decomposition of self-adjoint elements in positive and negative parts

decomposition of self-adjoint elements in positive and negative parts

\PMlinkescapephrase

decomposition

Every real valued

function

f

admits a well-known decomposition into its and parts:

f

=

f

+

-

f

-

. There is an analogous result for

self-adjoint elements

in a

C

*

-algebra (

http://planetmath.org/CAlgebra

) that we will now describe.

Theorem

-

Let

𝒜

be a

C

*

-algebra and

a

∈

𝒜

a self-adjoint element. Then there are unique

positive elements

a

+

and

a

-

in

𝒜

such that:

•

a

=

a

+

-

a

-

•

a

+

⁢

a

-

=

a

-

⁢

a

+

=

0

•

Both

a

+

and

a

-

belong to

C

*

-subalgebra generated by

a

.

•

∥

a

∥

=

max

⁡

{

∥

a

+

∥

,

∥

a

-

∥

}

Remark -

As a particular case, the result provides a decomposition of each

self-adjoint operator

T

on a

Hilbert space

as a

difference

of two positive operators

T

=

T

+

-

T

-

such that

Ran

⁢

T

-

⊆

Ker

⁢

T

+

and

Ran

⁢

T

+

⊆

Ker

⁢

T

-

, where

Ran

and

Ker

denote, respectively, the range and kernel of an

operator

.

Proof:

Let us some notation first:

•

σ

⁢

(

a

)

denotes the spectrum of

a

∈

𝒜

.

•

C

*

⁢

[

a

]

denotes the

C

*

-subalgebra generated by

a

.

•

C

0

⁢

(

σ

⁢

(

a

)

∖

{

0

}

)

denotes the algebra of

continuous functions

in

σ

⁢

(

a

)

∖

{

0

}

that

vanish at infinity

.

Let

f

,

f

+

,

f

-

∈

C

0

⁢

(

σ

⁢

(

a

)

∖

{

0

}

)

be the functions defined by

f

⁢

(

t

)

:=

t

f

+

⁢

(

t

)

:=

{

t

,

if

⁢

t

≥

0

0

,

if

⁢

t

≤

0

f

-

⁢

(

t

)

:=

{

0

,

if

⁢

t

≥

0

-

t

,

if

⁢

t

≤

0

Since

a

is ,

σ

⁢

(

a

)

⊆

ℝ

, so the above functions are well defined. It is clear that

f

=

f

+

-

f

-

⁢

and

⁢

f

+

⁢

f

-

=

f

-

⁢

f

+

=

0

⁢

and

⁢

f

+

,

f

-

⁢

are both positive

(1)

The

continuous functional calculus

gives an

isomorphism

C

*

⁢

[

a

]

≅

C

0

⁢

(

σ

⁢

(

a

)

∖

{

0

}

)

such that the

element

a

corresponds to the function

f

. Let

a

+

and

a

-

be the elements corresponding to

f

+

and

f

-

respectively. From the made in (1) it is now clear that

•

a

+

and

a

-

are both positive elements.

•

a

=

a

+

-

a

-

•

a

+

⁢

a

-

=

a

-

⁢

a

+

=

0

•

Both

a

+

and

a

-

belong to

C

*

⁢

[

a

]

.

From the fact the every

C

*

-isomorphism is

isometric

(see this entry (

http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous

)) and

∥

f

∥

=

max

⁡

{

∥

f

+

∥

,

∥

f

-

∥

}

it follows that

∥

a

∥

=

max

⁡

{

∥

a

+

∥

,

∥

a

-

∥

}

.

The uniqueness of the decomposition follows from the uniqueness of the decomposition of real valued functions in its positive and negative parts

f

=

f

+

-

f

-

(with

f

+

⁢

f

-

=

0

).

□

Title

decomposition of self-adjoint elements in positive and negative parts

Canonical name

DecompositionOfSelfadjointElementsInPositiveAndNegativeParts

Date of creation

2013-03-22 17:51:49

Last modified on

2013-03-22 17:51:49

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

12

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 47C15

Classification

msc 47B25

Classification

msc 47A60

Classification

msc 46L05

Related topic

CAlgebra

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