aarjaneiro/planetmath huggingface dataset:
spectral mapping theorem

spectral mapping theorem

Let

𝒜

be a unital

C

*

-algebra (

http://planetmath.org/CAlgebra

). Let

x

be a

normal element

in

𝒜

and

σ

⁢

(

x

)

be its spectrum.

The

continuous functional calculus

provides a

C

*

-

isomorphism

C

⁢

(

σ

⁢

(

x

)

)

⟶

𝒜

⁢

[

x

]

f

↦

f

⁢

(

x

)

between the

C

*

-algebra

C

⁢

(

σ

⁢

(

x

)

)

of complex valued

continuous functions

on

σ

⁢

(

x

)

and the

C

*

-

subalgebra

𝒜

⁢

[

x

]

⊆

𝒜

generated by

x

and the

identity

of

𝒜

.

Spectral Mapping Theorem

-

Let

x

∈

𝒜

be as above. Let

f

∈

C

⁢

(

σ

⁢

(

x

)

)

. Then

σ

⁢

(

f

⁢

(

x

)

)

=

f

⁢

(

σ

⁢

(

x

)

)

.

Proof :

Since

C

⁢

(

σ

⁢

(

x

)

)

and

𝒜

⁢

[

x

]

are isomorphic we must have

σ

⁢

(

f

)

=

σ

𝒜

⁢

[

x

]

⁢

(

f

⁢

(

x

)

)

where

σ

𝒜

⁢

[

x

]

⁢

(

f

⁢

(

x

)

)

denotes the spectrum of

f

⁢

(

x

)

relative to the subalgebra

𝒜

⁢

[

x

]

.

By the

spectral invariance theorem

we have

σ

𝒜

⁢

[

x

]

⁢

(

f

⁢

(

x

)

)

=

σ

⁢

(

f

⁢

(

x

)

)

. Hence

σ

⁢

(

f

)

=

σ

⁢

(

f

⁢

(

x

)

)

Thus, we only have to prove that

f

⁢

(

σ

⁢

(

x

)

)

=

σ

⁢

(

f

)

.

f

is defined on

σ

⁢

(

x

)

so

f

⁢

(

σ

⁢

(

x

)

)

is precisely the image of

f

.

Let

λ

∈

ℂ

. The

function

f

-

λ

is

invertible

if and only if

f

-

λ

has no zeros.

Equivalently,

f

-

λ

is not invertible if and only if

f

-

λ

has a zero, i.e.

f

⁢

(

λ

0

)

=

λ

for some

λ

0

.

The previous statement can be reformulated as:

λ

∈

σ

⁢

(

f

)

if and only if

λ

is in the image of

f

.

We conclude that

σ

⁢

(

f

)

=

f

⁢

(

σ

⁢

(

x

)

)

, and this proves the

theorem

.

□

Title

spectral mapping theorem

Canonical name

SpectralMappingTheorem

Date of creation

2013-03-22 17:30:08

Last modified on

2013-03-22 17:30:08

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L05

Classification

msc 47A60

Classification

msc 46H30

Generated on Fri Feb 9 21:43:45 2018 by

LaTeXML
state

state

A

state

Ψ

on a

C

*

-algebra

A

is a

positive linear functional

Ψ

:

A

→

ℂ

,

Ψ

⁢

(

a

*

⁢

a

)

≥

0

for all

a

∈

A

, with unit norm.
The norm of a positive linear functional is defined by

∥

Ψ

∥

=

sup

a

∈

A

⁡

{

|

Ψ

⁢

(

a

)

|

:

∥

a

∥

≤

1

}

.

(1)

For a unital

C

*

-algebra,

∥

Ψ

∥

=

Ψ

⁢

(

1

⁢

I

)

.

The space of states is a convex set.
Let

Ψ

1

and

Ψ

2

be states, then the

convex combination

λ

⁢

Ψ

1

+

(

1

-

λ

)

⁢

Ψ

2

,

λ

∈

[

0

,

1

]

,

(2)

is also a state.

A state is

pure

if it is not a convex combination of two other states.

Pure states

are the extreme points of the convex set of states.
A pure state on a commutative

C

*

-algebra is

equivalent

to a

character

.

A state is called a

tracial state

if it is also a trace.

