proof of quotients in C^*-algebras

proof of quotients in

C

*

-algebras

Proof:

We have that

ℐ

is

self-adjoint

(

http://planetmath.org/InvolutaryRing

), since it is a closed ideal of a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

) (see this entry (

http://planetmath.org/ClosedIdealsInCAlgebrasAreSelfAdjoint

)). Hence, the

involution

in

𝒜

induces a

well-defined

involution in

𝒜

/

ℐ

by

(

x

+

ℐ

)

*

:=

x

*

+

ℐ

.

Recall that, since

ℐ

is closed, the

quotient norm

is indeed a norm in

𝒜

/

ℐ

that makes

𝒜

/

ℐ

a

Banach algebra

(see this entry (

http://planetmath.org/QuotientsOfBanachAlgebras

)). Thus we only have to prove the

C

*

to prove that

𝒜

/

ℐ

is a

C

*

-algebra.

Recall that

C

*

-algebras have

approximate identities

(

http://planetmath.org/CAlgebrasHaveApproximateIdentities

). Notice that

ℐ

itself is a

C

*

-algebra and pick an approximate identity

(

e

λ

)

in

ℐ

such that

•

each

e

λ

is

positive

.

•

∥

e

λ

∥

≤

1

We will only prove the case when

𝒜

has an

identity element

e

. For the non-unital case, one can consider

𝒜

as a

C

*

-subalgebra of its

minimal unitization

and the same proof will still work.

Let

∥

⋅

∥

q

denote the quotient norm in

𝒜

/

ℐ

. We claim that for every

x

∈

𝒜

:

∥

x

+

ℐ

∥

q

=

lim

λ

⁡

∥

x

⁢

(

e

-

e

λ

)

∥

(1)

We will prove the above equality as a lemma at the end of the entry. Assuming this result, it follows that for every

a

∈

𝒜

∥

x

+

ℐ

∥

q

2

=

lim

⁡

∥

x

⁢

(

e

-

e

λ

)

∥

2

=

lim

⁡

∥

(

e

-

e

λ

)

⁢

x

*

⁢

x

⁢

(

e

-

e

λ

)

∥

≤

lim

⁡

∥

(

e

-

e

λ

)

∥

⁢

∥

x

*

⁢

x

⁢

(

e

-

e

λ

)

∥

Since each

e

λ

is positive and

∥

e

λ

∥

≤

1

we know that its spectrum lies on the interval

[

0

,

1

]

. Hence

e

-

e

λ

is also positive and its spectrum also lies on the interval

[

0

,

1

]

. Thus,

∥

e

-

e

λ

∥

≤

1

. Therefore:

∥

x

+

ℐ

∥

q

2

≤

lim

⁡

∥

(

e

-

e

λ

)

∥

⁢

∥

x

*

⁢

x

⁢

(

e

-

e

λ

)

∥

≤

lim

⁡

∥

x

*

⁢

x

⁢

(

e

-

e

λ

)

∥

=

∥

x

*

⁢

x

+

ℐ

∥

q

Since

𝒜

/

ℐ

is a Banach algebra, we also have

∥

x

*

⁢

x

+

ℐ

∥

q

≤

∥

x

+

ℐ

∥

q

2

and so

∥

x

+

ℐ

∥

q

2

=

∥

x

*

⁢

x

+

ℐ

∥

q

which proves that

𝒜

/

ℐ

is a

C

*

-algebra.

□

We now prove equality (1) as a lemma.

Lemma -

Suppose

𝒜

is a

C

*

-algebra with identity element

e

. Let

ℐ

⊂

𝒜

be a closed ideal and

(

e

λ

)

be an approximate identity in

ℐ

such that each

e

λ

is positive and

∥

e

λ

∥

≤

1

. Then

∥

x

+

ℐ

∥

q

=

lim

λ

⁡

∥

x

⁢

(

e

-

e

λ

)

∥

for every

x

in

𝒜

.

Proof:

Since

y

⁢

(

e

-

e

λ

)

⟶

0

for every

y

∈

ℐ

it follows that

lim sup

⁡

∥

x

⁢

(

e

-

e

λ

)

∥

=

lim sup

⁡

∥

x

-

x

⁢

e

λ

-

y

+

y

⁢

e

λ

∥

=

lim sup

⁡

∥

(

x

-

y

)

⁢

(

e

-

e

λ

)

∥

≤

∥

x

-

y

∥

Therefore, taking the infimum over all

y

∈

ℐ

we obtain:

lim sup

⁡

∥

x

⁢

(

e

-

e

λ

)

∥

≤

inf

y

∈

ℐ

⁡

∥

x

-

y

∥

=

∥

x

+

ℐ

∥

q

Also, since

x

⁢

e

λ

∈

ℐ

,

lim inf

⁡

∥

x

⁢

(

e

-

e

λ

)

∥

≥

inf

y

∈

ℐ

⁡

∥

x

-

y

∥

=

∥

x

+

ℐ

∥

q

and this proves the lemma.

□

Title

proof of

quotients

in

C

*

-algebras

Canonical

name

ProofOfQuotientsInCalgebras

Date of creation

2013-03-22 17:41:56

Last modified on

2013-03-22 17:41:56

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Proof

Classification

msc 46L05

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