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:
I
Actions and coactions of compact quantum groups.
I
Actions on C∗ -algebras and Hilbert modules.
I
Crossed products.
I
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:
I unital C∗ -algebra C(G),
I unital ∗ -homomorphism, comultiplication
∆ : C(G) → C(G) ⊗ C(G)
s.t.
I coassociativity:
(∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆,
I 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 7→ (∆(f ) : (g, h) 7→ 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:
I
finite dimensional Hilbert space H,
I
U ∈ B(H) ⊗ C(G)
s.t.
I
U unitary,
I
(id ⊗ ∆)(U ) = U12 U13 , where U12 = U ⊗ 1 etc.
U ∈ B(H) ⊗ C(G)
l
δ : H → H ⊗C(G),
s.t. . . . ?
ξ 7→ U (ξ ⊗ 1G )
CQG
Compact actions
G-representations
Definition
G compact quantum group.
(Continuous finite dimensional unitary left) G-representation π:
I finite dimensional Hilbert space Hπ ,
I linear map
δπ : Hπ → Hπ ⊗C(G)
s.t.
I right comodule: (id ⊗ ∆) ◦ δπ = (δπ ⊗ id) ◦ δπ ,
I isometric: δπ (ξ)∗ δπ (η) = hξ, ηi1C(G) ,
I density: [δπ (H)(1 ⊗ C(G))] = H ⊗ C(G).
I Here Hπ ∼
= B(C, Hπ ), so
(ξ ⊗ a)∗ (η ⊗ b) = ξ ∗ η ⊗ a∗ b ∼
= hξ, ηia∗ b.
I Density condition automatically satisfied.
I C(G)-corepresentations ↔ G-representations.
Non-compact actions
CQG
Compact actions
Classical representations
Example
Let G compact Hausdorff group. Then
G-representations as compact quantum group
l
G-representations as compact group
by
δπ : Hπ → Hπ ⊗C(G) ∼
= C(G, Hπ )
l
π : G × Hπ → Hπ ,
(g, ξ) 7→ π(g)ξ = δπ (ξ)(g).
Non-compact actions
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
I
(O(G), ∆) Hopf ∗ -algebra, (O(G), ∆, , S),
I
O(G) dense in C(G),
I
(O(G), ∆) unique dense Hopf ∗ -algebra,
δπ : H → H ⊗O(G) is O(G)-comodule:
I
I
I
(id ⊗ ∆) ◦ δπ = (δπ ⊗ id) ◦ δπ ,
(idH ⊗ )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) ⊗ (h(2) )1
= h ⊗ 1.
Hence
(Linear span) ∆(O(G))(1 ⊗ O(G)) = O(G) ⊗ O(G).
alg
CQG
Compact actions
Universal C∗ -algebra
Lemma
G CQG.
I
Universal C∗ -envelope C(Gu ) of O(G) exists.
I
CQG Gu by
∆u : C(Gu ) → C(Gu ) ⊗ C(Gu ).
Definition
Gu universal CQG (associated to G).
Non-compact actions
CQG
Compact actions
Non-compact actions
Right actions of compact quantum groups on C∗ -algebras
Definition (Podleś)
Right action X x G:
I
Compact quantum group G,
I
C∗ -algebra C(X) (with X ‘compact quantum space’),
I
Unital ∗ -homomorphism, right coaction
α : C(X) → C(X) ⊗ C(G)
s.t.
I
coaction property:
(α ⊗ idG ) ◦ α = (idX ⊗ ∆) ◦ α,
I
density (Podleś 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 x G by
∆ : C(G) → C(G) ⊗ C(G).
CQG
Compact actions
Half-classical case
Lemma (All C(G) commutative)
I
G compact Hausdorff group,
I
C∗ -algebra C(X),
I
G y C(X) continuous action:
α
I
I
I
I
(g, a) 7→ αg (a) continuous,
each αg ∗ -automorphism,
αgh = αg ◦ αh ,
αe = idX , for e ∈ G identity element.
⇒ X x G,
α : C(X) → C(X) ⊗ C(G) ∼
= C(G, C(X)),
a 7→ (α(a) : g 7→ αg (a)) .
Non-compact actions
CQG
Compact actions
Non-compact actions
Proof, Part I
I
Forgetting group structure:
I
Using partitions of unity on G:
I
I
I
∼
=
C(X) ⊗ C(G) → C(G, C(X)) by a ⊗ f 7→ (g 7→ f (g)a).
