Borel functional calculus

Borel functional calculus

Let

B

⁢

(

H

)

be the

algebra

(

http://planetmath.org/Algebra

) of

bounded operators

over a

complex

Hilbert space

H

and

T

∈

B

⁢

(

H

)

a normal operator.

The

Borel functional calculus

is a

functional calculus

which enables the

expression

f

⁢

(

T

)

to make sense as a bounded operator in

H

, for a

bounded

(

http://planetmath.org/Bounded

)

Borel function

f

.

In particular, it allows the

definition

of

operators

χ

S

⁢

(

T

)

for any

characteristic function

χ

S

, which are of significant importance on the of the of

T

.

The Borel functional calculus will be constructed by extending the

continuous functional calculus

for arbitrary bounded Borel functions.

1

Preliminary Facts

Let us set some notation first:

•

σ

⁢

(

T

)

will denote the spectrum (

http://planetmath.org/Spectrum

) of

T

.

•

C

⁢

(

σ

⁢

(

T

)

)

will denote the

C

*

-algebra (

http://planetmath.org/CAlgebra

) of

continuous functions

σ

⁢

(

T

)

→

ℂ

.

•

B

⁢

(

σ

⁢

(

T

)

)

will denote the

C

*

-algebra of bounded Borel functions

σ

⁢

(

T

)

→

ℂ

, endowed with the sup norm.

The continuous functional calculus for

T

allows the expression

f

⁢

(

T

)

to make sense for continuous functions

f

∈

C

⁢

(

σ

⁢

(

T

)

)

, by the assignment of a unital *-

homomorphism

π

:

C

⁢

(

σ

⁢

(

T

)

)

⟶

B

⁢

(

H

)

f

⟼

f

⁢

(

T

)

:=

π

⁢

(

f

)

that sends the

identity function

to

T

. This unital *-homomorphism is in fact uniquely determined by this

property

(see the entry on the continuous functional calculus (

http://planetmath.org/ContinuousFunctionalCalculus2

) for more details).

The objective is to extend

π

to a unital *-homomorphism

π

~

:

B

⁢

(

σ

⁢

(

T

)

)

⟶

B

⁢

(

H

)

.

Since

B

⁢

(

σ

⁢

(

T

)

)

is a much larger

C

*

-algebra than

C

⁢

(

σ

⁢

(

T

)

)

, there is no reson to presume that there is only one

extension

of

π

. Which extension would be the most natural then? It turns out that there is a unique extension that

satisfies

a good

continuity

property.

It is known that *-homomorphisms between

C

*

-algebras are

continuous

(see this entry (

http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous

)), so that whenever a net

f

i

∈

B

⁢

(

σ

⁢

(

T

)

)

converges

in the sup norm to a

function

f

∈

B

⁢

(

σ

⁢

(

T

)

)

we will have that

f

i

⁢

(

T

)

→

f

⁢

(

T

)

in the

operator norm

. All extensions of

π

will automaticaly satisfy this continuity property, but this can be improved in a satisfactory manner.

Notation -

Let

X

be a

compact

Hausdorff space

,

M

⁢

(

X

)

the space of all finite

regular

(

http://planetmath.org/OuterRegular

)

Borel measures

in

X

and

B

⁢

(

X

)

the

C

*

-algebra of all bounded Borel functions in

X

. The weakest

topology

in

B

⁢

(

X

)

for which integration against any

measure

ν

is continuous will be reffered to as the

μ

-topology.

This means that

f

i

→

f

in the

μ

-topology if and only if

∫

f

i

⁢

𝑑

ν

→

∫

f

⁢

𝑑

ν

for all

ν

∈

M

⁢

(

X

)

.

Notice that we can identify each function

f

∈

B

⁢

(

X

)

with a bounded

linear functional

(

http://planetmath.org/

Functional

)

ω

f

in

M

⁢

(

X

)

, given by

ω

f

⁢

(

ν

)

:=

∫

X

f

⁢

𝑑

ν

,

ν

∈

M

⁢

(

X

)

and the

μ

-topology corresponds exactly to the weak-* topology under this identification.

