對於巴拿赫代數
A
{\displaystyle {\mathcal {A}}}
a
{\displaystyle a}
譜
σ
(
a
)
{\displaystyle \sigma (a)}
多項式函數演算 推廣到譜上的連續函數，似乎有一個明顯的思路：依照魏爾施特拉斯逼近定理 用多項式來逼近連續函數，然後將多項式中的數換成
A
{\displaystyle {\mathcal {A}}}
a
{\displaystyle a}
a
{\displaystyle a}
序列 收斂為
A
{\displaystyle {\mathcal {A}}}
譜集
σ
(
a
)
⊂
C
{\displaystyle \sigma (a)\subset \mathbb {C} }
z
{\displaystyle z}
z
¯
{\displaystyle {\overline {z}}}
p
(
z
,
z
¯
)
=
∑
k
,
l
=
0
N
c
k
,
l
z
k
z
¯
l
(
c
k
,
l
∈
C
)
{\displaystyle p(z,{\overline {z}})=\sum _{k,l=0}^{N}c_{k,l}z^{k}{\overline {z}}^{l}\;\left(c_{k,l}\in \mathbb {C} \right)}
z
¯
{\displaystyle {\overline {z}}}
z
{\displaystyle z}
復共軛 ，而復共軛是複數 上的一個對合 。在將
z
{\displaystyle z}
a
{\displaystyle a}
z
¯
{\displaystyle {\overline {z}}}
A
{\displaystyle {\mathcal {A}}}
*-代數 ，即配備了一個對合運算
∗
{\displaystyle *}
z
¯
{\displaystyle {\overline {z}}}
a
∗
{\displaystyle a^{*}}
多項式環
C
[
z
,
z
¯
]
{\displaystyle \mathbb {C} [z,{\overline {z}}]}
交換環 ，為得到一個
C
[
z
,
z
¯
]
→
A
{\displaystyle {\mathbb {C} }[z,{\overline {z}}]\rightarrow {\mathcal {A}}}
代數同態 ，須限制在
A
{\displaystyle {\mathcal {A}}}
a
∗
a
=
a
a
∗
{\displaystyle a^{*}a=aa^{*}}
須保證：若多項式序列
(
p
n
(
z
,
z
¯
)
)
n
{\displaystyle (p_{n}(z,{\overline {z}}))_{n}}
σ
(
a
)
{\displaystyle \sigma (a)}
一致收斂 於一連續函數
f
{\displaystyle f}
A
{\displaystyle {\mathcal {A}}}
(
p
n
(
a
,
a
∗
)
)
n
{\displaystyle (p_{n}(a,a^{*}))_{n}}
f
(
a
)
∈
A
{\displaystyle f(a)\in {\mathcal {A}}}
連續函數演算
Φ
a
{\displaystyle \Phi _{a}}
a
{\displaystyle a}
e
{\displaystyle e}
C
∗
(
a
,
e
)
{\displaystyle C^{*}(a,e)}
等距同構 ，即：
∀
f
∈
C
(
σ
(
a
)
)
,
‖
Φ
a
(
f
)
‖
=
‖
f
‖
σ
(
a
)
.
