Đặc trưng Chern không giao hoán của C - Đại số đối với mặt cầu Sn

C
∗
−
∗
−
n
K−
∗
−
n
∗
−
n
X H
∗
(X) K
∗
(X)
K− X.
X
ch : K
∗
(X) −→ H
∗
(X).
G H
∗
D R
(G; Q)
Z/(2)− K
∗

(G) K− G.
G
ch : K
∗
(G) ⊗ Q −→ H
∗
D R
(G; Q).
C
∗
−
C
∗
−
C
∗
− S
n
.
C
∗
− K−
C
∗
−
S
n
K− S
n
.

S
n
.
C
∗
− K−
K− C
∗
− S
n
.
C
∗
−
G G
dg. L
1
(G)
L
1
(G) =

f : G −→ C |

G
|f(x)|dx < ∞

,
(f ∗ g)(x) :=


G
f(y)g(y
−1
x)dy,
f
∗
(x) := f(x
−1
)
L
1
−
∥f∥ :=

G
|f(x)|dx.
L
1
(G)
∥f ∗ g∥ ̸= ∥f∥.∥g∥.
L
1
(G).
∥f∥
C
∗
:= sup
π∈

G

∥π(f)∥, (1)

G
G. L
1
(G) (1)
C
C
∗
−
G

G
G.

G G
G
π G ∗− C
∗
(G)

f(π) = π(f) :=

G
π(x)f(x)dx.
1 − 1 π G ∗−
C
∗
(G).
π

n
∈

G n ∈ N. ∀f ∈ C
∗
(G) c = c
f
∥

f(π
n
) − c
f
.Id∥ −→ 0 n −→ ∞,
Id
′
∞

i=1
Mat
n
i
(C) =


f | ∥

f(π
n
) − c

f
.Id∥ −→ 0 khi n −→ ∞

.
′
∞

i=1
Mat
n
i
(C) Mat
n
i
(C)
n
i
.
C
∗
(G)
∼
=
′
∞

i=1
Mat
n
i

(C).
G C
∗
(G) C
∗
− G. I
N
:=
N

i=1
Mat
n
i
(C).
I
N
C
∗
(G) C
∗
−
G I
N
,
C
∗
(G) = lim
−→
I

N
.
C
∗
− A (π
1
, π
2
, F ),
π
1
, π
2
: A −→ £(H
B
) ∗− F ∈ F(H
B
)
F C
∗
− H
B
= l
2
B
C
∗
− B,
π
1

(a)F − Fπ
2
(a) ∈ K(B),
K(B) H
B
).
K− KK
∗
(A, B)).
G A = C
∗
(G) B = C
K
∗
(C
∗
(G))
∼
=
KK
∗
(C
∗
(G), C).
KK
0
(C
∗
(G), C)
∼

=
K
0
(C
∗
(G)) KK
1
(C
∗
(G), C)
∼
=
K
1
(C
∗
(G)),
K
∗
(C
∗
(G)) K− C
∗
(G).
A
HE
∗
(A) K
∗
(A)×HE

∗
(A) −→ C.
HE
∗
(A).
HE
∗
(A) A {I
α
}
α∈Γ
A
τ
α
: I
α
−→ C ad
A
−
A {I
α
}
α∈Γ
A
τ
α
: I
α
−→ C,
τ

α
∥τ
α
∥ = 1,
τ
α
(aa
∗
) ≥ 0 ∀a ∈ I
α
,
τ
α
(aa
∗
) = 0 a = 0 α ∈ Γ,
τ
α
ad
A
− τ
α
(xa) = τ
α
(ax) ∀x ∈ A, a ∈ I
α
.
α ∈ Γ, τ
α
I

α
⟨a, b⟩
τ
α
:= τ
α
(ab
∗
) ∀a, b ∈ I
α
.
I
α
I
α
Γ
α  β  γ ⇐⇒ I
α
⊆ I
β
⊆ I
γ
, ∀α, β, γ ∈ Γ.
{I
β
, j
β
α
} Γ : ∀α, β, γ ∈ Γ, α  β  γ
j

β
α
: I
α
−→ I
β
j
γ
β
j
β
α
= j
γ
α
: I
α
−→ I
γ
j
α
α
= id.
I
α
⊗(n+1)
(n + 1)− I
α
. I
α

