continuous functional calculus

continuous functional calculus

Let

B

⁢

(

H

)

be the

algebra

of

bounded operators

over a complex

Hilbert space

H

. Let

T

∈

B

⁢

(

H

)

be a normal operator.

The

continuous functional calculus

is a

functional calculus

which enables the expression

f

⁢

(

T

)

to make sense as a bounded operator in

H

, for

continuous functions

f

.

More generally, when

𝒜

is a

C

*

-algebra (

http://planetmath.org/CAlgebra

) with

identity element

e

, and

x

is a

normal element

of

𝒜

, the

continuous functional calculus

allows one to define

f

⁢

(

x

)

∈

𝒜

when

f

is a continuous function.

More precisely, if

σ

⁢

(

x

)

denotes the spectrum of

x

and

C

⁢

(

σ

⁢

(

x

)

)

denotes the

C

*

-algebra of complex valued continuous functions on

σ

⁢

(

x

)

, we will define a continuous

homomorphism

C

⁢

(

σ

⁢

(

x

)

)

⟶

𝒜

f

↦

f

⁢

(

x

)

that the functional calculus (

http://planetmath.org/FunctionalCalculus

) .

There are several reasons to require the continuity of

f

on the spectrum

σ

⁢

(

x

)

.

For example, suppose

λ

0

∈

σ

⁢

(

x

)

. The

function

f

⁢

(

λ

)

=

1

λ

-

λ

0

is clearly not continuous in

λ

0

. By the functional calculus we would obtain

f

⁢

(

x

)

=

1

x

-

λ

0

⁢

e

=

(

x

-

λ

0

⁢

e

)

-

1

but

x

-

λ

0

⁢

e

is not

invertible

since

λ

0

∈

σ

⁢

(

x

)

.

The abstraction towards

C

*

-algebras is almost . Indeed,

C

*

-algebras are the appropriate where to and prove the continuous functional calculus. The

conclusions

towards

B

⁢

(

H

)

then follow as a particular case.

1

Preliminary construction

Let

𝒜

be a unital

C

*

-algebra and

x

a normal element in

𝒜

. Let

ℬ

⊆

𝒜

be the

C

*

-

subalgebra generated by

x

and the

identity

e

of

𝒜

.

Thus,

ℬ

is the norm

closure

of the algebra generated by

x

,

x

*

and

e

.

Moreover, since

x

is ,

x

*

⁢

x

=

x

⁢

x

*

, it follows that

ℬ

is

commutative

and

ℬ

consists of those

elements

y

∈

𝒜

that can be approximated by

polynomials

p

⁢

(

x

,

x

*

)

in

x

and

x

*

.

Recall the following facts:

•

Since

ℬ

is a commutative unital

C

*

-algebra, the set

△

of

multiplicative linear functionals

on

ℬ

is a

compact

Hausdorff space

.

•

Let

C

⁢

(

△

)

denote the

C

*

-algebra of complex valued continuous functions on

△

. The

Gelfand transform

G

:

ℬ

⟶

C

⁢

(

△

)

is a

C

*

-

isomorphism

.

The following result is perhaps the for the definition of the continuous functional calculus.

Theorem

1 -

△

and

σ

⁢

(

x

)

are

homeomorphic

topological spaces

. Moreover, the

mapping

S

:

△

→

σ

(

x

)

defined by

S

⁢

(

ϕ

)

:=

ϕ

⁢

(

x

)

is such an homeomorphism.

:

We need to check that

S

is well defined, i.e.

ϕ

⁢

(

x

)

∈

σ

⁢

(

x

)

for all

ϕ

∈

△

.

From the identity

ϕ

⁢

(

x

-

ϕ

⁢

(

x

)

⁢

e

)

=

ϕ

⁢

(

x

)

-

ϕ

⁢

(

x

)

⁢

ϕ

⁢

(

e

)

=

ϕ

⁢

(

x

)

-

ϕ

⁢

(

x

)

=

0

follows that

x

-

ϕ

⁢

(

x

)

⁢

e

cannot be invertible in

ℬ

(recall that

ϕ

is a multiplicative linear functional on

ℬ

).

