In functional analysis , every C* -algebra is isomorphic to a subalgebra of the C* -algebra
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
of bounded linear operators on some Hilbert space
H
.
{\displaystyle H.}
This article describes the spectral theory of closed normal subalgebras of
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
. A subalgebra
A
{\displaystyle A}
of
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
is called normal if it is commutative and closed under the
∗
{\displaystyle \ast }
operation: for all
x
,
y
∈
A
{\displaystyle x,y\in A}
, we have
x
∗
∈
A
{\displaystyle x^{\ast }\in A}
and that
x
y
=
y
x
{\displaystyle xy=yx}
.[ 1]
Resolution of identity [ edit ]
Throughout,
H
{\displaystyle H}
is a fixed Hilbert space .
A projection-valued measure on a measurable space
(
X
,
Ω
)
,
{\displaystyle (X,\Omega ),}
where
Ω
{\displaystyle \Omega }
is a σ-algebra of subsets of
X
,
{\displaystyle X,}
is a mapping
π
:
Ω
→
B
(
H
)
{\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}
such that for all
ω
∈
Ω
,
{\displaystyle \omega \in \Omega ,}
π
(
ω
)
{\displaystyle \pi (\omega )}
is a self-adjoint projection on
H
{\displaystyle H}
(that is,
π
(
ω
)
{\displaystyle \pi (\omega )}
is a bounded linear operator
π
(
ω
)
:
H
→
H
{\displaystyle \pi (\omega ):H\to H}
that satisfies
π
(
ω
)
=
π
(
ω
)
∗
{\displaystyle \pi (\omega )=\pi (\omega )^{*}}
and
π
(
ω
)
∘
π
(
ω
)
=
π
(
ω
)
{\displaystyle \pi (\omega )\circ \pi (\omega )=\pi (\omega )}
) such that
π
(
X
)
=
Id
H
{\displaystyle \pi (X)=\operatorname {Id} _{H}\quad }
(where
Id
H
{\displaystyle \operatorname {Id} _{H}}
is the identity operator of
H
{\displaystyle H}
) and for every
x
,
y
∈
H
,
{\displaystyle x,y\in H,}
the function
Ω
→
C
{\displaystyle \Omega \to \mathbb {C} }
defined by
ω
↦
⟨
π
(
ω
)
x
,
y
⟩
{\displaystyle \omega \mapsto \langle \pi (\omega )x,y\rangle }
is a complex measure on
M
{\displaystyle M}
(that is, a complex-valued countably additive function).
A resolution of identity on a measurable space
(
X
,
Ω
)
{\displaystyle (X,\Omega )}
is a function
π
:
Ω
→
B
(
H
)
{\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}
such that for every
ω
1
,
ω
2
∈
Ω
{\displaystyle \omega _{1},\omega _{2}\in \Omega }
:
π
(
∅
)
=
0
{\displaystyle \pi (\varnothing )=0}
;
π
(
X
)
=
Id
H
{\displaystyle \pi (X)=\operatorname {Id} _{H}}
;
for every
ω
∈
Ω
,
{\displaystyle \omega \in \Omega ,}
π
(
ω
)
{\displaystyle \pi (\omega )}
is a self-adjoint projection on
H
{\displaystyle H}
;
for every
x
,
y
∈
H
,
{\displaystyle x,y\in H,}
the map
π
x
,
y
:
Ω
→
C
{\displaystyle \pi _{x,y}:\Omega \to \mathbb {C} }
defined by
π
x
,
y
(
ω
)
=
⟨
π
(
ω
)
x
,
y
⟩
{\displaystyle \pi _{x,y}(\omega )=\langle \pi (\omega )x,y\rangle }
is a complex measure on
Ω
{\displaystyle \Omega }
;
π
(
ω
1
∩
ω
2
)
=
π
(
ω
1
)
∘
π
(
ω
2
)
{\displaystyle \pi \left(\omega _{1}\cap \omega _{2}\right)=\pi \left(\omega _{1}\right)\circ \pi \left(\omega _{2}\right)}
;
if
ω
1
∩
ω
2
=
∅
{\displaystyle \omega _{1}\cap \omega _{2}=\varnothing }
then
π
(
ω
1
∪
ω
2
)
=
π
(
ω
1
)
+
π
(
ω
2
)
{\displaystyle \pi \left(\omega _{1}\cup \omega _{2}\right)=\pi \left(\omega _{1}\right)+\pi \left(\omega _{2}\right)}
;
If
Ω
{\displaystyle \Omega }
is the
σ
{\displaystyle \sigma }
-algebra of all Borels sets on a Hausdorff locally compact (or compact) space, then the following additional requirement is added:
for every
x
,
y
∈
H
,
{\displaystyle x,y\in H,}
the map
π
x
,
y
:
Ω
→
C
{\displaystyle \pi _{x,y}:\Omega \to \mathbb {C} }
is a regular Borel measure (this is automatically satisfied on compact metric spaces).