I
α
⊗(n+1)

I
α
= I
α
⊕ C, ∀α ∈ Γ I
α
C
n
(

I
α
) =

φ : (

I
α
)
⊗(n+1)
−→ C | φ (n + 1) −

(n + 1)−

I
α

C
n
(

I
α
)
α, β, γ ∈ Γ α  β  γ,
D
β
α
: C
n
(

I
α
) −→ C
n
(

I
β
),
D
β
α
j
β
α

C
n
(

I
α
)
D
γ
β
D
β
α
= D
γ
α
, D
α
α
= id.
{C
n
(

I
α
), D
β
α
}

α∈Γ
Q = lim
−→
C
n
(

I
α
).
Q = lim
−→
C
n
(

I
α
)
α ∈ Γ
b
′
: C
n
(

I
α
) −→ C
n+1

(

I
α
)
(b
′
φ)(a
0
, a
1
, , a
n+1
) =
n

j=0
(−1)
j
φ(a
0
, , a
j
a
j+1
, , a
n+1
),
b : C
n

(

I
α
) −→ C
n+1
(

I
α
)
(bφ)(a
0
, a
1
, , a
n+1
) =
n

j=0
(−1)
j
φ(a
0
, , a
j
a
j+1
, , a

n+1
)
+(−1)
n+1
φ(a
n+1
a
0
, , a
n−1
, a
n+1
),
(a
0
, , a
j
a
j+1
, , a
n+1
) ∈

I
α
⊗(n+1)
;
λ : C
n
(


I
α
) −→ C
n
(

I
α
)
(λφ)(a
0
, a
1
, , a
n
) = (−1)
n
φ(a
n
, a
0
, , a
n−1
),
S : C
n+1
(

I

α
) −→ C
n
(

I
α
),
(Sφ)(a
0
α
, a
1
α
, , a
n
α
) = φ(1, a
0
, , a
n
),

I
α
= I
α
⊕ C I
α
b, b

′
b
2
= (b
′
)
2
= 0. b, b
′
b, b
′
, λ S
b, b
′
: lim
−→
C
n
(

I
α
) −→ lim
−→
C
n+1
(

I
α

),
λ : lim
−→
C
n
(

I
α
) −→ lim
−→
C
n
(

I
α
),
S : lim
−→
C
n+1
(

I
α
) −→ lim
−→
C
n

(

I
α
).
N = 1 + λ + λ + + λ
n
, N λ
k
.
N
N : lim
−→
C
n
(

I
α
) −→ lim
−→
C
n
(

I
α
).
A {I
α

}
α∈Γ
A, τ
α
: A
α
−→ C ad
A
−
C
n
(A) := Hom(lim
−→
C
n
(

I
α
), C)
lim
−→
C
n
(

I
α
)
b, b

′
, λ, S
b
∗
, (b
′
)
∗
, λ
∗
, S
∗
b, b
′
, λ, S. b, b
′
, λ, S b
∗
, (b
′
)
∗
, λ
∗
, S
∗
b
2
= b
′

2
= 0 N(1 − λ) = (1 − λ)N = 0
(b
∗
)
2
= (b
′
∗
)
2
= 0 N
∗
(1 − λ
∗
) = (1 − λ
∗
)N
∗
= 0.
b
∗
, (b
′
)
∗
, λ
∗
, N
∗

A
C(A))
(−b
′
)
∗



b
∗



(−b
′
)
∗



←−−
1−λ
∗
C
1
(A) ←−−
N
∗
C

1
(A) ←−−
1−λ
∗
C
1
(A) ←−−
N
∗
· · ·
C(A)
(−b
′
)
∗



b
∗



(−b
′
)
∗




←−−
1−λ
∗
C
0
(A) ←−−
N
∗
C
0
(A) ←−−
1−λ
∗
C
0
(A) ←−−
N
∗
· · ·
b
∗
, (−b
′
)
∗
∗
C(A) C(A)
T ot(C(A))
even
= T ot(C(A))

odd
:=

n≥0
C
n
(A).
C(A) 2.
∂ :

n≥0
C
n
(A) 

n≥0
C
n
(A),
∂ = d
v
+ d
h
d
v
d
h
A
C(A)
A, HP

∗
(A).
(f
n
)
n≥0
∈ C(A)

n≥0
n!

