characterization of maximal ideals of the algebra of continuous functions on a compact set

characterization of maximal ideals of the algebra of continuous functions on a compact set

Let

X

be a

compact

topological space

and let

C

⁢

(

X

)

be the algebra of

continuous

real-valued functions on this space. In this entry, we shall examine the

maximal
ideals

of this algebra.

Theorem 1

.

Let

X

be a compact topological space and

I

be an ideal of

C

⁢

(

X

)

.
Then either

I

=

C

⁢

(

x

)

or there exists a point

p

∈

X

such that

f

⁢

(

p

)

=

0

for
all

f

∈

I

.

Proof.

Assume that, for every point

p

∈

X

, there exists a

continuous function

f

∈

I

such that

f

⁢

(

p

)

≠

0

. Then, by

continuity

, there must exist an

open set

U

containing

p

so that

f

⁢

(

q

)

≠

0

for all

q

∈

U

. Thus, we may
assign to each point

p

∈

X

a continuous function

f

∈

I

and
an open set

U

of

X

such that

f

⁢

(

q

)

≠

0

for all

q

∈

U

. Since this

collection

of open sets covers

X

, which is compact, there must exists a finite

subcover

which also covers

X

. Call this subcover

U

1

,

…

,

U

n

and the
corresponding

functions

f

1

,

…

⁢

f

n

. Consider the function

g

defined as

g

⁢

(

x

)

=

(

f

1

⁢

(

x

)

)

2

+

⋯

+

(

f

n

⁢

(

x

)

)

2

. Since

I

is an ideal,

g

∈

I

. For every
point

p

∈

X

, there exists an integer

i

between

1

and

n

such that

f

i

⁢

(

p

)

≠

0

. This implies that

g

⁢

(

p

)

≠

0

. Since

g

is a continuous function on
a compact set, it must attain a

minimum

. By construction of

g

, the value of

g

at its minimum cannot be

negative

; by what we just showed, it cannot equal zero either.
Hence being

bounded from below

by a positive number,

g

has a continuous

inverse

.
But, if an ideal contains an invertible element, it must be the whole algebra. Hence,
we conclude that either there exists a point

p

∈

x

such that

f

⁢

(

p

)

=

0

for all

f

∈

I

or

I

=

C

⁢

(

x

)

.
∎

Theorem 2

.

Let

X

be a compact

Hausdorff topological space

. Then an ideal is maximal if
and only if it is the ideal of all points which go zero at a given point.

Proof.

By the previous

theorem

, every non-trivial ideal must be a subset of an ideal of
functions which vanish at a given point. Hence, it only remains to prove that
ideals of functions vanishing at a point is maxiamal.

Let

p

be a point of

X

. Assume that the ideal of functions vanishing at

p

is properly

contained

in ideal

I

. Then there must exist a function

f

∈

I

such that

f

⁢

(

p

)

≠

0

(otherwise, the

inclusion

would not be proper). Since

f

is continuous, there will exist an open neighborhood

U

of

p

such that

f

⁢

(

x

)

≠

0

when

x

∈

U

. By Urysohn’s theorem, there exists a continuous
function

h

:

X

→

ℝ

such that

f

⁢

(

p

)

=

0

and

f

⁢

(

x

)

=

0

for
all

x

∈

X

∖

U

. Since

I

was assumed to contain all functions vanishing
at

p

, we must have

f

∈

I

. Hence, the function

g

defined by

g

⁢

(

x

)

=

(

f

⁢

(

x

)

)

2

+

(

h

⁢

(

x

)

)

2

must also lie in

I

. By construction,

g

⁢

(

g

)

>

0

when

x

∈

U

and when

g

⁢

(

x

)

∈

X

∖

U

. Because

X

is compact,

g

must attain a minimum somewhere, hence is bounded from below by a
positive number. Thus

g

has a continuous inverse, so

I

=

C

⁢

(

X

)

, hence the
ideal of functions vanishing at

p

is maximal.
∎

Title

characterization

of maximal ideals of the algebra of continuous functions on a compact set

Canonical name

CharacterizationOfMaximalIdealsOfTheAlgebraOfContinuousFunctionsOnACompactSet

Date of creation

2013-03-22 17:45:08

Last modified on

2013-03-22 17:45:08

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 46L05

Classification

msc 46J20

Classification

msc 46J10

Classification

msc 16W80

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