Actions of Compact Quantum Groups III Reduced and universal actions - - PowerPoint PPT Presentation

Actions of Compact Quantum Groups III

Reduced and universal actions

Kenny De Commer (VUB, Brussels, Belgium)

Universal actions Hilbert modules Reduced actions

Outline

Universal actions Hilbert modules Reduced actions

Universal actions Hilbert modules Reduced actions

Universal completions

Proposition (H. Li)

Let X

α

• G. Then OG(X) admits universal C∗-completion C0(Xu).

Universal actions Hilbert modules Reduced actions

Proof

◮ Take T ∈ Mor(π, α). ◮ Let {ei} o.n. basis of Hπ, δπ(ei) = j ej ⊗ uji. ◮ Then

α

i

T(ei)T(ei)∗ =

• i,j,k

T(ej)T(ek)∗ ⊗ ujiu∗

ki

=

• j

T(ej)T(ej)∗ ⊗ 1.

◮ Hence xT = i T(ei)T(ei)∗ ∈ C0(X/G), so

λ(T(ei)) ≤ xT , λ ∗-representation of OG(X).

◮ OG(X) = span{Tξ | π, T ∈ Mor(π, α), ξ ∈ Hπ}, so

∀a ∈ OG(X), a = sup{λ(a) | λ ∗-representation} < ∞.

Universal actions Hilbert modules Reduced actions

Universal coaction

Theorem (H. Li)

Let X

α

• G. Then αalg extends to injective right coaction

αu : C0(Xu) → C0(Xu) ⊗ C(Gu). Moreover

◮ C0(Yu) = C0(Y), ◮ OGu(Xu) = OG(X).

Remark: We will use the corresponding result for X = G.

Universal actions Hilbert modules Reduced actions

Proof (Part I)

(Cf. universal construction first lecture.)

◮ Existence αu: trivial. ◮ Coaction property: trivial. ◮ Density condition: via Hopf algebra theory (antipode)

α(OG(X))(1 ⊗ O(G)) = OG(X) ⊗

alg O(G). ◮ Injectivity: counit extends to C(Gu).

Universal actions Hilbert modules Reduced actions

Proof (Part II)

◮ Let λu : C0(Xu) → C0(X), λu : C(Gu) → C(G). ◮ Then (λu ⊗ λu) ◦ αu = α ◦ λu. ◮ Hence: λu(C0(Xu))π = C0(X)π. ◮ To show: λu injective on each C0(Xu)π. ◮ If an → b ∈ C0(Yu) with an ∈ OG(X),

bn = EY(an) = (id ⊗ ϕ)α(an) → (id ⊗ ϕ)αu(b) = b. But bn ∈ C0(Y) C∗-algebra, so b ∈ C0(Y).

◮ Assume a ∈ C0(Xu)π, λu(a) = 0. Then

0 = α(λu(a∗a)) = (λu ⊗ λu)αu(a∗a) ∈ C(Xu) ⊗

alg O(G). ◮ Apply (id ⊗ ϕ),

λu

• EYu(a∗a)
• = 0

⇒ EYu(a∗a) = 0.

◮ But EYu faithful on OGu(Xu), so a = 0.

Universal actions Hilbert modules Reduced actions

Right C∗-algebra valued inner products

Definition

Let C0(X) C∗-algebra. Let Γ(E) (unital) right C0(X)-module. Right C0(X)-valued inner product on Γ(E): · , · : Γ(E) × Γ(E) → C0(X), (s, t) → s, t s.t.

◮ · , · linear in second, anti-linear in first argument. ◮ s, ta = s, ta, ◮ s, t∗ = t, s, ◮ s, s ≥ 0, ◮ s, s = 0 ⇒ s = 0.

Universal actions Hilbert modules Reduced actions

Right pre-Hilbert modules

Definition

Right pre-Hilbert C0(X)-module:

◮ C∗-algebra C0(X), ◮ right C0(X)-module Γ(E) ◮ right C0(X)-valued inner product on Γ(E).

Lemma

If Γ(E) right pre-Hilbert C0(X)-module, then norm s = s, s1/2, s ∈ Γ(E).

Definition

Γ(E) right Hilbert C0(X)-module if Γ(E) complete. Γ(E) Right pre-Hilbert ⇒ completion Γ(E) Hilbert.

Universal actions Hilbert modules Reduced actions

Hilbert bundles

Example (Classical bundles)

X compact Hausdorff space, E ։

π X locally trivial Hilbert bundle:

◮ E is locally compact Hausdorff space, ◮ each Ex = π−1(x) is finite dimensional Hilbert space. ◮ the map E ×

X E = {(e, f) | π(e) = π(f)} → C,

(e, f) → e, f is continuous. ◮ E is locally trivial: π−1(U) ∼ = U × Cn, Then Γ(E) = Γ(E) = {continuous sections X → E} is Hilbert C(X)-module by (sf)(x) = s(x)f(x), s, t(x) = s(x), t(x).

Universal actions Hilbert modules Reduced actions

Example

Example (Trivial Hilbert modules)

Let C0(X) C∗-algebra, I set. Γ(E) = l2(I, C0(X)) = {(ai)i∈I |

• i

a∗

i ai norm-convergent}

is Hilbert C0(X)-module by  

a1 a2 . . .

  · a =  

a1a a2a . . .

 , s, t = s∗t.

Universal actions Hilbert modules Reduced actions

Tensor product with Hilbert space

Example (Tensor product with Hilbert space)

Let

◮ C0(X) C∗-algebra, ◮ Γ(E) right Hilbert C0(X)-module, ◮ H Hilbert space.