When a

C

*

-algebra is represented on a

Hilbert space

ℋ

,
every

unit vector

ψ

∈

ℋ

determines a (not necessarily pure) state in the form of an

expectation value

,

Ψ

⁢

(

a

)

=

⟨

ψ

,

a

⁢

ψ

⟩

.

(3)

In physics, it is common to refer to such states by their vector

ψ

rather than the

linear functional

Ψ

.
The

converse

is not always true; not every state need be given by
an expectation value.
For example, delta functions (which are

distributions

not

functions

)
give pure states on

C

0

⁢

(

X

)

,
but they do not correspond to any vector in a Hilbert space
(such a vector would not be square-integrable).

References

1

G. Murphy,

C

*

-Algebras and Operator Theory

.

Academic Press, 1990.

Title

state

Canonical name

State

Date of creation

2013-03-22 13:50:18

Last modified on

2013-03-22 13:50:18

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

8

Author

mhale (572)

Entry type

Definition

Classification

msc 46L05

Related topic

ExtensionAndRestrictionOfStates

Related topic

AlgebraicQuantumFieldTheoriesAQFT

Defines

pure state

Defines

tracial state

Generated on Fri Feb 9 21:43:47 2018 by

LaTeXML
topologically irreducible representations are algebrically irreducible for C^*-algebras

topologically irreducible representations are algebrically irreducible for

C

*

-algebras

Theorem -

Every

topologically irreducible

representation

(

http://planetmath.org/BanachAlgebraRepresentation

) of a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

) is algebraically irreducible.

R

⁢

e

⁢

m

⁢

a

⁢

r

⁢

k

-

It follows easily from the definition that algebraically irreducible of any Banach *-algebra are always topologically irreducible. The above theorem says that for

C

*

-algebras the converse is also true.

Title

topologically irreducible representations are algebrically irreducible for

C

*

-algebras

Canonical name

TopologicallyIrreducibleRepresentationsAreAlgebricallyIrreducibleForCalgebras

Date of creation

2013-03-22 17:27:47

Last modified on

2013-03-22 17:27:47

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L05

Generated on Fri Feb 9 21:43:48 2018 by

LaTeXML
uniform continuity over locally compact quantum groupoids

uniform continuity over locally compact quantum groupoids

0.1

Uniform Continuity over Locally Compact Quantum Groupoids

Let us consider

Locally Compact

Quantum Groupoids

(

L

⁢

C

⁢

Q

⁢

G

⁢

n

) defined as

locally compact groupoids

endowed with a

Haar system

,

ν

,

(

𝒢

,

ν

)

:=

(

[

𝔾

,

G

2

,

μ

]

,

ν

)

, or as derived from a

(

non-commutative

)

weak Hopf algebra

(WHA), with the additional condition of

uniform continuity

over

𝔾

defined as follows . Let us also consider a space

L

⁢

U

⁢

C

⁢

(

𝔾

)

of left uniformly continuous

elements

in

L

∞

⁢

(

𝔾

)

defined over

G

2

, which is endowed with the

induced

product topology

from the subset

G

2

of

composable pairs

in the

topological groupoid

𝔾

. This step

completes

the construction of uniform continuity over

L

⁢

C

⁢

Q

⁢

G

⁢

n

that can be then compared with the results obtained from ‘quantum groupoids’ derived from a weak Hopf algebra.

0.1.1

C*-algebra Comparison and Example

Consider

L

⁢

C

⁢

G

to be a

locally compact quantum group

. Then consider the space

L

⁢

U

⁢

C

⁢

(

G

)

of left uniformly continuous elements in

L

∞

⁢

(

G

)

introduced in ref.

[

2

]

. (The definition according to

V. Runde

(

loc. cit.) covers both the space of left uniformly continuous functions on a

locally compact group

and (Granirer’s) uniformly continuous

functionals

on the Fourier algebra.) Also consider

L

⁢

U

⁢

C

⁢

(

G

)

which is then an

operator system containing the C*-

algebra

C

o

⁢

(

G

)

. One may compare the

groupoid

C*-convolution algebra,

G

C

⁢

A

– obtained in the general case– with the C*-algebra (

http://planetmath.org/CAlgebra3

)

C

o

⁢

(

G

)

obtained from

L

⁢

U

⁢

C

⁢

(

G

)

in the particular case of uniform continuity over a locally compact (

http://planetmath.org/LocallyCompact

) group.

References

1

M. Buneci. 2003.

Groupoid Representations

, Publs: Ed. Mirton, Timishoara.

