Groupes quantiques discrets et algèbres d`opérateurs

K
K
C∗
L2

K
L2

C∗
Ar=C∗
r( ) = Cr( )
K
= Γ C∗
r(Γ)
Γ
2(Γ)
G C∗
r( ) C(G) = ˆ
G
G q q
C∗Au(Q)
Ao(Q)
FU(Q)FO(Q)N≥3
FN

K
C∗
⊗
C∗⊗max
C∗(x⊗y7→ y⊗x)
E⊗EΣE σ E σ = Ad(Σ)
L(E⊗E)E L(E)E
E B(E)E
ΓC[Γ] Γ
x=Pg∈Γxgg
Γ
C[Γ] K
ΓKΓC[Γ]
∆ : C[Γ] →C[Γ]⊗C[Γ]
∆(g) = g⊗g x =gΓ
C[Γ] ∆(x) = x⊗x
C[Γ] C∗
λ:C[Γ] →B(2(Γ)) Γ λ(g)ξ=
(h7→ ξ(g−1h)) C∗C∗
r(Γ) λ(C[Γ])
B(2(Γ)) C∗
C∗
p(Γ) C[Γ]
kxkp= sup{kπ(x)kB(K)|π:C[Γ] →B(K)∗ }.
C[Γ] ∗C∗
p(Γ)
Γ∗C∗
p(Γ)
λ∗λ:C∗
p(Γ) →C∗ r(Γ)

λΓ
πi: Γ →B(Hi)i= 1 2 Γ
π1⊗π2: Γ →B(H1⊗H2)g7→ π1(g)⊗π2(g)
C∗∆
∗∆p:C∗
p(Γ) →C∗ p(Γ)⊗C∗ p(Γ)
λΓ
λ C∗
(λ⊗id) ◦∆p∗∆0
r:C∗ r(Γ) →C∗ r(Γ)⊗C∗ p(Γ)
id⊗λ C∗
∆r:C∗
r(Γ) →C∗ r(Γ)⊗C∗ r(Γ) ∆r(λ(g)) = λ(g)⊗λ(g)
C∗
C∗C∗
A∗∆ : A→A⊗A
(id⊗∆) ◦∆ = (∆⊗id) ◦∆
∆(A)(1⊗A) ∆(A)(A⊗1) A⊗A
C∗∗Φ : A→B
(Φ⊗Φ) ◦∆A= ∆B◦Φ
(A, ∆) C∗
h:A→CA(h⊗id) ◦∆ = (id⊗h)◦∆ =
1·h A h A
λ C∗A
Ar=λ(A) (λ⊗λ)◦∆λ
∗∆r:Ar→Ar⊗ArAr
C∗h A λ
ArC∗A
∆r= ∆
C∗Γ
C∗∆(C[Γ])(1⊗C[Γ]) =
C[Γ]⊗C[Γ] C∗(A, ∆)
σ◦∆=∆ A
π(C∗
p(Γ)) Γ π
Γπ⊗π π
h
h(π(g)) = δg,e eΓ
h 2(Γ) λ π h
C∗
C∗
r(Γ)
C∗C∗
r( ) C∗
A=C∗( )
C∗Ar
C∗
C∗C∗
p( ) C∗