Novembre 2005

σ(X∗, X)X∗X
C(X)
(X, O)C(X)
X C(X)
kfk:= sup
x∈X
|f(x)|, f ∈C(X).
C(X)
kfgk ≤ kfk×kgk,∀f, g ∈C(X).
C(X)χ:C(X)→C
χ
χ(fg) = χ(f)χ(g)χ(f) = χ(f),∀f, g ∈C(X).
χ C(X)
χ(1X) = 1 1X1X
f∈C(X)χ(f) = 0 f
X
χ(f)f
χ1
\
C(X)C(X)\ C(X)∗− σ(C(X)∗, C(X))
\
C(X)
x∈X ϕx:C(X)→C
ϕx(f) := f(x),∀f∈C(X),
C(X)
ϕ:X→\ C(X)x∈X ϕx
X d O
(x, y)∈X2x6=y f ∈C(X)f(x) = 1
f(y) = 0
ϕ
ϕ χ ∈\ C(X)x∈X
fx∈Ker(χ)fx(x)6= 0
fx(x) = 1 fx
X g X
g(y)>0,∀y∈X g ∈Ker(χ).
ϕ
ϕ X \ C(X)

Z
X E =Cb(X)
Rkfk∞= supx∈X|f(x)|ϕ:E→R
ϕ(f)≥0f≥0
ϕ(1X) = 1
1Xx7→ 1XR
ϕ ϕ
f≤g ϕ(f)≤ϕ(g)
ϕ
X
E∗E
E M X M σ(E∗, E)
M⊂E∗
f∈E{ϕ∈E∗|ϕ(f)≥0} {ϕ∈E∗|ϕ(1X) = 1}
M
X=Zα:E→E
(α(f))(x) = f(x−1) Fn={ϕ∈M| kϕ−ϕ◦αk ≤ 1
n}ϕ◦α
Xkϕ−ϕ◦αkE
ϕ7→ ϕ−ϕ◦α E∗E∗E∗
σ(E∗, E)
(Fn)n≥1M
σ(E∗, E)
Fnϕnϕn(f) = 1
2nP2n
j=1 f(j)
ϕ ϕ ∈M ϕ =ϕ◦α
ϕ
ϕ(δn) = ϕ(δ0)n∈Zδn∈E1n
ϕ(δn) = 0 n
ϕ