C(X) ⊗ C(G) ⊗ C(G) ∼
= C(G × G, C(X)), etc.
α
continuous G y C(X) by unital ∗ -endomorphisms
⇔ α : C(X) → C(G, C(X)) unital ∗ -homomorphism.
I ((id ⊗ ∆)α)(a)(g, h) = ((α ⊗ id)α)(a)(g, h) ⇔ αgh (a) = αg (αh (a)).
Conclusion: one-to-one correspondence between
I
α with coaction property, and
I
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
I ∗ -homomorphism
α
e : C(X) ⊗ C(G) → C(X) ⊗ C(G),
I
I
Density ⇔ α
e surjective.
On level of C(G, C(X)) ∼
= C(X) ⊗ C(G):
∀F ∈ C(G, C(X)),
I
a ⊗ f 7→ α(a)(1 ⊗ f ).
α
e(F )(g) = αg (F (g)).
e
Assume αe = idC(X) . Then α
e has inverse β,
e )(g) = αg−1 (F (g)).
β(F
I
I
I
I
Hence range α
e dense.
If αe 6= idC(X) ⇒ αe non-trivial idempotent ∗ -endomorphism.
Put C(Xe ) = αe (C(X)) 6= C(X).
∀g ∈ G: αg (C(X)) = αe (αg (C(X))) ⊆ C(Xe ).
⇒ If a ∈
/ C(Xe ), then g 7→ a not in range α
e.
CQG
Compact actions
Non-compact actions
Classical
Example (All C(G) and C(X) commutative)
G compact Hausdorff group, X compact Hausdorff space,
X x G continuous
⇒
G y C(X),
αg (f )(x) = f (x · g).
Example
Consider sphere
S N −1 = {z = (z1 , . . . , zN ) ∈ RN |
X
i
Then S N −1 x O(N ) by
(z, g) 7→ zg.
zi2 = 1}.
CQG
Compact actions
Non-compact actions
Example: Half-classical I
Example
Cuntz algebras,
On = C ∗ (V1 , . . . , Vn | Vi∗ Vj = δij ,
X
i
Then U (n) y On by
αu (Vi ) =
X
uji Vj .
j
In particular, S 1 y On by
αz (Vi ) = zVi .
Vi Vi∗ = 1).
CQG
Compact actions
Non-compact actions
Example: Half-classical II
Example (Banica)
Free spheres,
N −1
C(S+
) =< V1 , . . . , VN | Vi = Vi∗ ,
X
i
N −1
Then O(N ) y C(S+
) by
αg (Vi ) =
X
j
gji Vj .
Vi2 = 1}.
CQG
Compact actions
Non-compact actions
Left actions of compact quantum groups on C∗ -algebras
Definition (Podleś)
Left action G y H:
I
Compact quantum group G,
I
C∗ -algebra C(X),
I
Unital ∗ -homomorphism, left coaction
α : C(X) → C(G) ⊗ C(X)
s.t.
I
coaction property:
(idG ⊗ α) ◦ α = (∆ ⊗ idX ) ◦ α,
I
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 7→ h ⊗ g.
Lemma
α
GyX
↔
αop
X x 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):
I Concrete: For C0 (X) ⊆ B(H) with [C0 (X) H] = H:
Cb (X) = {T ∈ B(H) | ∀a ∈ C0 (X), T a, aT ∈ C0 (X)}.
I Abstract: Cb (X) collection maps T : C0 (X) → C0 (X) s.t.
∃T ∗ , ∀a, b ∈ C0 (X),
a∗ (T b) = (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.
I Non-degenerate maps C0 (Y ) → Cb (X) ⇔ continuous maps X → Y .
I Non-degenerate maps C0 (Y ) → C0 (X) ⇔ continuous proper maps X → Y .
I 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 x G:
I Compact quantum group G,
I C∗ -algebra C0 (X),
I non-degenerate ∗ -homomorphism, right coaction
α : C0 (X) → Cb (X × G)
s.t.
I coaction property:
I density:
(α ⊗ idG ) ◦ α = (idX ⊗ ∆) ◦ α,
[α(C0 (X))(1X ⊗ C(G))] = C0 (X) ⊗ C(G).
In particular...
α : C0 (X) → C0 (X) ⊗ C(G)
(proper action).