2

V. Runde. 2008. Uniform continuity over locally compact quantum groups.

http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.2053v4.pdf

(math.OA -arxiv/0802.2053v4).

Title

uniform continuity over locally compact quantum groupoids

Canonical name

UniformContinuityOverLocallyCompactQuantumGroupoids

Date of creation

2013-03-22 18:21:21

Last modified on

2013-03-22 18:21:21

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

24

Author

bci1 (20947)

Entry type

Topic

Classification

msc 46L05

Classification

msc 81R50

Classification

msc 46M20

Classification

msc 18B40

Classification

msc 22A22

Related topic

QuantumGroups

Related topic

QuantumGroupoids2

Related topic

LocallyCompact

Related topic

LocallyCompactGroupoids

Related topic

UniformContinuity

Related topic

RepresentationsOfLocallyCompactGroupoids

Related topic

HomomorphismsOfCAlgebrasAreContinuous

Related topic

CAlgebra

Defines

uniform continuity over topological groupoids

Generated on Fri Feb 9 21:43:50 2018 by

LaTeXML
commutant

commutant

Definition

Let

H

be an

Hilbert Space

,

B

⁢

(

H

)

the

algebra

of

bounded operators

in

H

and

ℱ

⊂

B

⁢

(

H

)

.

The

commutant

of

ℱ

, usually denoted

ℱ

′

, is the subset of

B

⁢

(

H

)

consisting of all
elements that commute with every element of

ℱ

, that is

ℱ

′

=

{

T

∈

B

⁢

(

H

)

:

T

⁢

S

=

S

⁢

T

,

∀

S

∈

ℱ

}

The

double commutant

of

ℱ

is just

(

ℱ

′

)

′

and is usually denoted

ℱ

′′

.

Properties:

•

If

ℱ

1

⊆

ℱ

2

, then

ℱ

2

′

⊆

ℱ

1

′

.

•

ℱ

⊆

ℱ

′′

.

•

If

𝒜

is a

subalgebra

of

B

⁢

(

H

)

, then

𝒜

∩

𝒜

′

is the center (

http://planetmath.org/CenterOfARing

) of

𝒜

.

•

If

ℱ

is

self-adjoint

then

ℱ

′

is self-adjoint.

•

ℱ

′

is always a subalgebra of

B

⁢

(

H

)

that contains the

identity operator

and is closed in the

weak operator topology

.

•

If

ℱ

is self-adjoint then

ℱ

′

is a

von Neumann algebra

.

Remark:

The commutant is a particular case of the more general definition of

centralizer

.

Title

commutant

Canonical name

Commutant

Date of creation

2013-03-22 17:21:53

Last modified on

2013-03-22 17:21:53

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

11

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46L10

Related topic

VonNeumannAlgebra

Defines

double commutant

Generated on Fri Feb 9 21:43:52 2018 by

LaTeXML
commutant is a weak operator closed subalgebra

commutant is a weak operator closed subalgebra

Let

H

be a

Hilbert space

and

B

⁢

(

H

)

the algebra of

bounded operators

in

H

. Recall that the

commutant

of a subset

ℱ

⊂

B

⁢

(

H

)

is the set of all bounded operators that commute with those of

ℱ

, i.e.

ℱ

′

:=

{

T

∈

B

⁢

(

H

)

:

T

⁢

S

=

S

⁢

T

,

∀

S

∈

ℱ

}

.

-

If

ℱ

⊂

B

⁢

(

H

)

, then

ℱ

′

is a

subalgebra

of

B

⁢

(

H

)

that contains the

identity operator

and is closed in the

weak operator topology

.

:

It is clear that

ℱ

′

contains the identity operator, since it commutes with all

operators

in

B

⁢

(

H

)

and in particular with those of

ℱ

.

Let us now see that

ℱ

′

is a subalgebra of

B

⁢

(

H

)

. Let

T

1

,

T

2

∈

ℱ

′

and

λ

∈

ℂ

. We have that, for all

S

∈

ℱ

,

S

⁢

(

T

1

+

T

2

)

=

S

⁢

T

1

+

S

⁢

T

2

=

T

1

⁢

S

+

T

2

⁢

S

=

(

T

1

+

T

2

)

⁢

S

S

⁢

(

λ

⁢

T

1

)

=

λ

⁢

S

⁢

T

1

=

λ

⁢

T

1

⁢

S

S

⁢

(

T

1

⁢

T

2

)

=

T

1

⁢

S

⁢

T

2

=

T

1

⁢

T

2

⁢

S

thus,

T

1

+

T

2

,

λ

⁢

T

1

and

T

1

⁢

T

2

all belong to

ℱ

′

, and therefore

ℱ

′

is a subalgebra of

B

⁢

(

H

)

.