Conditions 2, 3, and 4 imply that
π
{\displaystyle \pi }
is a projection-valued measure.
Throughout, let
π
{\displaystyle \pi }
be a resolution of identity.
For all
x
∈
H
,
{\displaystyle x\in H,}
π
x
,
x
:
Ω
→
C
{\displaystyle \pi _{x,x}:\Omega \to \mathbb {C} }
is a positive measure on
Ω
{\displaystyle \Omega }
with total variation
‖
π
x
,
x
‖
=
π
x
,
x
(
X
)
=
‖
x
‖
2
{\displaystyle \left\|\pi _{x,x}\right\|=\pi _{x,x}(X)=\|x\|^{2}}
and that satisfies
π
x
,
x
(
ω
)
=
⟨
π
(
ω
)
x
,
x
⟩
=
‖
π
(
ω
)
x
‖
2
{\displaystyle \pi _{x,x}(\omega )=\langle \pi (\omega )x,x\rangle =\|\pi (\omega )x\|^{2}}
for all
ω
∈
Ω
.
{\displaystyle \omega \in \Omega .}
For every
ω
1
,
ω
2
∈
Ω
{\displaystyle \omega _{1},\omega _{2}\in \Omega }
:
π
(
ω
1
)
π
(
ω
2
)
=
π
(
ω
2
)
π
(
ω
1
)
{\displaystyle \pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right)}
(since both are equal to
π
(
ω
1
∩
ω
2
)
{\displaystyle \pi \left(\omega _{1}\cap \omega _{2}\right)}
).
If
ω
1
∩
ω
2
=
∅
{\displaystyle \omega _{1}\cap \omega _{2}=\varnothing }
then the ranges of the maps
π
(
ω
1
)
{\displaystyle \pi \left(\omega _{1}\right)}
and
π
(
ω
2
)
{\displaystyle \pi \left(\omega _{2}\right)}
are orthogonal to each other and
π
(
ω
1
)
π
(
ω
2
)
=
0
=
π
(
ω
2
)
π
(
ω
1
)
.
{\displaystyle \pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=0=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right).}
π
:
Ω
→
B
(
H
)
{\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}
is finitely additive.
If
ω
1
,
ω
2
,
…
{\displaystyle \omega _{1},\omega _{2},\ldots }
are pairwise disjoint elements of
Ω
{\displaystyle \Omega }
whose union is
ω
{\displaystyle \omega }
and if
π
(
ω
i
)
=
0
{\displaystyle \pi \left(\omega _{i}\right)=0}
for all
i
{\displaystyle i}
then
π
(
ω
)
=
0.
{\displaystyle \pi (\omega )=0.}
However,
π
:
Ω
→
B
(
H
)
{\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}
is countably additive only in trivial situations as is now described: suppose that
ω
1
,
ω
2
,
…
{\displaystyle \omega _{1},\omega _{2},\ldots }
are pairwise disjoint elements of
Ω
{\displaystyle \Omega }
whose union is
ω
{\displaystyle \omega }
and that the partial sums
∑
i
=
1
n
π
(
ω
i
)
{\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)}
converge to
π
(
ω
)
{\displaystyle \pi (\omega )}
in
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
(with its norm topology) as
n
→
∞
{\displaystyle n\to \infty }
; then since the norm of any projection is either
0
{\displaystyle 0}
or
≥
1
,
{\displaystyle \geq 1,}
the partial sums cannot form a Cauchy sequence unless all but finitely many of the
π
(
ω
i
)
{\displaystyle \pi \left(\omega _{i}\right)}
are
0.