n
2

!
∥f
n
∥z
n
z ∈ C.
C(A)
C
e
(A). C
e
(A)
C(A).
C
e
(A) C(A)

T ot(C
e
(A))
even
= T ot(C
e
(A))
odd
:=

n≥0
C
e
n
(A),
C
e
n
(A) n−
∂ 2,
∂ :

n≥0
C
e
n
(A) 

n≥0
C

e
n
(A),
∂ = d
v
+ d
h
d
v
, d
h
A
C
e
(A)
A, HE
∗
(A).
HE
∗
(A) :
A, B {A
λ
}
λ∈Γ
,
{B
λ
}
λ∈Γ

A, B φ
t
= (φ
λ
t
)
λ
∈ Γ,
φ
λ
t
: A
λ
−→ B
λ
, t ∈ [0, 1]
δ
t
= (δ
λ
t
)
λ∈Γ
,
δ
λ
t
: A
λ
−→ B

λ
φ
t
.
φ
1∗
= φ
0∗
: HE
∗
(A) −→ HE
∗
(B).
A {A
λ
}
∈Γ
A. i = (i
λ
)
λ∈Γ
, i
λ
i
λ
:

A
λ
−→ Mat

q
(

A
λ
)
a
λ
−→




a
λ
0 . . . 0
0 0 . . . 0
0 0 . . . 0




q ≥ 1, ∀λ ∈ Γ. i
λ
HE
∗
(A).
G H
∗
D R

(G, Q)
G
K
∗
(G)
K− G
G
ch : K
∗
(G) ⊗ Q −→ H
∗
D R
(G; Q)
K− G.
C
∗
−
ch : K
∗
(C
∗
(G)) −→ HP
∗
(C
∗
(G))
K−
HP
∗
HE

∗
KK−
C
∗
− G.
e M
k
(A)
k ∈ N) φ = ∂ψ φ ∈ C
n
(

I
α
) ψ ∈ C
n+1
(

I
α
) n
< e, φ >=

n≥0
(−1)
n
n!
φ(e, e, , e) = 0.
A
ch

C
∗
: K
∗
(A) −→ HE
∗
(A).
K
n
(A) × C
n
(A) −→ C.
C
n
(A), Hom(C
n
(A), C)
C
n
(A).
K
n
(A) × C
n
(A) −→ C
K
n
(A)
C
n

−→ Hom(C
n
(A), C).
A {I
α
}
α∈Γ
A α ∈ Γ
K
n
(A)
C
α
n
−→ Hom(C
n
(

I
α
), C),
{I
α
}
α∈Γ
K
n
(A)
C
n

−→ Hom(lim
→α
(C
n
(

I
α
), C).
HE
∗
(A)
A, C
n
ch
C
∗
: K
∗
(A) −→ HE
∗
(A).
A
A
ch
C
∗
: K
∗
(A) −→ HE

∗
(A).
A = C
∗
(G) C
∗
−
G.
ch
C
∗
: K
∗
(C
∗
(G)) −→ HE
∗
(C
∗
(G))
C
∗
(G).
T W = N
T
/T
G, N
T
T G.
ch

C
∗
: K
∗
(C
∗
(G)) −→ HE
∗
(C
∗
(G))
C
∗
(G) ch
C
∗
ch : K
W
∗
(C(T )) −→ HE
W
∗
(C(T )),
C(T ) T
C, K
W
∗
(C(T )) HE
W
∗

(C(T )) K− HE
∗
C(T ).
C
∗
(G)
∼
=
′
∞

i=1
Mat
n
i
(C)
K− K
∗
(C
∗
(G))
K
∗
(C
∗
(G))
∼
=
K
∗


′
∞

i=1
Mat
n
i
(C)

∼
=
lim
−→
K
∗

N

i=1
Mat
n
i
(C)

∼
=
K
∗



λ
C
λ

∼
=
K
W
∗
(C(T )),
C
λ
= C λ
G.
K
∗
(C
∗
(G))
∼
=
K
W
∗
(C(T )).
HE
∗
(C
∗

(G))
HE
∗
(C
∗
(G))
∼
=
HE
∗

′
∞

i=1
Mat
n
i
(C)

∼
=
lim
−→
HE
∗

N

i=1

Mat
n
i
(C)

