bounded operators on a Hilbert space form a C^*-algebra

bounded operators on a Hilbert space form a

C

*

-algebra

In this entry we show how the

algebra

B

⁢

(

H

)

of

bounded linear operators

on an

Hilbert space

H

is one of the most natural examples of

C

*

-algebras (

http://planetmath.org/CAlgebra

). In fact, by the

Gelfand-Naimark representation theorem

, every

C

*

-algebra is

isomorphic

to a *-

subalgebra

of

B

⁢

(

H

)

for some Hilbert space

H

.

Lemma

If

H

is a Hilbert space, then

B

⁢

(

H

)

, the algebra of bounded linear operators on

H

, is a

*

-algebra.

Proof:

Let

H

be a Hilbert space. We must prove that the adjugation is an involution.
Let

{

P

,

Q

}

⊂

B

⁢

(

H

)

and

l

∈

𝐂

. For every

{

x

,

y

}

⊂

H

we have

1.

⟨

P

*

*

⁢

x

|

y

⟩

=

⟨

x

|

P

*

⁢

y

⟩

=

⟨

P

⁢

x

|

y

⟩

so

P

*

*

=

P

,

2.

⟨

(

P

⁢

Q

)

*

⁢

x

|

y

⟩

=

⟨

x

|

P

⁢

Q

⁢

y

⟩

=

⟨

P

*

⁢

x

|

Q

⁢

y

⟩

=

⟨

Q

*

⁢

P

*

⁢

x

|

y

⟩

so

(

P

⁢

Q

)

*

=

Q

*

⁢

P

*

and

3.

⟨

(

l

⁢

P

+

Q

)

*

⁢

x

|

y

⟩

=

⟨

x

|

(

l

⁢

P

+

Q

)

⁢

y

⟩

=

l

⁢

⟨

x

|

P

⁢

y

⟩

+

⟨

x

|

Q

⁢

y

⟩

=

l

⁢

⟨

P

*

⁢

x

|

y

⟩

+

⟨

Q

*

⁢

x

|

y

⟩

=

⟨

(

l

*

⁢

P

*

+

Q

*

)

⁢

x

|

y

⟩

so

(

l

⁢

P

+

Q

)

*

=

l

*

⁢

P

*

+

Q

*

,

so we see that the adjugation is an involution and thus

B

⁢

(

H

)

is a

*

-algebra.

□

Lemma

If

H

is a Hilbert space, then

B

⁢

(

H

)

is a

Banach algebra

.

Proof:

Let

H

be a Hilbert space and let

{

P

,

Q

}

⊂

B

⁢

(

H

)

. We have

∥

P

⁢

Q

∥

=

sup

x

∈

H

∖

{

0

}

⁡

∥

P

⁢

Q

⁢

x

∥

H

∥

x

∥

H

≤

sup

x

∈

H

∖

{

0

}

⁡

∥

P

∥

⁢

∥

Q

⁢

x

∥

H

∥

x

∥

H

=

∥

P

∥

⁢

∥

Q

∥

,

so we see that

B

⁢

(

H

)

is a Banach algebra.

□

Lemma

If

H

is a Hilbert space, then

B

⁢

(

H

)

is a

C

*

-algebra.

Proof:

Let

H

be a Hilbert space and let

P

∈

B

⁢

(

H

)

. We have

∥

P

∥

2

=

sup

x

∈

H

∖

{

0

}

⁡

∥

P

⁢

x

∥

H

2

∥

x

∥

H

2

=

sup

x

∈

H

∖

{

0

}

⁡

⟨

P

⁢

x

|

P

⁢

x

⟩

∥

x

∥

H

2

=

sup

x

∈

H

∖

{

0

}

⁡

⟨

P

*

⁢

P

⁢

x

|

x

⟩

∥

x

∥

H

2

≤

sup

x

∈

H

∖

{

0

}

⁡

∥

P

*

⁢

P

⁢

x

∥

H

⁢

∥

x

∥

H

∥

x

∥

H

2

=

sup

x

∈

H

∖

{

0

}

⁡

∥

P

*

⁢

P

⁢

x

∥

H

∥

x

∥

H

=

∥

P

*

⁢

P

∥

so

∥

P

∥

2

≤

∥

P

*

⁢

P

∥

and because of the previous two lemmas say

B

⁢

(

H

)

is a Banach algebra with involution it is a

C

*

-algebra.

□

Lemma

If

H

is a Hilbert space, then every closed

*

-subalgebra of

B

⁢

(

H

)

is a

C

*

-algebra.

Proof:

Let

A

be a closed

*

-subalgebra of

B

⁢

(

H

)

. Because

A

is a closed subspace of a

Banach space

it is itself a Banach space and thus a Banach algebra with an involution and also a

C

*

-algebra.

□

Title

bounded operators

on a Hilbert space form a

C

*

-algebra

Canonical name

BoundedOperatorsOnAHilbertSpaceFormACalgebra

Date of creation

2013-03-22 14:47:12

Last modified on

2013-03-22 14:47:12

Owner

HkBst (6197)

Last modified by

HkBst (6197)

Numerical id

9

Author

HkBst (6197)

Entry type

Result

Classification

msc 46L05

Related topic

RepresentationOfAC_cG_dTopologicalAlgebra

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