{\displaystyle 0.}
For any fixed
x
∈
H
,
{\displaystyle x\in H,}
the map
π
x
:
Ω
→
H
{\displaystyle \pi _{x}:\Omega \to H}
defined by
π
x
(
ω
)
:=
π
(
ω
)
x
{\displaystyle \pi _{x}(\omega ):=\pi (\omega )x}
is a countably additive
H
{\displaystyle H}
-valued measure on
Ω
.
{\displaystyle \Omega .}
Here countably additive means that whenever
ω
1
,
ω
2
,
…
{\displaystyle \omega _{1},\omega _{2},\ldots }
are pairwise disjoint elements of
Ω
{\displaystyle \Omega }
whose union is
ω
,
{\displaystyle \omega ,}
then the partial sums
∑
i
=
1
n
π
(
ω
i
)
x
{\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)x}
converge to
π
(
ω
)
x
{\displaystyle \pi (\omega )x}
in
H
.
{\displaystyle H.}
Said more succinctly,
∑
i
=
1
∞
π
(
ω
i
)
x
=
π
(
ω
)
x
.
{\displaystyle \sum _{i=1}^{\infty }\pi \left(\omega _{i}\right)x=\pi (\omega )x.}
In other words, for every pairwise disjoint family of elements
(
ω
i
)
i
=
1
∞
⊆
Ω
{\displaystyle \left(\omega _{i}\right)_{i=1}^{\infty }\subseteq \Omega }
whose union is
ω
∞
∈
Ω
{\displaystyle \omega _{\infty }\in \Omega }
, then
∑
i
=
1
n
π
(
ω
i
)
=
π
(
⋃
i
=
1
n
ω
i
)
{\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)=\pi \left(\bigcup _{i=1}^{n}\omega _{i}\right)}
(by finite additivity of
π
{\displaystyle \pi }
) converges to
π
(
ω
∞
)
{\displaystyle \pi \left(\omega _{\infty }\right)}
in the strong operator topology on
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
: for every
x
∈
H
{\displaystyle x\in H}
, the sequence of elements
∑
i
=
1
n
π
(
ω
i
)
x
{\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)x}
converges to
π
(
ω
∞
)
x
{\displaystyle \pi \left(\omega _{\infty }\right)x}
in
H
{\displaystyle H}
(with respect to the norm topology).
L∞ (π) - space of essentially bounded function[ edit ]
The
π
:
Ω
→
B
(
H
)
{\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}
be a resolution of identity on
(
X
,
Ω
)
.
{\displaystyle (X,\Omega ).}
Essentially bounded functions [ edit ]
Suppose
f
:
X
→
C
{\displaystyle f:X\to \mathbb {C} }
is a complex-valued
Ω
{\displaystyle \Omega }
-measurable function. There exists a unique largest open subset
V
f
{\displaystyle V_{f}}
of
C
{\displaystyle \mathbb {C} }
(ordered under subset inclusion) such that
π
(
f
−
1
(
V
f
)
)
=
0.
{\displaystyle \pi \left(f^{-1}\left(V_{f}\right)\right)=0.}
To see why, let
D
1
,
D
2
,
…
{\displaystyle D_{1},D_{2},\ldots }
be a basis for
C
{\displaystyle \mathbb {C} }
's topology consisting of open disks and suppose that
D
i
1
,
D
i
2
,
…
{\displaystyle D_{i_{1}},D_{i_{2}},\ldots }
is the subsequence (possibly finite) consisting of those sets such that
π
(
f
−
1
(
D
i
k
)
)
=
0
{\displaystyle \pi \left(f^{-1}\left(D_{i_{k}}\right)\right)=0}
; then
D
i
1
∪
D
i
2
∪
⋯
=
V
f
.