∼
=
HE
∗


λ
C
λ

∼
=
HE
W
∗
(C(T )),
C
λ
= C λ
G.
HE
∗
(C
∗
(G))

∼
=
HE
W
∗
(C(T )).
G
W.
K
∗
W
(T )
∼
=
K
∗
W
(BT), BT
T.
K
∗
(C
∗
(G))
η
−−→ K
W
∗
(C(T ))
ch

C
∗



ch



HE
∗
(C
∗
(G))
δ
−−→ HE
W
∗
(C(T ))
η : K
∗
(C
∗
(G)) −→ K
W
∗
(C(T ))
δ : HE
∗
(C

∗
(G)) −→ HE
W
∗
(C(T ))
ch : K
W
∗
(C(T )) −→ HE
W
∗
(C(T ))
C(T ).
ch : K
W
∗
(C(T )) −→ HE
W
∗
(C(T )).
ch
C
∗
= δ
−1
◦ ch ◦ η : K
∗
(C
∗
(G)) −→ HE

∗
(C
∗
(G))
SU(n + 1), SO(2n + 1), SU(2n) Sp(n)
K
∗
(G)
∼
=
K
W
∗
(C(T ))
∼
=
K
∗
W
(T )
∼
=
K
∗
(C
∗
(G))
∼
=
HE

∗
(C
∗
(G))
∼
=
HE
W
∗
(C(T ))
∼
=
H
W
∗
(C(T ))
∼
=
H
∗
W
(T )
∼
=
H
∗
(G)
∼
=
HP

∗
(C
∗
(G)).
G R[G]
G.
Z/(2)− K
∗
(G) G C,
K
∗
(G) = Λ
C
(β(ρ
1
), β(ρ
2
), , β(ρ
n
)),
ρ
i
i = 1, 2, 3, , n) G
β : R[G] −→ K
∗
(G)
G = SU(n + 1) SO(2n + 1)
K
∗
(SU(n + 1)) = Λ

C
(β(ρ
1
), β(ρ
2
), , β(ρ
n
))
K
∗
(SO(2n + 1)) = Λ
C
(β(ρ
1
), β(ρ
2
), , β(ρ
n
), ϵ
2n+1
)
ϵ
2n+1
∈ K
∗
(SO(2n + 1))
p
∗
: K
∗

(SO(2n + 1)) −→ K
∗
(Spin(2n + 1))
p
∗
(ϵ
2n+1
) = 2β(∆
2n+1
), ∆
2n+1
: Spin(2n+ 1) −→ U(2
n
)
G T G,
i : T −→ G δ
∗
: H
∗
(BG, R) −→ H
∗−1
(G, R)
H
∗
(BT; R)
W (G)
H
∗
(BT; R)
W.

Z/(2)−
SU(2n), SO(2n + 1), SU(2n + 1) Sp(n)
H
∗
(SU(2n))
∼
=
Λ
C
(x
3
, x
5
, , x
4n−1
)
H
∗
(Sp(n))
∼
=
Λ
C
(x
3
, x
5
, , x
4n−1
)

H
∗
(SU(2n + 1))
∼
=
Λ
C
(x
3
, x
5
, , x
4n+1
)
H
∗
(SO(2n + 1))
∼
=
Λ
C
(x
3
, x
5
, , x
4n−1
).
ϕ : N × N × N −→ Z
ϕ(n, k, q) =

k

i=1
(−1)
i−1

n
k − i

i
q−1
n ≥ 1.
G = SU(n + 1)
ch : K
∗
(SU(n + 1)) −→ H
∗
(SU(n + 1))
ch(β(ρ
k
)) =
n

i=1
(−1)
n
i!
ϕ(n + 1, k, i + 1)x
2i+1
, ∀k ≥ 1

β : R(SU(n + 1)) −→ K
∗
(SU(n + 1)) ρ
k
SU(n + 1) U(n + 1).
G = SO(2n + 1)
ch : K
∗
(SO(2n + 1)) −→ H
∗
(SO(2n + 1))
ch(β(λ
k
)) =
n

i=1
2.(−1)
i−1
(2i − 1)!
ϕ(2n + 1, k, 2i)x
4i−1
ch(ε
2n+1
) =
n

i=1
(−1)
i−1

2
n−1
.(2i − 1)!
ϕ(2n + 1, k, 2i)x
4i−1
β : R(SO(2n + 1)) −→ K
∗
(SO(2n + 1))
ε
2n+1
∈ K
∗
(SO(2n + 1)).
G C
∗
(G)
C
∗
− G,
K
∗
(G)
∼
=
K
∗
(C
∗
(G))
∼

