over orbits, EY (f)(xG) =

• G

f(xg)dµ(g).

Example

S1 On: EY(Vi1 . . . ViN V ∗

j1 . . . V ∗ jM )

=

• S1 zN−M(Vi1 . . . ViN V ∗

j1 . . . V ∗ jM )dz

= δM,NVi1 . . . ViN V ∗

j1 . . . V ∗ jN .

Examples Coinvariants Isotypical components Algebraic actions

Proof (of properties EY)

◮ Range ⊆ C0(Y):

α(EY(a)) = α

• (id ⊗ ϕ)α(a)
• =

(id ⊗ id ⊗ ϕ)((α ⊗ id)α(a)) = (id ⊗ id ⊗ ϕ)((id ⊗ ∆)α(a)) = (id ⊗ ϕ)(α(a)) ⊗ 1G = EY(a) ⊗ 1G.

◮ Trivially, EY(b) = b for b ∈ C0(Y). ◮ Trivially, EY completely positive (state + ∗-homs c.p.). ◮ Trivially, EY C0(Y)-bimodular. ◮ Non-degenerate: uα bounded approximate unit C0(X), ∀b ∈ C0(X), EY(uα)b = (id ⊗ ϕ)(α(uα)(b ⊗ 1)) → b,

since b ⊗ 1 ∈ [α(C0(X))(1 ⊗ C(G))].

Examples Coinvariants Isotypical components Algebraic actions

Results on G-representations

Definition

◮ π indecomposable: π ≇ π1 ⊕ π2. ◮ π irreducible: T ∈ Mor(π′, π), T ∗T = id ⇒ TT ∗ = id or 0.

Proposition

Let G compact quantum group.

◮ G-representation indecomposable ⇔ irreducible. ◮ G-representation ∼

= direct sum irreducibles.

Examples Coinvariants Isotypical components Algebraic actions

Isotypical components

Definition

X

α

G, π G-representation. Intertwiner space Mor(π, α): Mor(π, α) = {T : Hπ → C0(X) | α(Tξ) = (T ⊗ id)δπ(ξ)}. π irreducible: π-isotypical component (or π-spectral subspace) C0(X)π = {Tξ | ξ ∈ Hπ, T ∈ Mor(π, α)} ⊆ C0(X). Note: Each C0(X)π is C0(Y)-bimodule.

Examples Coinvariants Isotypical components Algebraic actions

The Podle´ s subalgebra

Theorem (Podle´ s)

Let X

α

• G. Then ∗-algebra (unital if X compact)

OG(X) = linear span {C0(X)π | π irreducible} ⊆ C0(X).

Definition

OG(X) Podle´ s subalgebra of C0(X).

Examples Coinvariants Isotypical components Algebraic actions

Example

Example

O(G) = OG(G) for G

∆

G.

Proof.

⊆ For ω ∈ H∗

π, put

Rω : H → C(G), ξ → (ω ⊗ id)δπ(ξ). Then (Rω ⊗ id)δπ(ξ) = (ω ⊗ id ⊗ id)((δπ ⊗ id)δπ(ξ)) = (ω ⊗ id ⊗ id)((id ⊗ ∆)δπ(ξ)) = ∆(Rω(ξ)), hence Rω ∈ Mor(π, ∆), and δπ : Hπ → Hπ ⊗C(G)π. ⊇ O(G)π unitary G-representation for ∆ and g, h = ϕ(g∗h).

Examples Coinvariants Isotypical components Algebraic actions

Tensor products of unitary left comodules

Notation (Unsummed Heyneman-Sweedler notation)

For h ∈ O(G): ∆(h) = h(1) ⊗ h(2), ∆(2)(h) = h(1) ⊗ h(2) ⊗ h(3), . . . For π and ξ ∈ Hπ: δπ(ξ) = ξ(0) ⊗ ξ(1), (id ⊗ ∆)δπ(ξ) = ξ(0) ⊗ ξ(1) ⊗ ξ(2), . . .

Definition

π1, π2 G-representations: tensor product π1 ⊗ π2, (H1 ⊗ H2, δπ1⊗π2), (δπ1⊗π2)(ξ ⊗ η) = ξ(0) ⊗ η(0) ⊗ ξ(1)η(1).

Examples Coinvariants Isotypical components Algebraic actions

Contragredient representation

Lemma

π G-representation:

◮ ξ∗, η∗ = (Tr ⊗ ϕ)(δ(ξ)δ(η)∗) inner product on H∗ π. ◮ δc π : H∗ π → H∗ π ⊗C(G),

δc

π(ξ∗) = δπ(ξ)∗

unitary G-representation.

Definition

πc =

• (H∗

π, ·, ·), δc π

• contragredient of π.

Examples Coinvariants Isotypical components Algebraic actions

Proof OG(X) is unital ∗-algebra

◮ Unitality: trivial left representation

η : C → C ⊗ C(G), 1 → 1 ⊗ 1G.