{\displaystyle \forall f\in {\mathcal {C}}(\sigma (a)),\quad \left\|\Phi _{a}(f)\right\|=\left\|f\right\|_{\sigma (a)}.}
Φ
a
{\displaystyle \Phi _{a}}
Φ
a
(
C
(
σ
(
a
)
)
)
=
C
∗
(
a
,
e
)
⊆
A
,
{\displaystyle \Phi _{a}\left({\mathcal {C}}(\sigma (a))\right)=C^{*}(a,e)\subseteq {\mathcal {A}},}
C
∗
(
a
,
e
)
{\displaystyle C^{*}(a,e)}
由於
a
{\displaystyle a}
A
{\displaystyle {\mathcal {A}}}
a
{\displaystyle a}
e
{\displaystyle e}
交換代數 。特別地，
f
(
a
)
{\displaystyle f(a)}
對易 。
全純函數演算 可無歧義地擴張 為連續函數演算。因此，連續函數演算在多項式
p
(
z
,
z
¯
)
{\displaystyle p(z,{\overline {z}})}
∀
c
k
,
l
∈
C
,
Φ
a
(
p
(
z
,
z
¯
)
)
=
p
(
a
,
a
∗
)
=
∑
k
,
l
=
0
N
c
k
,
l
a
k
(
a
∗
)
l
,
{\displaystyle \forall c_{k,l}\in \mathbb {C} ,\quad \Phi _{a}(p(z,{\overline {z}}))=p(a,a^{*})=\sum _{k,l=0}^{N}c_{k,l}a^{k}(a^{*})^{l},}
p
(
z
,
z
¯
)
=
∑
k
,
l
=
0
N
c
k
,
l
z
k
z
¯
l
{\displaystyle p(z,{\overline {z}})=\sum _{k,l=0}^{N}c_{k,l}z^{k}{\overline {z}}^{l}}
對於
σ
(
a
)
{\displaystyle \sigma (a)}
f
∈
C
(
σ
(
a
)
)
{\displaystyle f\in {\mathcal {C}}(\sigma (a))}
f
n
∈
C
(
σ
(
a
)
)
{\displaystyle f_{n}\in {\mathcal {C}}(\sigma (a))}
f
n
(
a
)
{\displaystyle f_{n}(a)}
f
(
a
)
{\displaystyle f(a)}
σ
(
a
)
{\displaystyle \sigma (a)}
絕對 且一致 地收斂的冪級數
f
(
z
)
=
∑
n
=
0
∞
c
n
z
n
{\textstyle f(z)=\sum _{n=0}^{\infty }c_{n}z^{n}}
f
(
a
)
=
∑
n
=
0
∞
c
n
a
n
{\textstyle f(a)=\sum _{n=0}^{\infty }c_{n}a^{n}}
若有
f
∈
C
(
σ
(
a
)
)
{\displaystyle f\in {\mathcal {C}}(\sigma (a))}
g
∈
C
(
σ
(
f
(
a
)
)
)
{\displaystyle g\in {\mathcal {C}}(\sigma (f(a)))}
複合 滿足
(
g
∘
f
)
(
a
)
=
g
(
f
(
a
)
)
{\displaystyle (g\circ f)(a)=g(f(a))}
設有兩個正規元
a
,
b
∈
A
N
{\displaystyle a,b\in {\mathcal {A}}_{N}}
f
(
a
)
=
f
(
b
)
{\displaystyle f(a)=f(b)}
σ
(
a
)
{\displaystyle \sigma (a)}
σ
(
b
)
{\displaystyle \sigma (b)}
g
{\displaystyle g}
f
{\displaystyle f}
反函數 ，那麼必然有
a
=
b
{\displaystyle a=b}
a
=
(
f
∘
g
)
(
a
)
=
f
(
g
(
a
)
)
=
f
(
g
(
b
)
)
=
(
f
∘
g
)
(
b
)
=
b
{\displaystyle a=(f\circ g)(a)=f(g(a))=f(g(b))=(f\circ g)(b)=b}
譜映射定理
∀
f
∈
C
(
σ
(
a
)
)
,
σ
(
f
(
a
)
)
=
f
(
σ
(
a
)
)
{\displaystyle \forall f\in {\mathcal {C}}(\sigma (a)),\quad \sigma (f(a))=f(\sigma (a))}
對於
b
∈
A
{\displaystyle b\in {\mathcal {A}}}
a
b
=
b
a
{\displaystyle ab=ba}
∀
f
∈
C
(
σ
(
a
)
)
,
f
(
a
)
b
=
b
f
(
a
)
.