{\displaystyle D_{i_{1}}\cup D_{i_{2}}\cup \cdots =V_{f}.}
Note that, in particular, if
D
{\displaystyle D}
is an open subset of
C
{\displaystyle \mathbb {C} }
such that
D
∩
Im
f
=
∅
{\displaystyle D\cap \operatorname {Im} f=\varnothing }
then
π
(
f
−
1
(
D
)
)
=
π
(
∅
)
=
0
{\displaystyle \pi \left(f^{-1}(D)\right)=\pi (\varnothing )=0}
so that
D
⊆
V
f
{\displaystyle D\subseteq V_{f}}
(although there are other ways in which
π
(
f
−
1
(
D
)
)
{\displaystyle \pi \left(f^{-1}(D)\right)}
may equal 0 ). Indeed,
C
∖
cl
(
Im
f
)
⊆
V
f
.
{\displaystyle \mathbb {C} \setminus \operatorname {cl} (\operatorname {Im} f)\subseteq V_{f}.}
The essential range of
f
{\displaystyle f}
is defined to be the complement of
V
f
.
{\displaystyle V_{f}.}
It is the smallest closed subset of
C
{\displaystyle \mathbb {C} }
that contains
f
(
x
)
{\displaystyle f(x)}
for almost all
x
∈
X
{\displaystyle x\in X}
(that is, for all
x
∈
X
{\displaystyle x\in X}
except for those in some set
ω
∈
Ω
{\displaystyle \omega \in \Omega }
such that
π
(
ω
)
=
0
{\displaystyle \pi (\omega )=0}
). The essential range is a closed subset of
C
{\displaystyle \mathbb {C} }
so that if it is also a bounded subset of
C
{\displaystyle \mathbb {C} }
then it is compact.
The function
f
{\displaystyle f}
is essentially bounded if its essential range is bounded, in which case define its essential supremum , denoted by
‖
f
‖
∞
,
{\displaystyle \|f\|^{\infty },}
to be the supremum of all
|
λ
|
{\displaystyle |\lambda |}
as
λ
{\displaystyle \lambda }
ranges over the essential range of
f
.
{\displaystyle f.}
Space of essentially bounded functions [ edit ]
Let
B
(
X
,
Ω
)
{\displaystyle {\mathcal {B}}(X,\Omega )}
be the vector space of all bounded complex-valued
Ω
{\displaystyle \Omega }
-measurable functions
f
:
X
→
C
,
{\displaystyle f:X\to \mathbb {C} ,}
which becomes a Banach algebra when normed by
‖
f
‖
∞
:=
sup
x
∈
X
|
f
(
x
)
|
.
{\displaystyle \|f\|_{\infty }:=\sup _{x\in X}|f(x)|.}
The function
‖
⋅
‖
∞
{\displaystyle \|\,\cdot \,\|^{\infty }}
is a seminorm on
B
(
X
,
Ω
)
,
{\displaystyle {\mathcal {B}}(X,\Omega ),}
but not necessarily a norm.
The kernel of this seminorm,
N
∞
:=
{
f
∈
B
(
X
,
Ω
)
:
‖
f
‖
∞
=
0
}
,
{\displaystyle N^{\infty }:=\left\{f\in {\mathcal {B}}(X,\Omega ):\|f\|^{\infty }=0\right\},}
is a vector subspace of
B
(
X
,
Ω
)
{\displaystyle {\mathcal {B}}(X,\Omega )}
that is a closed two-sided ideal of the Banach algebra
(
B
(
X
,
Ω
)
,
‖
⋅
‖
∞
)
.