◮ By linearity and semisimplicity:

OG(X) = {Tξ | π, ξ ∈ Hπ, T ∈ Mor(π, α)} ⊆ C0(X).

◮ If a = Tξ, b = Sη, then with m multiplication map,

ab = m(Tξ ⊗ Sη), where m ◦ (T ⊗ S) in Mor(π1 ⊗ π2, α) since α hom.

◮ If a = Tξ, then

a∗ = T c(ξ∗), where T c : η∗ → (Tη)∗ is in Mor(πc, α) since α ∗-preserving.

Examples Coinvariants Isotypical components Algebraic actions

An algebraisation of actions

Proposition

Let X

α

• G. Then α restricts to Hopf ∗-algebraic right coaction

αalg : OG(X) → OG(X) ⊗

alg O(G).

This means: (αalg ⊗ id)αalg = (id ⊗ ∆)αalg, (id ⊗ ǫ)αalg = idC(X), with ǫ counit O(G).

Examples Coinvariants Isotypical components Algebraic actions

Proof

◮ For a = Tξ with ξ ∈ Hπ, T ∈ Mor(π, α):

◮ a ∈ C0(X)π and

α(a) = α(Tξ) = (T ⊗ id)δπ(ξ) ∈ C0(X)π ⊗

alg C(G).

◮ Hence

α(a) ∈ C0(X)π ⊗

alg C(G)π ⊆ OG(X) ⊗ alg O(G).

◮ αalg coaction property: immediate. ◮ αalg counital: for a = Tξ with ξ ∈ Hπ, T ∈ Mor(π, α):

(id ⊗ ǫ)α(Tξ) = T((id ⊗ ǫ)δπ(ξ)) = Tξ.

Examples Coinvariants Isotypical components Algebraic actions

Density

Theorem (Podle´ s)

Let X

α

• G. Then OG(X) is dense in C0(X).

Remark: density O(G) in C(G) used in proof.

Examples Coinvariants Isotypical components Algebraic actions

More on unitary representations

Lemma

◮ Any finite-dimensional O(G)-comodule is unitarizable. (Take any ·, · and define ξ, η = (id ⊗ ϕ)(δ(ξ)∗δ(η)).) ◮ C(G)π finite dimensional coalgebra. (In fact C(G)π matrix coefficients δπ.) ◮ C(G)π for non-isomorphic π mutually ⊥ for g, h = ϕ(g∗h). (In fact ξη∗ → (id ⊗ ϕ)(δ(ξ)δ(η)∗) in Mor(π1, π2).) ◮ If h ∈ C(G)π, g ∈ C(G), then

(id ⊗ ϕ)((1 ⊗ h∗)∆(g)) ∈ C(G)π.

(In fact, follows from density O(G) and previous properties.)

Examples Coinvariants Isotypical components Algebraic actions

Proof of density Podle´ s algebra

◮ [α(C0(X))(1 ⊗ C(G))] = C0(X) ⊗ C(G) ⊇ C0(X) ⊗ C. ◮ O(G) dense in C(G). ◮ C0(X) = [(id ⊗ ϕ)((1 ⊗ h∗)α(a)) | a ∈ C0(X), π, h ∈ C(G)π]. ◮ For a ∈ C0(X), put

Va = {(id ⊗ ϕ)((1 ⊗ h∗)α(a)) | h ∈ C(G)π}. Then (Va, α) finite dimensional right O(G)-comodule, α(Va) ⊆ Va ⊗ C(G)π.

Examples Coinvariants Isotypical components Algebraic actions

Faithfulness of EY on OG(X)

Lemma

Let X

α

• G. Then EY faithful on OG(X):

∀a ∈ OG(X), EY(a∗a) = 0 ⇒ a = 0.

Proof.

◮ Assume a ∈ OG(X), EY(a∗a) = 0. ◮ For ω positive on C0(X): 0 = ω(EY(a∗a)) = ϕ((ω ⊗ id)α(a∗a)). ◮ Since (ω ⊗ id)α(a∗a) ∈ O(G) positive in C(G), (ω ⊗ id)α(a∗a) = 0. ◮ Hence α(a∗a) = 0, and a = 0.

Examples Coinvariants Isotypical components Algebraic actions

An orthogonality result

Lemma

◮ EY(a∗b) = 0 if a ∈ C0(X)π, b ∈ C0(X)π′, π ≇ π′. ◮ OG(X) = πirrep ⊕ C0(X)π.

Proof.

Exercise.

Examples Coinvariants Isotypical components Algebraic actions

A nuissance

Question

Let X

α

• G. Is Ker(α) = {0}?

Unfortunately, no.

Example (So ltan)

Γ non-amenable, C∗(Γ) ≇ C∗

red(Γ). Then

∆ : C∗(Γ) → C∗(Γ) ⊗ C∗

red(Γ),

λg → λg ⊗ λg not injective by Fell absorption.