{\displaystyle \forall f\in {\mathcal {C}}(\sigma (a)),\quad f(a)b=bf(a).}
b
{\displaystyle b}
a
{\displaystyle a}
a
{\displaystyle a}
f
(
a
)
{\displaystyle f(a)}
設
Ψ
:
A
→
B
{\displaystyle \Psi \colon {\mathcal {A}}\rightarrow {\mathcal {B}}}
A
{\displaystyle {\mathcal {A}}}
B
{\displaystyle {\mathcal {B}}}
保單位元 的*-同態，那麼
Ψ
{\displaystyle \Psi }
∀
f
∈
C
(
σ
(
a
)
)
,
Ψ
(
f
(
a
)
)
=
f
(
Ψ
(
a
)
)
.
{\displaystyle \forall f\in C(\sigma (a)),\quad \Psi (f(a))=f(\Psi (a)).}
利用譜映射定理，具有某些性質的函數可以直接關聯到C*-代數成員的某些性質：
f
(
a
)
{\displaystyle f(a)}
可逆元 若且唯若
f
{\displaystyle f}
σ
(
a
)
{\displaystyle \sigma (a)}
零點 。於是有
f
(
a
)
−
1
=
1
f
(
a
)
{\textstyle f(a)^{-1}={\tfrac {1}{f}}(a)}
f
(
a
)
{\displaystyle f(a)}
自伴元 若且唯若
f
{\displaystyle f}
σ
(
a
)
⊆
{
0
,
1
}
{\displaystyle \sigma (a)\subseteq \{0,1\}}
f
(
σ
(
a
)
)
⊆
R
{\displaystyle f(\sigma (a))\subseteq \mathbb {R} }
f
(
a
)
{\displaystyle f(a)}
正元 （
f
(
a
)
≥
0
{\displaystyle f(a)\geq 0}
f
≥
0
{\displaystyle f\geq 0}
f
(
σ
(
a
)
)
⊆
[
0
,
∞
)
{\displaystyle f(\sigma (a))\subseteq [0,\infty )}
f
(
a
)
{\displaystyle f(a)}
么正元 ，若
f
{\displaystyle f}
復單位圓 中。也就是說，
f
(
σ
(
a
)
)
⊆
T
=
{
λ
∈
C
∣
‖
λ
‖
=
1
}
.
{\displaystyle f(\sigma (a))\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid \left\|\lambda \right\|=1\}.}
f
(
a
)
{\displaystyle f(a)}
投影 ，若
f
{\displaystyle f}
0
{\displaystyle 0}
1
{\displaystyle 1}
f
(
σ
(
a
)
)
⊆
{
0
,
1
}
{\displaystyle f(\sigma (a))\subseteq \{0,1\}}
這些斷言的基礎是關於特定元素的譜的結論，這些結論會在§ 應用 一節中展示。
在
A
{\displaystyle {\mathcal {A}}}
H
{\displaystyle H}
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
T
∈
B
(
H
)
{\displaystyle T\in {\mathcal {B}}(H)}
特徵值
λ
∈
σ
(
T
)
{\displaystyle \lambda \in \sigma (T)}
特徵向量
v
∈
H
{\displaystyle v\in H}
f
(
T
)
{\displaystyle f(T)}
f
(
λ
)
∈
σ
(
f
(
T
)
)
{\displaystyle f(\lambda )\in \sigma (f(T))}
T
v
=
λ
v
{\displaystyle Tv=\lambda v}
∀
f
∈
σ
(
T
)
,
f
(
T
)
v
=
f
(
λ
)
v
{\displaystyle \forall f\in \sigma (T),\quad f(T)v=f(\lambda )v}
下面給出連續函數演算的眾多應用中一些典型且非常簡單的例子。
設
A
{\displaystyle {\mathcal {A}}}
a
∈
A
N
{\displaystyle a\in {\mathcal {A}}_{N}}
σ
(
a
)
{\displaystyle \sigma (a)}
a
{\displaystyle a}
σ
(
a
)
⊆
R
.