It remains to see that

ℱ

′

is weak operator closed. Suppose

(

T

i

)

is a net in

ℱ

′

that

converges

to

T

in the weak operator topology. Then, for all

x

,

y

∈

H

we have that

⟨

T

i

⁢

x

,

y

⟩

→

⟨

T

⁢

x

,

y

⟩

. Thus, for all

S

∈

ℱ

, we have

⟨

(

T

⁢

S

-

S

⁢

T

)

⁢

x

,

y

⟩

=

⟨

T

⁢

S

⁢

x

,

y

⟩

-

⟨

T

⁢

x

,

S

*

⁢

y

⟩

=

lim

⁡

(

⟨

T

i

⁢

S

⁢

x

,

y

⟩

-

⟨

T

i

⁢

x

,

S

*

⁢

y

⟩

)

=

lim

⁡

⟨

(

T

i

⁢

S

-

S

⁢

T

i

)

⁢

x

,

y

⟩

=

lim

⁡

⟨

(

T

i

⁢

S

-

T

i

⁢

S

)

⁢

x

,

y

⟩

=

0

Hence,

T

⁢

S

-

S

⁢

T

=

0

, so that

T

∈

ℱ

′

. We conclude that

ℱ

′

is closed in the weak operator topology.

□

Title

commutant is a weak operator closed subalgebra

Canonical name

CommutantIsAWeakOperatorClosedSubalgebra

Date of creation

2013-03-22 18:39:32

Last modified on

2013-03-22 18:39:32

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L10

Generated on Fri Feb 9 21:43:55 2018 by

LaTeXML
commutant of B⁢(H) is ℂ⁢I

commutant of

B

⁢

(

H

)

is

ℂ

⁢

I

Let

H

be a

Hilbert space

and

B

⁢

(

H

)

its algebra of

bounded operators

. We denote by

I

the

identity operator

of

B

⁢

(

H

)

and by

ℂ

⁢

I

the set of all multiples of

I

, that is

ℂ

⁢

I

:=

{

λ

⁢

I

:

λ

∈

ℂ

}

. Let

B

⁢

(

H

)

′

denote the

commutant

of

B

⁢

(

H

)

, which is precisely the center of

B

⁢

(

H

)

.

Theorem -

We have that

B

⁢

(

H

)

′

=

ℂ

⁢

I

.

As a particular case, we see that the center of the matrix algebra

M

⁢

a

⁢

t

n

×

n

⁢

(

ℂ

)

consists solely of the multiples of the

identity matrix

, i.e. a matrix in

M

⁢

a

⁢

t

n

×

n

⁢

(

ℂ

)

that commutes with all other matrices is necessarily a multiple of the identity matrix.

:

For each

x

,

y

∈

H

we denote by

T

x

,

y

the

operator

given by

T

x

,

y

⁢

z

:=

⟨

z

,

x

⟩

⁢

y

,

z

∈

H

Let

S

∈

B

⁢

(

H

)

′

. We must have

S

⁢

T

x

,

y

=

T

x

,

y

⁢

S

for all

x

,

y

∈

H

, hence

⟨

z

,

x

⟩

⁢

S

⁢

y

=

⟨

S

⁢

z

,

x

⟩

⁢

y

,

∀

x

,

y

,

z

∈

H

(1)

Choosing a non-zero

x

and taking

z

=

x

, we see that

S

⁢

y

=

⟨

S

⁢

x

,

x

⟩

⟨

x

,

x

⟩

⁢

y

,

y

∈

H

,

x

∈

H

∖

{

0

}

Hence,

⟨

S

⁢

x

,

x

⟩

⟨

x

,

x

⟩

must be constant for all

x

∈

H

∖

{

0

}

. Denote by

λ

∈

ℂ

this constant.

We have that

S

⁢

y

=

λ

⁢

y

for all

y

∈

H

, which simply means that

S

=

λ

⁢

I

. Thus,

B

⁢

(

H

)

′

⊆

ℂ

⁢

I

.