=
HE
∗
(C
∗
(G))
∼
=
HE
∗
(G).
K
∗
(G)
η
−−→ K
∗
(C
∗
(G))
ch



ch
C
∗




H(G)
δ
−−→ HE
∗
(C
∗
(G))
ch
C
∗
= δ ◦ ch ◦ η
−1
: K
∗
(C
∗
(G)) −→ HE
∗
(C
∗
(G)).
C
∗
−
∗
−
n
C
∗
−

S
n
.
ch
C
∗
: K
∗
(C
∗
(S
n
)) −→ HE
∗
(C
∗
(S
n
)).
K−
∗
−
n
S
n
= O(n + 1)/O(n)
ch : K
∗
(S
n

) ⊗ Q −→ H
∗
D R
(S
n
, Q)
C
∗
(S
n
),
ch
C
∗
: K
∗
(C
∗
(S
n
)) −→ HE
∗
(C
∗
(S
n
)).
C
∗
− G

ch
C
∗
: K
∗
(C
∗
(G)) −→ HE
∗
(C
∗
(G)).
C
∗
(S
n
) C
∗
− S
n
.
O(n) O(n+1) S
n
= O(n+1)/O(n)
C
∗
− S
n
C
∗

−
C
∗
(S
n
)
∼
=
C
∗
(O(n)) ⊗ K(L
2
(S
n
)).
K
∗
HE
∗
C
∗
(S
n
) C
∗
−
O(n) O(n + 1).
T
n
O(n) N

T
n
T
n
O(n). C
∗
(S
n
)
HE
∗
(C
∗
(S
n
))
∼
=
H
W
D R
(T
n
)
H
W
D R
(T
n
)

T
n
.
S
n
= O(n + 1)/O(n)
C
∗
(S
n
) C
∗
−
C
∗
(S
n
) = C
∗

O(n + 1)/O(n)

∼
=
C
∗
(O(n)) ⊗ K

L
2

(O(n + 1)/O(n))

K

L
2
(O(n + 1)/O(n)

C
∗
−
L
2
(O(n + 1)/O(n)).
HE
∗
(C
∗
(S
n
)) = HE
∗

C
∗
(O(n + 1)/O(n))

∼
=
HE

∗

C
∗
(O(n)) ⊗ K(L
2
(O(n + 1)/O(n))

∼
=
HE
∗

C
∗
(O(n))

∼
=
HE
∗
(C(N
T
n
))
HE
∗
(C
∗
(S

n
))
∼
=
HE
∗
(C(N
T
n
)).
N
T
n
T
n
O(n), C(N
T
n
) C−
HP
∗
(C(N
T
n
))
∼
=
H
∗
D R

(N
T
n
).
HP
∗
(C(N
T
n
))
∼
=
HE
∗
(C(N
T
n
)).
HE(C
∗
(S
n
))
∼
=
HE
∗
(C(N
T
n

))
∼
=
HP
∗
(C(N
T
n
))
∼
=
H
∗
D R
(N
T
n
)
∼
=
H
W
D R
(T
n
)
∼
=
H
∗

(SO(n))
HE
∗
(C
∗
(S
n
))
∼
=
H
W
D R
(T
n
)
∼
=
H
∗
(SO(n)).
SO(n),
H
∗
(SO(n))
H
∗
(SO(n))
∼
=

Λ
C
(x
′
3
, x
′
7
, , x
′
2n+3
),
x
′
2i+3
∈ σ
∗
(p
i
) ∈ H
2n+3
(SO(n)),
σ
∗
: H
∗
(BSO(n), R) −→ H
∗
(BSO(n), R), R
p

i
= σ
i
(t
′
1
, t
′
2
, , t
′
n
) ∈ H
∗
(BT
n
; Z)
HE
∗
(C
∗
(S
n
))
∼
=
Λ
C
(x
′

3
, x
′
7
, , x
′
2n+3
).
T
n
O(n) N
T
n
T
n
O(n). K− C
∗
(S
n
)
K
∗
(C
∗
(S
n
))
∼
=
K

