Math 7330: Functional Analysis
Fall 2002
Homework 5: Commutative Banach Algebras II
A. Sengupta
We work with a complex commutative Banach algebra
B
.
It had been shown that the set
G
(
B
) of all invertible elements in
B
is an open subset
of
B
. A proper ideal
I
in
B
cannot contain any invertible elements (for if
x
∈
I
is invertible
then for any
y
∈
B
we would have
y
= (
yx

1
)
x
∈
I
, which would mean
I
=
B
), i.e. is a
subset of the closed set
G
(
B
)
c
.
Zorn’s lemma shows that every proper ideal of
B
is contained in a maximal ideal.
1. Let
J
be an ideal of
B
.
(i) Check that the closure
J
is also an ideal.
(ii) Show that if
J
is a proper ideal then so is its closure
J
.
(iii) Show that if
J
is a maximal ideal then
J
is closed. Hint: Consider the ideal
J
. It
is an ideal which contains
J
. Since
J
, being maximal, is proper, (ii) implies that
J
is a proper ideal.
1
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A mapping
φ
:
B
→
C
is a
complex homomorphism
if
f
is linear and satisfies
f
(
xy
) =
f
(
x
)
f
(
y
) for all
x, y
∈
B
.
Note that then
f
(
x
) =
f
(
xe
) =
f
(
x
)
f
(
e
) for every
x
∈
B
,
and so either
f
(
e
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 Spring '08
 Davidson,M
 Algebra, Vector Space, Compact space, Banach space, C*algebra

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