{\displaystyle \left({\mathcal {B}}(X,\Omega ),\|\cdot \|_{\infty }\right).}
Hence the quotient of
B
(
X
,
Ω
)
{\displaystyle {\mathcal {B}}(X,\Omega )}
by
N
∞
{\displaystyle N^{\infty }}
is also a Banach algebra, denoted by
L
∞
(
π
)
:=
B
(
X
,
Ω
)
/
N
∞
{\displaystyle L^{\infty }(\pi ):={\mathcal {B}}(X,\Omega )/N^{\infty }}
where the norm of any element
f
+
N
∞
∈
L
∞
(
π
)
{\displaystyle f+N^{\infty }\in L^{\infty }(\pi )}
is equal to
‖
f
‖
∞
{\displaystyle \|f\|^{\infty }}
(since if
f
+
N
∞
=
g
+
N
∞
{\displaystyle f+N^{\infty }=g+N^{\infty }}
then
‖
f
‖
∞
=
‖
g
‖
∞
{\displaystyle \|f\|^{\infty }=\|g\|^{\infty }}
) and this norm makes
L
∞
(
π
)
{\displaystyle L^{\infty }(\pi )}
into a Banach algebra.
The spectrum of
f
+
N
∞
{\displaystyle f+N^{\infty }}
in
L
∞
(
π
)
{\displaystyle L^{\infty }(\pi )}
is the essential range of
f
.
{\displaystyle f.}
This article will follow the usual practice of writing
f
{\displaystyle f}
rather than
f
+
N
∞
{\displaystyle f+N^{\infty }}
to represent elements of
L
∞
(
π
)
.
{\displaystyle L^{\infty }(\pi ).}
Theorem — Let
π
:
Ω
→
B
(
H
)
{\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}
be a resolution of identity on
(
X
,
Ω
)
.
{\displaystyle (X,\Omega ).}
There exists a closed normal subalgebra
A
{\displaystyle A}
of
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
and an isometric * -isomorphism
Ψ
:
L
∞
(
π
)
→
A
{\displaystyle \Psi :L^{\infty }(\pi )\to A}
satisfying the following properties:
⟨
Ψ
(
f
)
x
,
y
⟩
=
∫
X
f
d
π
x
,
y
{\displaystyle \langle \Psi (f)x,y\rangle =\int _{X}f\operatorname {d} \pi _{x,y}}
for all
x
,
y
∈
H
{\displaystyle x,y\in H}
and
f
∈
L
∞
(
π
)
,
{\displaystyle f\in L^{\infty }(\pi ),}
which justifies the notation
Ψ
(
f
)
=
∫
X
f
d
π
{\displaystyle \Psi (f)=\int _{X}f\operatorname {d} \pi }
;
‖
Ψ
(
f
)
x
‖
2
=
∫
X
|
f
|
2
d
π
x
,
x
{\displaystyle \|\Psi (f)x\|^{2}=\int _{X}|f|^{2}\operatorname {d} \pi _{x,x}}
for all
x
∈
H
{\displaystyle x\in H}
and
f
∈
L
∞
(
π
)
{\displaystyle f\in L^{\infty }(\pi )}
;
an operator
R
∈
B
(
H
)
{\displaystyle R\in \mathbb {B} (H)}
commutes with every element of
Im
π
{\displaystyle \operatorname {Im} \pi }
if and only if it commutes with every element of
A
=
Im
Ψ
.
{\displaystyle A=\operatorname {Im} \Psi .}
if
f
{\displaystyle f}
is a simple function equal to
f
=
∑
i
=
1
n
λ
i
1
ω
i
,
{\displaystyle f=\sum _{i=1}^{n}\lambda _{i}\mathbb {1} _{\omega _{i}},}
where
ω
1
,
…
ω
n
{\displaystyle \omega _{1},\ldots \omega _{n}}
is a partition of
X
{\displaystyle X}
and the
λ
i
{\displaystyle \lambda _{i}}
are complex numbers, then
Ψ
(
f
)
=
∑
i
=
1
n
λ
i
π
(
ω
i
)
{\displaystyle \Psi (f)=\sum _{i=1}^{n}\lambda _{i}\pi \left(\omega _{i}\right)}
(here
1
{\displaystyle \mathbb {1} }
is the characteristic function);
if
f
{\displaystyle f}
is the limit (in the norm of
L
∞
(
π
)
{\displaystyle L^{\infty }(\pi )}
) of a sequence of simple functions
s
1
,
s
2
,
…
{\displaystyle s_{1},s_{2},\ldots }
in
L
∞
(
π
)
{\displaystyle L^{\infty }(\pi )}
then
(
Ψ
(
s
i
)
)
i
=
1
∞
{\displaystyle \left(\Psi \left(s_{i}\right)\right)_{i=1}^{\infty }}
converges to
Ψ
(
f
)
{\displaystyle \Psi (f)}
in
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
and
‖
Ψ
(
f
)
‖
=
‖
f
‖
∞
{\displaystyle \|\Psi (f)\|=\|f\|^{\infty }}
;
(
‖
f
‖
∞
)
2
=
sup
‖
h
‖
≤
1
∫
X
d
π
h
,
h
{\displaystyle \left(\|f\|^{\infty }\right)^{2}=\sup _{\|h\|\leq 1}\int _{X}\operatorname {d} \pi _{h,h}}
for every
f
∈
L
∞
(
π
)
.