∗
(N
T
n
)
∼
=
K
∗
(SO(n + 1))/T or.
C
∗
(S
n
)
∼
=
C
∗
(O(n + 1)/O(n)).
K
∗
(C
∗
(S
n
))
∼
=
K

∗
(C
∗
(O(n + 1)/O(n)))
∼
=
K
∗
(C
∗
(O(n))) ⊗ K(L
2
(O(n + 1)/O(n))))
∼
=
K
∗
(C
∗
(O(n)))
∼
=
K
∗
(C(N
T
n
)
∼
=

K
∗
(N
T
n
)
K
∗
(C
∗
(S
n
))
∼
=
K
∗
(N
T
n
).
K
∗
(N
T
n
)
∼
=
K

∗
(SO(n + 1))/T or.
K
∗
(C
∗
(S
n
))
∼
=
K
∗
(SO(n + 1))/T or.
K
∗
(C
∗
(S
n
)), K
∗
(SO(n + 1))/T or.
G = SU(n + 1)
λ
1
: SU(n + 1) −→ U(n + 1), λ
1
∈ R(SU(n + 1)) λ
1

ω
1
= t
1
. {t
i
\ i = 1, 2, , n + 1}
λ
1
. λ
k
= λ
k
(λ
1
), λ
k
λ− k λ−
K−
R(SU(n + 1)) = Z[λ
1
, λ
2
, , λ
n
].
s
∗
1
: R(SU(n + 1)) −→ R(SU(n + 1)),

s
∗
1
(λ
k
) = λ
n+1−k
k = 1, 2, , n.
λ
′
1
= λ
1
i
1
: SO(n + 1) −→ U(n + 1)
i
1
: U(1) −→ U
λ
′
1
∈ R(SO(n + 1)) λ
′
1
ω
′
1
= t
′

1
, {±t
′
i
, 0 \ i = 1, 2, , n}
λ
′
1
λ
′
k
= λ
k
(λ
′
1
)
R(SO(n + 1)) = Z[λ
′
1
, λ
′
2
, , λ
′
n
].
i
∗
1

: R(SU(n + 1)) −→ R(SO(n + 1))
i
∗
1
(λ
k
) = λ
′
k
= i
∗
(λ
n+1−k
), k = 1, 2, , n.
Spin(n + 1) SO(n + 1),
K
∗
(Spin(n + 1)) = Λ
C
(β(α
1
), β(α
2
), , β(α
n−1
), β(∆
n
)), (1)
∆
n

: Spin(n) −→ U(2
n
)
β : R(SO(n)) −→

K
−1
(SO(n))
ρ : SO(n) −→ U(n),
β(ρ) = [i
n
ρ] ∈ [SO, U] =

K
−1
(SO(n)).
(1), ε
n+1
∈ K
−1
(SO(n)) ξ
n+1
∈ K
0
(SO(n))
K
∗
(SO(n + 1)) = [Λ
C
(β(λ

′
1
), β(λ
′
2
), , β(λ
′
n−1
), ε
n+1
) ⊗ T
n+1
]/(ε
n+1
⊗ ξ
n+1
),
T
2n+1
= Z{1} ⊕ Z(2
n
){ε
n+1
}.
K
∗
(SO(n + 1))/T or = Λ
C

β(λ

′
1
), β(λ
′
2
), , β(λ
′
n−1
), ε
n+1

.
K
∗
(C
∗
(S
n
))
∼
=
K
∗
(SO(n + 1))/T or
= Λ
C

β(λ
′
1

), β(λ
′
2
), , β(λ
′
n−1
), ε
n+1

.
∗
−
n
T
n
O(n), T
n
SO(n) W = N
T
n
/T
n
.
C
∗
(S
n
)
ch : K
∗

(C
∗
(S
n
)) −→ HE
∗
(C
∗
(S
n
))
i) ch
C
∗

β(λ
′
k
)

=
n

i=1

(−1)
i−1
2/(2i − 1)!

ϕ(2n + 1, k, 2i)x

′
2i+3
, k = 1, n − 1.
ii) ch
C
∗
(ε
n+1
) =
n

i=1

(−1)
i−1
2/(2i − 1)!

1
2
n
n

k=1
ϕ(2n + 1, k, 2i

x
′
2i+3
,