{\displaystyle \sigma (a)\subseteq \mathbb {R} .}
a
{\displaystyle a}
σ
(
a
)
⊆
T
=
{
λ
∈
C
∣
‖
λ
‖
=
1
}
.
{\displaystyle \sigma (a)\subseteq \mathbb {T} =\{\lambda \in \mathbb {C} \mid \left\|\lambda \right\|=1\}.}
a
{\displaystyle a}
σ
(
a
)
⊆
{
0
,
1
}
{\displaystyle \sigma (a)\subseteq \{0,1\}}
證明
正規元
a
∈
A
{\displaystyle a\in {\mathcal {A}}}
Φ
a
{\displaystyle \Phi _{a}}
Id
∈
C
(
σ
(
a
)
)
{\displaystyle \operatorname {Id} \in {\mathcal {C}}(\sigma (a))}
a
{\displaystyle a}
Id
{\displaystyle \operatorname {Id} }
∀
z
∈
σ
(
a
)
,
z
=
Id
(
z
)
=
Id
¯
(
z
)
=
z
¯
,
{\displaystyle \forall z\in \sigma (a),\quad z={\text{Id}}(z)={\overline {\text{Id}}}(z)={\overline {z}},}
σ
(
a
)
{\displaystyle \sigma (a)}
Id
{\displaystyle {\text{Id}}}
∀
z
∈
σ
(
a
)
,
1
=
Id
(
z
)
Id
¯
(
z
)
=
z
z
¯
=
|
z
|
2
,
{\displaystyle \forall z\in \sigma (a),\quad 1={\text{Id}}(z){\overline {\operatorname {Id} }}(z)=z{\overline {z}}=|z|^{2},}
σ
(
a
)
⊆
{
λ
∈
C
|
‖
λ
‖
=
1
}
{\displaystyle \sigma (a)\subseteq \{\lambda \in \mathbb {C} \ |\ \left\|\lambda \right\|=1\}}
Id
{\displaystyle {\text{Id}}}
(
Id
(
z
)
)
2
=
Id
(
z
)
=
Id
(
z
)
,
¯
{\displaystyle (\operatorname {Id} (z))^{2}=\operatorname {Id} }(z)={\overline {\operatorname {Id} (z),}}
∀
z
∈
σ
(
a
)
,
z
2
=
z
=
z
¯
,
{\displaystyle \forall z\in \sigma (a),\quad z^{2}=z={\overline {z}},}
σ
(
a
)
⊆
{
0
,
1
}
{\displaystyle \sigma (a)\subseteq \{0,1\}}
設
a
{\displaystyle a}
A
{\displaystyle {\mathcal {A}}}
n
∈
N
{\displaystyle n\in \mathbb {N} }
b
∈
A
+
{\displaystyle b\in {\mathcal {A}}_{+}}
b
n
=
a
{\displaystyle b^{n}=a}
n
{\displaystyle n}
若
a
∈
A
s
a
{\displaystyle a\in {\mathcal {A}}_{sa}}
n
∈
N
{\displaystyle n\in \mathbb {N} }
b
∈
A
s
a
{\displaystyle b\in {\mathcal {A}}_{sa}}
b
n
=
a
{\displaystyle b^{n}=a}
類似地，對於C*-代數
A
{\displaystyle {\mathcal {A}}}
a
{\displaystyle a}
α
≥
0
{\displaystyle \alpha \geq 0}
a
α
{\displaystyle a^{\alpha }}
C
∗
(
a
)
{\displaystyle C^{*}(a)}
∀
α
,
β
≥
0
,
a
α
a
β
=
a
α
+
β
.
