Math 7330: Functional Analysis
Fall 2002
Notes 7: The Spectral Theorem
A. Sengupta
Let
B
be a complex, commutative
B
*
algebra, with Δ its Gelfand spectrum. Then,
as we have seen in class,
(i) the Gelfand transform
B
→
C
(Δ) :
x
7→
ˆ
x
satisﬁes
ˆ
x
*
=
ˆ
x
for every
x
∈
B
;
(ii) the spectral radius
ρ
(
x
) equals the norm

x

for every
x
∈
B
.
Fact (ii) was proven ﬁrst for hermitian elements in any
B
*
algebra and then, using the
Gelfand transform, for all elements in a commutative
B
*
algebra. If
a
∈
B
is hermitian
then
ρ
(
a
) = lim
n
→∞

a
n

1
/n
while

a
2

=

aa
*

=

a

2
which implies

a
2
k

=

a

2
k
, and so, letting
n
→ ∞
through powers
of 2 we get
ρ
(
a
) =

a

for every hermitian
a
in any
B
*
algebra. For a commutative
B
*
algebra
B
we have for a
general
x
∈
B
,
ρ
(
xx
*
) =

ˆ
xx
*
)

sup
≤ 
ˆ
x

sup

ˆ
x
*

sup
=
ρ
(
x
)
ρ
(
x
*
)
≤
ρ
(
x
)

x
*

Since
xx
*
is hermitian,
ρ
(
xx
*
) =

xx
*

, which is equal to

x

x
*

. So we have

x
 ≤
ρ
(
x
)
But we already know the opposite inequality. So
ρ
(
x
) =

x

.
By (i) and (ii) and other properties we have studied before, the Gelfand transform
is a
*
–algebra homomorphism and is also an isometry. Its image
ˆ
B
in
C
(Δ) is therefore
a subalgebra of
C
(Δ) which is preserved under conjugation. Moreover, since the Gelfand
transform is an isometry it follows that
ˆ
B
is a
closed
subset of
C
(Δ): for if
x
n
∈
B
are such
that ˆ
x
n
→
f
for some
f
∈
C
(Δ) then (ˆ
x
n
)
n
is Cauchy in
C
(Δ) and so, by isometricity,
(
x
n
)
n
is Cauchy in
B
and so is convergent, say to
x
and then by continuity ofˆit follows
that
f
= ˆ
x
, and so
f
is in the image of the Gelfand transform. Finally,
ˆ
B
separates points
of Δ because if
h
1
and
h
2
are distinct elements of Δ, then, by deﬁnition of Δ, there must
be some
x
∈
B
for which
h
1
(
x
)
6
=
h
2
(
x
), i.e. ˆ
x
(
h
1
)
6
=
ˆ
h
(
x
2
).
The StoneWeierstrass theorem now implies that
ˆ
B
=
C
(Δ)
This proves the
GelfandNaimark
theorem:
Theorem
. For a complex commutative
B
*
–algebra
B
, the Gelfand transform is an
isometric isomorphism of
B
onto
C
(Δ), where Δ is the Gelfand spectrum of
B
.
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