Thus,

ϕ

⁢

(

x

)

∈

σ

ℬ

⁢

(

x

)

. By the

spectral invariance theorem

, we see that

ϕ

⁢

(

x

)

∈

σ

⁢

(

x

)

=

σ

ℬ

⁢

(

x

)

, and so

S

is well defined.

•

S

is continuous -

Suppose

ϕ

α

is a net in

△

such that

ϕ

α

⟶

ϕ

. Recall that the

topology

in

△

is the weak-* topology, so

ϕ

α

⁢

(

x

)

⟶

ϕ

⁢

(

x

)

.

Thus,

S

⁢

(

ϕ

α

)

⟶

S

⁢

(

ϕ

)

and so

S

is continuous.

•

S

is -

Suppose

S

⁢

(

ϕ

1

)

=

S

⁢

(

ϕ

2

)

. Then,

ϕ

1

⁢

(

x

)

=

ϕ

2

⁢

(

x

)

. Since

ϕ

i

⁢

(

x

*

)

=

G

⁢

(

x

*

)

⁢

(

ϕ

i

)

=

G

⁢

(

x

)

⁢

(

ϕ

i

)

¯

=

ϕ

i

⁢

(

x

)

¯

we must also have

ϕ

1

⁢

(

x

*

)

=

ϕ

2

⁢

(

x

*

)

.

This clearly implies that

ϕ

1

⁢

(

p

⁢

(

x

,

x

*

)

)

=

ϕ

2

⁢

(

p

⁢

(

x

,

x

*

)

)

for every polynomial in two

variables

p

.

Recall that the ”polynomials”

p

⁢

(

x

,

x

*

)

are

dense in

ℬ

. So we must have

ϕ

1

⁢

(

y

)

=

ϕ

2

⁢

(

y

)

for every

y

∈

ℬ

, i.e.

ϕ

1

=

ϕ

2

.

•

S

is -

Let

λ

∈

σ

⁢

(

x

)

=

σ

ℬ

⁢

(

x

)

. Then

x

-

λ

⁢

e

is not invertible.

Since

ℬ

is commutative,

x

-

λ

⁢

e

is

contained

in a

maximal ideal

ℳ

.

As

ℳ

is maximal ideal, the

quotient

ℬ

/

ℳ

is a

division algebra

, and so by the

Gelfand-Mazur theorem

,

ℬ

/

ℳ

must ne isomorphic to

ℂ

.

Therefore the quotient homomorphism

ϕ

:

ℬ

⟶

ℬ

/

ℳ

=

ℂ

is a multiplicative linear functional such that

ϕ

⁢

(

x

-

λ

⁢

e

)

=

0

, i.e.

ϕ

⁢

(

x

)

=

λ

, i.e.

S

⁢

(

ϕ

)

=

λ

.

Therefore,

S

is

surjective

.

Since

S

is a continuous

bijective function

from the compact Hausdorff space

△

to

σ

⁢

(

x

)

, it follows that it must be a homeomorphism.

□

2

Definition of the continuous functional calculus

If

S

is the homeomorphism between

△

and

σ

⁢

(

x

)

defined as above, then the mapping

f

↦

f

∘

S

is a *-isomorphism between the algebras

C

⁢

(

σ

⁢

(

x

)

)

and

C

⁢

(

△

)

. Since the Gelfand transform

G

:

ℬ

⟶

C

⁢

(

△

)

is a also a *-isomorphism, we obtain a *-isomorphism

Γ

:

C

⁢

(

σ

⁢

(

x

)

)

⟶

ℬ

by setting

Γ

⁢

(

f

)

:=

G

-

1

⁢

(

f

∘

S

)

.