{\displaystyle \forall \alpha ,\beta \geq 0,\quad a^{\alpha }a^{\beta }=a^{\alpha +\beta }.}
a
{\displaystyle a}
α
{\displaystyle \alpha }
若
a
∈
A
{\displaystyle a\in {\mathcal {A}}}
a
∗
a
{\displaystyle a^{*}a}
|
a
|
=
a
∗
a
{\displaystyle |a|={\sqrt {a^{*}a}}}
設
a
{\displaystyle a}
A
{\displaystyle {\mathcal {A}}}
a
+
,
a
−
∈
A
+
{\displaystyle a_{+},a_{-}\in {\mathcal {A}}_{+}}
a
=
a
+
−
a
−
{\displaystyle a=a_{+}-a_{-}}
a
+
a
−
=
a
−
a
+
=
0
{\displaystyle a_{+}a_{-}=a_{-}a_{+}=0}
a
+
{\displaystyle a_{+}}
a
−
{\displaystyle a_{-}}
正部和負部 。此外還有
|
a
|
=
a
+
+
a
−
{\displaystyle |a|=a_{+}+a_{-}}
證明
函數
f
+
(
z
)
=
max
(
z
,
0
)
{\displaystyle f_{+}(z)=\max(z,0)}
f
−
(
z
)
=
−
min
(
z
,
0
)
{\displaystyle f_{-}(z)=-\min(z,0)}
σ
(
a
)
⊆
R
{\displaystyle \sigma (a)\subseteq \mathbb {R} }
Id
(
z
)
=
z
=
f
+
(
z
)
−
f
−
(
z
)
,
{\displaystyle \operatorname {Id} (z)=z=f_{+}(z)-f_{-}(z),}
f
+
(
z
)
f
−
(
z
)
=
f
−
(
z
)
f
+
(
z
)
=
0.
{\displaystyle f_{+}(z)f_{-}(z)=f_{-}(z)f_{+}(z)=0.}
設
a
+
=
f
+
(
a
)
,
a
−
=
f
−
(
a
)
{\displaystyle a_{+}=f_{+}(a),a_{-}=f_{-}(a)}
a
+
{\displaystyle a_{+}}
a
−
{\displaystyle a_{-}}
a
=
Id
(
a
)
=
(
f
+
−
f
−
)
(
a
)
=
f
+
(
a
)
−
f
−
(
a
)
=
a
+
−
a
−
,
{\displaystyle a=\operatorname {Id} (a)=(f_{+}-f_{-})(a)=f_{+}(a)-f_{-}(a)=a_{+}-a_{-},}
a
+
a
−
=
f
+
(
a
)
f
−
(
a
)
=
(
f
+
f
−
)
(
a
)
=
0
=
(
f
−
f
+
)
(
a
)
=
f
−
(
a
)
f
+
(
a
)
=
a
−
a
+
.
{\displaystyle a_{+}a_{-}=f_{+}(a)f_{-}(a)=(f_{+}f_{-})(a)=0=(f_{-}f_{+})(a)=f_{-}(a)f_{+}(a)=a_{-}a_{+}.}
此外，
f
+
(
z
)
+
f
−
(
z
)
=
|
z
|
=
z
∗
z
=
z
2
,
{\displaystyle f_{+}(z)+f_{-}(z)=|z|={\sqrt {z^{*}z}}={\sqrt {z^{2}}},}
a
+
+
a
−
=
f
+
(
a
)
+
f
−
(
a
)
=
|
a
|
=
a
∗
a
=
a
2
.