We will see in the next that there is an unique extension of

π

that is continuous from the

μ

-topology to the

weak operator topology

.

Just like the

Stone-Weierstrass theorem

(

http://planetmath.org/StoneWeierstrassTheoremComplexVersion

) allowed the passage from the

polynomial functional calculus

to the continuous functional calculus, the

Riesz representation theorem

(

http://planetmath.org/RieszRepresentationTheoremOfLinearFunctionalsOnFunctionSpaces

) will allow the passage from the latter to the Borel functional calculus.

2

Definition

The following result is the key for the definition of the Borel functional calculus.

Theorem

1 -

Let

T

be a normal operator in

B

⁢

(

H

)

and

π

:

C

⁢

(

σ

⁢

(

T

)

)

⟶

B

⁢

(

H

)

the unital *-homomorphism corresponding to the continuous functional calculus for

T

. Then,

π

extends uniquely to a *-homomorphism

π

~

:

B

⁢

(

σ

⁢

(

T

)

)

⟶

B

⁢

(

H

)

that is continuous from the

μ

-topology to the weak operator topology. Moreover, each operator

π

⁢

(

f

)

lies in strong operator (

http://planetmath.org/OperatorTopologies

)

closure

(

http://planetmath.org/Closure

) of the unital *-algebra

generated by

T

.

:

See this attached entry (

http://planetmath.org/ProofOfBorelFunctionalCalculus

)

We are now able to define the Borel functional calculus:

Definition -

Let

T

be a normal operator in

B

⁢

(

H

)

. Let

π

~

:

B

⁢

(

σ

⁢

(

T

)

)

⟶

B

⁢

(

H

)

be the unique *-homomorphism defined in Theorem 1. This *-homomorphism is denoted by

f

⟼

f

⁢

(

T

)

,

f

∈

B

⁢

(

σ

⁢

(

T

)

)

and it is called the

Borel functional calculus

for

T

.

Since this functional calculus extends the polynomial functional calculus, we have that for any

polynomial

p

⁢

(

z

)

:=

∑

c

n

,

m

⁢

z

n

⁢

z

¯

m

,

p

⁢

(

T

)

=

∑

c

n

,

m

⁢

T

n

⁢

(

T

*

)

m

Moreover, since

f

⁢

(

T

)

lies in the strong operator closure of the unital *-algebra generated by

T

, for any function

f

∈

B

⁢

(

σ

⁢

(

T

)

)

, we see that

f

⁢

(

T

)

is the strong operator limit of polynomials

∑

c

n

,

m

⁢

T

n

⁢

(

T

*

)

m

.

3

Borel Calculus in von Neumann Algebras

The Borel functional calculus is in fact applicable for any normal operator

T

in any

von Neumann algebra

ℳ

.

That is due to the fact, expressed in Theorem 1, that for every

f

∈

B

⁢

(

σ

⁢

(

T

)

)

the operator

f

⁢

(

T

)

belongs to the strong operator closure of the unital *-algebra generated by

T

. Being a von Neumann algebra,

ℳ

is closed (

http://planetmath.org/ClosedSet

) in the strong operator topology, and therefore all operators

f

⁢

(

T

)

belong to

ℳ

.

Thus, by

restriction

, we have in fact a *-homomorphism

π

~

:

B

⁢

(

σ

⁢

(

T

)

)

⟶

ℳ

f

⟼

f

⁢

(

T

)

satisfying the properties of Theorem 1, i.e. we have a Borel functional calculus for normal operators of a von Neumann algebra.

References

1

W. Arveson,

A Short Course on

Spectral Theory

, Graduate Texts in Mathematics, 209, Springer, New York, 2002

2

N. Weaver,

Mathematical

Quantization

, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001

Title

Borel functional calculus

Canonical

name

BorelFunctionalCalculus

Date of creation

2013-03-22 18:48:44

Last modified on

2013-03-22 18:48:44

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Feature

Classification

msc 47A60

Classification

msc 46L10

Classification

msc 46H30

Related topic

FunctionalCalculus

Related topic

PolynomialFunctionalCalculus

Related topic

ContinuousFunctionalCalculus2

Defines

Borel functions of a normal operator

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