Then right pre-Hilbert C0(X)-module Γ(E) ⊗

alg H

with inner product s ⊗ ξ, t ⊗ η = ξ, ηs, t. ⇒ Completion Γ(E) ⊗ H. When Γ(E) = C0(X) and H = l2(I), l2(I, C0(X)) ∼ = C0(X) ⊗ l2(I).

Universal actions Hilbert modules Reduced actions

Hilbert modules from conditional expectations

Example

Let C0(X) C∗-algebra. Let EY faithful conditional expectation, EY : C0(X) → C0(Y) ⊆ C0(X). Then C0(X) pre-Hilbert C0(Y)-module by a, bY = EY(a∗b).

Remark: For EY not faithful: first divide out submodule {a ∈ C0(X) | EY(a∗a) = 0}.

Notation

L2

Y(X): completed Hilbert C0(Y)-module of (C0(X), · , · Y)

Universal actions Hilbert modules Reduced actions

Adjointable maps

Definition

Let Γ(E) and Γ(F) Hilbert C0(X)-modules. Linear map T : Γ(E) → Γ(F) adjointable if ∃T ∗ : Γ(F) → Γ(E) s.t. s, Tt = T ∗s, t, ∀s, t. Then L(Γ(E), Γ(F)) = {T : Γ(E) → Γ(F) | T adjointable}.

Universal actions Hilbert modules Reduced actions

Properties of adjointable maps

Lemma

◮ Adjointable maps are bounded (⇐ Banach-Steinhaus). ◮ T adjointable ⇒ T module map, T(ξa) = T(ξ)a. ◮ L(Γ(E), Γ(F)) is a Banach space. ◮ L(Γ(E)) = L(Γ(E), Γ(E)) is C∗-algebra. ◮ U : Γ(E) → Γ(F) surjective linear isometry iff

U ∈ L(Γ(E), Γ(F)) and unitary. Remark: U linear isometry U ∈ L(Γ(E), Γ(F)).

Universal actions Hilbert modules Reduced actions

Left Hilbert modules

Definition

Left pre-Hilbert C0(X)-module:

◮ left C0(X)-module Γ(E), ◮ left C0(X)-valued inner product on Γ(E),

◮ · , · linear in first, anti-linear in second argument. ◮ as, t = as, t, ◮ s, t∗ = t, s, ◮ s, s ≥ 0, ◮ s, s = 0 ⇒ s = 0.

Universal actions Hilbert modules Reduced actions

Examples

Example

For EY conditional expectation, x, yY = EY(xy∗).

Example

Let Γ(E) right Hilbert C(X)-module. Then Γ(E∗) = Γ(E)∗ = L(Γ(E), C(X)) = {L∗

ξ : η → ξ, η | ξ ∈ Γ(E)}

left Hilbert C0(X)-module by (aL)(s) = a(L(s)), L, M = LM∗, where we use L(C(X)) ∼ = C(X) by T → T(1X).

Universal actions Hilbert modules Reduced actions

An equivariance property

Lemma

Let X

α

G with Y = X/G. Then (EY ⊗ id)α(a) = EY(a) ⊗ 1, a ∈ C0(X).

Proof.

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

Universal actions Hilbert modules Reduced actions

The implementing unitary

Lemma

Let X

α

G with Y = X/G. Then OG(X) ⊗

alg O(G) → OG(X) ⊗ alg O(G),

a ⊗ g → α(a)(1 ⊗ g) completes to a unitary map Uα : L2

Y(X) ⊗ L2(G) → L2 Y(X) ⊗ L2(G).

Proof.

◮ Isometric: α(a)(1 ⊗ g), α(b)(1 ⊗ h) = (EY ⊗ ϕ)((1X ⊗ g∗)α(a∗b)(1X ⊗ h)) = ϕ(g∗h)EY(a∗b) = a ⊗ g, b ⊗ h. ◮ Surjective: range dense by algebraic surjectivity.

Universal actions Hilbert modules Reduced actions

The reduced C∗-algebra

Lemma

The non-degenerate ∗-homomorphisms πred : C0(X) → L(L2

Y(X)),

πred : C(G) → B(L2(G)) by left multiplication satisfy (πred ⊗ πred)(α(a)) = Uα(πred(a) ⊗ 1)U ∗

α.

Moreover, πred is injective on OG(X).

Proof.

◮ πred well-defined and non-degenerate: basic (positivity EY). ◮ Uα implements α: check on OG(X). ◮ EY faithful on OG(X), so πred injective on OG(X).

Universal actions Hilbert modules Reduced actions

The reduced coaction

Theorem (H. Li)

Let X

α

G, C0(Xred) = πred(C0(X)), C(Gred) = πred(C(G)). Then αred : C0(Xred) → C0(Xred) ⊗ C(Gred) ⊆ L(L2

Y(X) ⊗ L2(G)),

a → Uα(πred(a) ⊗ 1)U ∗

α

defines injective right coaction Xred

αred

Gred. Moreover, OGred(Xred) = OG(X) and C0(Yred) = C0(Y).

Universal actions Hilbert modules Reduced actions

Proof

◮ Uα implements coaction αred: check on OG(X). ◮ αred injective: obvious. ◮ For λred : C0(X) → C0(Xred),

(λred ⊗ λred) ◦ α = αred ◦ λred.

◮ OGred(Xred) = OG(X): faithfulness EY on OG(X). ◮ Hence C0(Yred) = C0(Y).