{\displaystyle f\in L^{\infty }(\pi ).}
The maximal ideal space of a Banach algebra
A
{\displaystyle A}
is the set of all complex homomorphisms
A
→
C
,
{\displaystyle A\to \mathbb {C} ,}
which we'll denote by
σ
A
.
{\displaystyle \sigma _{A}.}
For every
T
{\displaystyle T}
in
A
,
{\displaystyle A,}
the Gelfand transform of
T
{\displaystyle T}
is the map
G
(
T
)
:
σ
A
→
C
{\displaystyle G(T):\sigma _{A}\to \mathbb {C} }
defined by
G
(
T
)
(
h
)
:=
h
(
T
)
.
{\displaystyle G(T)(h):=h(T).}
σ
A
{\displaystyle \sigma _{A}}
is given the weakest topology making every
G
(
T
)
:
σ
A
→
C
{\displaystyle G(T):\sigma _{A}\to \mathbb {C} }
continuous. With this topology,
σ
A
{\displaystyle \sigma _{A}}
is a compact Hausdorff space and every
T
{\displaystyle T}
in
A
,
{\displaystyle A,}
G
(
T
)
{\displaystyle G(T)}
belongs to
C
(
σ
A
)
,
{\displaystyle C\left(\sigma _{A}\right),}
which is the space of continuous complex-valued functions on
σ
A
.
{\displaystyle \sigma _{A}.}
The range of
G
(
T
)
{\displaystyle G(T)}
is the spectrum
σ
(
T
)
{\displaystyle \sigma (T)}
and that the spectral radius is equal to
max
{
|
G
(
T
)
(
h
)
|
:
h
∈
σ
A
}
,
{\displaystyle \max \left\{|G(T)(h)|:h\in \sigma _{A}\right\},}
which is
≤
‖
T
‖
.
{\displaystyle \leq \|T\|.}
Theorem — Suppose
A
{\displaystyle A}
is a closed normal subalgebra of
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
that contains the identity operator
Id
H
{\displaystyle \operatorname {Id} _{H}}
and let
σ
=
σ
A
{\displaystyle \sigma =\sigma _{A}}
be the maximal ideal space of
A
.
{\displaystyle A.}
Let
Ω
{\displaystyle \Omega }
be the Borel subsets of
σ
.
{\displaystyle \sigma .}
For every
T
{\displaystyle T}
in
A
,
{\displaystyle A,}
let
G
(
T
)
:
σ
A
→
C
{\displaystyle G(T):\sigma _{A}\to \mathbb {C} }
denote the Gelfand transform of
T
{\displaystyle T}
so that
G
{\displaystyle G}
is an injective map
G
:
A
→
C
(
σ
A
)
.