{\displaystyle a_{+}+a_{-}=f_{+}(a)+f_{-}(a)=|a|={\sqrt {a^{*}a}}={\sqrt {a^{2}}}.}
若
a
{\displaystyle a}
e
{\displaystyle e}
A
{\displaystyle {\mathcal {A}}}
u
=
e
i
a
{\displaystyle u=\mathrm {e} ^{\mathrm {i} a}}
i
{\displaystyle \mathrm {i} }
虛數單位 。反過來，若
u
∈
A
U
{\displaystyle u\in {\mathcal {A}}_{U}}
真子集 （即
σ
(
u
)
⊊
T
{\displaystyle \sigma (u)\subsetneq \mathbb {T} }
a
∈
A
s
a
{\displaystyle a\in {\mathcal {A}}_{sa}}
u
=
e
i
a
{\displaystyle u=\mathrm {e} ^{\mathrm {i} a}}
設
A
{\displaystyle {\mathcal {A}}}
a
∈
A
N
{\displaystyle a\in {\mathcal {A}}_{N}}
n
{\displaystyle n}
不相交的 閉 子集
σ
k
⊂
C
,
(
1
≤
k
≤
n
)
{\displaystyle \sigma _{k}\subset \mathbb {C} ,\ (1\leq k\leq n)}
σ
(
a
)
=
σ
1
⊔
⋯
⊔
σ
n
{\displaystyle \sigma (a)=\sigma _{1}\sqcup \cdots \sqcup \sigma _{n}}
p
1
,
…
,
p
n
∈
A
{\displaystyle p_{1},\ldots ,p_{n}\in {\mathcal {A}}}
j
≥
1
,
k
≤
n
{\displaystyle j\geq 1,k\leq n}
投影的譜滿足
σ
(
p
k
)
=
σ
k
.
{\displaystyle \sigma (p_{k})=\sigma _{k}.}
投影與
a
{\displaystyle a}
p
k
a
=
a
p
k
.
{\displaystyle p_{k}a=ap_{k}.}
投影是正交 的，即
p
j
p
k
=
δ
j
k
p
k
.
{\displaystyle p_{j}p_{k}=\delta _{jk}p_{k}.}
投影之和為單位元，即
∑
k
=
1
n
p
k
=
e
.
{\displaystyle \sum _{k=1}^{n}p_{k}=e.}
特別是，有分解
a
=
∑
k
=
1
n
a
k
{\textstyle a=\sum _{k=1}^{n}a_{k}}
∀
1
≤
k
≤
n
,
σ
(
a
k
)
=
σ
k
.
{\displaystyle \forall 1\leq k\leq n,\sigma (a_{k})=\sigma _{k}.}
Blackadar, Bruce. Operator Algebras. Theory of C*-Algebras and von Neumann Algebras.. Berlin/Heidelberg: Springer. 2006. ISBN 3-540-28486-9 Deitmar, Anton; Echterhoff, Siegfried. Principles of Harmonic Analysis. Second Edition.. Springer. 2014. ISBN 978-3-319-05791-0 Dixmier, Jacques. Les C*-algèbres et leurs représentations. Gauthier-Villars. 1969 （法語） . Dixmier, Jacques. C*-algebras . 由Jellett, Francis翻譯. Amsterdam/New York/Oxford: North-Holland. 1977. ISBN 0-7204-0762-1 Les C*-algèbres et leurs représentations. Gauthier-Villars. 1969 （法語） . Kaballo, Winfried. Aufbaukurs Funktionalanalysis und Operatortheorie.. Berlin/Heidelberg: Springer. 2014. ISBN 978-3-642-37794-5（德語） . Kadison, Richard V.; Ringrose, John R. Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory.. New York/London: Academic Press. 1983. ISBN 0-12-393301-3 Kaniuth, Eberhard. A Course in Commutative Banach Algebras. . Springer. 2009. ISBN 978-0-387-72475-1 Schmüdgen, Konrad. Unbounded Self-adjoint Operators on Hilbert Space.. Springer. 2012. ISBN 978-94-007-4752-4 Reed, Michael; Simon, Barry. Methods of modern mathematical physics. vol. 1. Functional analysis. San Diego, CA: Academic Press. 1980. ISBN 0-12-585050-6 Takesaki, Masamichi. Theory of Operator Algebras I.. Heidelberg/Berlin: Springer. 1979. ISBN 3-540-90391-7 
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