Definition -

Suppose

x

is a normal element in a unital

C

*

-algebra

A

. For every

f

∈

C

⁢

(

σ

⁢

(

x

)

)

we define

f

⁢

(

x

)

:=

Γ

⁢

(

f

)

∈

ℬ

⊆

𝒜

The mapping

Γ

, such that

f

↦

f

⁢

(

x

)

, is called the

continuous functional calculus

for

x

.

We now prove the functional calculus (

http://planetmath.org/FunctionalCalculus

) for the continuous functional calculus and show its uniqueness:

Theorem 2 -

Let

A

be a unital

C

*

-algebra,

x

∈

A

a normal element and

id

the

identity function

in

C

. The continuous functional calculus for

x

is the unique unital *-homomorphism between

C

⁢

(

σ

⁢

(

x

)

)

and

A

which sends

id

to

x

. In particular, for every polynomial

p

in

C

of the form

p

⁢

(

λ

)

:=

∑

c

n

,

m

⁢

λ

n

⁢

λ

¯

m

, we have

p

⁢

(

x

)

=

∑

c

n

,

m

⁢

x

n

⁢

(

x

*

)

m

.

:

We have seen that the continuous functional calculus

Γ

for

x

is a *-homomorphism between

C

⁢

(

σ

⁢

(

x

)

)

and

𝒜

. Recall that

Γ

was defined by

Γ

⁢

(

f

)

:=

G

-

1

⁢

(

f

∘

S

)

. It is clear by the definition that

Γ

is unital. Also,

G

∘

Γ

⁢

(

f

)

=

f

∘

S

for every

f

∈

C

⁢

(

△

)

. Taking the identity function

id

we obtain that for every

ϕ

∈

△

G

∘

Γ

⁢

(

id

)

⁢

(

ϕ

)

=

id

⁢

(

S

⁢

(

ϕ

)

)

=

ϕ

⁢

(

x

)

=

G

⁢

(

x

)

⁢

(

ϕ

)

Since the Gelfand transform is a *-isomorphism, we must have

Γ

⁢

(

id

)

=

x

.

Now, let

p

:

ℂ

→

ℂ

be a polynomial of the form

p

⁢

(

λ

)

:=

∑

c

n

,

m

⁢

λ

n

⁢

λ

¯

m

. Notice that

p

=

∑

c

n

,

m

⁢

id

n

⁢

id

¯

m

. If

F

is any unital *-homomorphism such that

F

⁢

(

id

)

=

x

, then one must have

F

⁢

(

p

)

=

∑

c

n

,

m

⁢

x

n

⁢

(

x

*

)

m

. Thus all such unital *-homomorphisms coincide on the

subspace

of polynomials of the above form. By the

Stone-Weierstrass theorem

(

http://planetmath.org/StoneWeierstrassTheoremComplexVersion

), this subspace is dense in

C

⁢

(

σ

⁢

(

x

)

)

. Thus, all such unital *-homomorphisms coincide in

C

⁢

(

σ

⁢

(

x

)

)

, and uniqueness is proven.

□

3

Properties

•

The

spectral mapping theorem

assures that for

f

∈

C

⁢

(

σ

⁢

(

x

)

)

σ

⁢

(

f

⁢

(

x

)

)

=

f

⁢

(

σ

⁢

(

x

)

)

•

When

σ

⁢

(

x

)

⊂

ℝ

+

the continuous functional calculus assures the

existence

of a square root

x

of

x

, since

λ

is defined and continuous on

λ

∈

σ

⁢

(

x

)

.

Title

continuous functional calculus

Canonical name

ContinuousFunctionalCalculus

Date of creation

2013-03-22 17:30:02

Last modified on

2013-03-22 17:30:02

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

11

Author

asteroid (17536)

Entry type

Feature

Classification

msc 47A60

Classification

msc 46L05

Classification

msc 46H30

Related topic

FunctionalCalculus

Related topic

PolynomialFunctionalCalculus

Related topic

BorelFunctionalCalculus

Defines

continuous functions of normal operators

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