{\displaystyle G:A\to C\left(\sigma _{A}\right).}
There exists a unique resolution of identity
π
:
Ω
→
A
{\displaystyle \pi :\Omega \to A}
that satisfies:
⟨
T
x
,
y
⟩
=
∫
σ
A
G
(
T
)
d
π
x
,
y
for all
x
,
y
∈
H
and all
T
∈
A
;
{\displaystyle \langle Tx,y\rangle =\int _{\sigma _{A}}G(T)\operatorname {d} \pi _{x,y}\quad {\text{ for all }}x,y\in H{\text{ and all }}T\in A;}
the notation
T
=
∫
σ
A
G
(
T
)
d
π
{\displaystyle T=\int _{\sigma _{A}}G(T)\operatorname {d} \pi }
is used to summarize this situation.
Let
I
:
Im
G
→
A
{\displaystyle I:\operatorname {Im} G\to A}
be the inverse of the Gelfand transform
G
:
A
→
C
(
σ
A
)
{\displaystyle G:A\to C\left(\sigma _{A}\right)}
where
Im
G
{\displaystyle \operatorname {Im} G}
can be canonically identified as a subspace of
L
∞
(
π
)
.
{\displaystyle L^{\infty }(\pi ).}
Let
B
{\displaystyle B}
be the closure (in the norm topology of
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
) of the linear span of
Im
π
.
{\displaystyle \operatorname {Im} \pi .}
Then the following are true:
B
{\displaystyle B}
is a closed subalgebra of
B
(
H
)
{\displaystyle {\mathcal {B}}(H)}
containing
A
.
{\displaystyle A.}
There exists a (linear multiplicative) isometric * -isomorphism
Φ
:
L
∞
(
π
)
→
B
{\displaystyle \Phi :L^{\infty }(\pi )\to B}
extending
I
:
Im
G
→
A
{\displaystyle I:\operatorname {Im} G\to A}
such that
Φ
f
=
∫
σ
A
f
d
π
{\displaystyle \Phi f=\int _{\sigma _{A}}f\operatorname {d} \pi }
for all
f
∈
L
∞
(
π
)
.
{\displaystyle f\in L^{\infty }(\pi ).}
Recall that the notation
Φ
f
=
∫
σ
A
f
d
π
{\displaystyle \Phi f=\int _{\sigma _{A}}f\operatorname {d} \pi }
means that
⟨
(
Φ
f
)
x
,
y
⟩
=
∫
σ
A
f
d
π
x
,
y
{\displaystyle \langle (\Phi f)x,y\rangle =\int _{\sigma _{A}}f\operatorname {d} \pi _{x,y}}
for all
x
,
y
∈
H
{\displaystyle x,y\in H}
;
Note in particular that
T
=
∫
σ
A
G
(
T
)
d
π
=
Φ
(
G
(
T
)
)
{\displaystyle T=\int _{\sigma _{A}}G(T)\operatorname {d} \pi =\Phi (G(T))}
for all
T
∈
A
.
{\displaystyle T\in A.}
Explicitly,
Φ
{\displaystyle \Phi }
satisfies
Φ
(
f
¯
)
=
(
Φ
f
)
∗
{\displaystyle \Phi \left({\overline {f}}\right)=(\Phi f)^{*}}
and
‖
Φ
f
‖
=
‖
f
‖
∞
{\displaystyle \|\Phi f\|=\|f\|^{\infty }}
for every
f
∈
L
∞
(
π
)
{\displaystyle f\in L^{\infty }(\pi )}
(so if
f
{\displaystyle f}
is real valued then
Φ
(
f
)
{\displaystyle \Phi (f)}
is self-adjoint).
If
ω
⊆
σ
A
{\displaystyle \omega \subseteq \sigma _{A}}
is open and nonempty (which implies that
ω
∈
Ω
{\displaystyle \omega \in \Omega }
) then
π
(
ω
)
≠
0.
{\displaystyle \pi (\omega )\neq 0.}
A bounded linear operator
S
∈
B
(
H
)
{\displaystyle S\in {\mathcal {B}}(H)}
commutes with every element of
A
{\displaystyle A}
if and only if it commutes with every element of
Im
π
.
{\displaystyle \operatorname {Im} \pi .}
The above result can be specialized to a single normal bounded operator.
^ Rudin, Walter (1991). Functional Analysis (2nd ed.). New York: McGraw Hill. pp. 292–293. ISBN 0-07-100944-2 .
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