It is clear that the multiples of the identity operator commute with all operators, hence we also have

ℂ

⁢

I

⊆

B

⁢

(

H

)

′

.

We conclude that

B

⁢

(

H

)

′

=

ℂ

⁢

I

.

□

Title

commutant of

B

⁢

(

H

)

is

ℂ

⁢

I

Canonical name

CommutantOfBHIsmathbbCI

Date of creation

2013-03-22 18:39:35

Last modified on

2013-03-22 18:39:35

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L10

Synonym

center of

B

⁢

(

H

)

Generated on Fri Feb 9 21:43:57 2018 by

LaTeXML
von Neumann algebras contain the range projections of its elements

von Neumann algebras contain the range projections of its elements

-

Let

T

be an

operator

in a

von Neumann algebra

ℳ

acting on an

Hilbert space

H

. Then the orthogonal projection onto the range of

T

and the orthogonal projection onto the kernel of

T

both belong to

ℳ

.

Proof :

Let

T

=

V

⁢

R

be the

polar decomposition

of

T

with

K

⁢

e

⁢

r

⁢

V

=

K

⁢

e

⁢

r

⁢

R

.

By the result on the parent entry (

http://planetmath.org/PolarDecompositionInVonNeumannAlgebras

) we see that

V

∈

ℳ

.

As

V

is a

partial isometry

,

V

⁢

V

*

is the ()

projection

onto the range of

T

, and

I

-

V

*

⁢

V

is the () projection onto the kernel of

T

, where

I

is the

identity operator

in

ℳ

.

Therefore the () projections onto the range and kernel of

T

both belong to

ℳ

.

□

Title

von Neumann algebras contain the range projections of its elements

Canonical name

VonNeumannAlgebrasContainTheRangeProjectionsOfItsElements

Date of creation

2013-03-22 17:28:57

Last modified on

2013-03-22 17:28:57

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Result

Classification

msc 46L10

Classification

msc 47A05

Generated on Fri Feb 9 21:44:02 2018 by

LaTeXML
von Neumann algebras of dimension greater than one contain non-trivial projections

von Neumann algebras of dimension greater than one contain non-trivial projections

-

Let

ℳ

be a

von Neumann algebra

of

dimension

greater than one acting on an

Hilbert space

H

. Then

ℳ

contains a non-trivial

projection

, i.e. a projection that is not the zero (

http://planetmath.org/ZeroMap

) the identity operator.

N

⁢

o

⁢

t

⁢

e

- As in the theory of

C

*

-

algebras

, for a projection we mean an orthogonal projection.

Title

von Neumann algebras of dimension greater than one contain non-trivial projections

Canonical

name

VonNeumannAlgebrasOfDimensionGreaterThanOneContainNontrivialProjections

Date of creation

2013-03-22 17:29:00

Last modified on

2013-03-22 17:29:00

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Result

Classification

msc 46L10

Generated on Fri Feb 9 21:44:04 2018 by

LaTeXML
extension and restriction of states

extension and restriction of states

\PMlinkescapephrase

restriction

0.1

Restriction of States

Let

𝒜

be a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

) and

ℬ

⊂

𝒜

a

C

*

-

subalgebra

, both having the same

identity element

.

-

Given a state

ϕ

of

𝒜

, its restriction (

http://planetmath.org/RestrictionOfAFunction

)

ϕ

|

ℬ

to

ℬ

is also a state of

ℬ

.

Remark -

Note that the requirement that the

C

*

-algebras

𝒜

and

ℬ

have a (common) identity element is necessary.

For example, let

X

be a

compact space

and

C

⁢

(

X

)

the

C

*

-algebra of

continuous functions

X

→

ℂ

. Pick a point

x

0

∈

X

and consider the

C

*

-subalgebra of continuous functions

X

→

ℂ

which vanish at

x

0

. Notice that this subalgebra never has the same identity element of

C

⁢

(

X

)

(the

constant function

that equals

1

). In fact, this subalgebra may not have an

identity

at all.

Now the evaluation

mapping

at

x

0

, i.e. the

function

ev

x

0

:

C

⁢

(

X

)

→

ℂ

ev

x

0

⁢

(

f

)

:=

f

⁢

(

x

0

)

is a state of

C

⁢

(

X

)

. Of course, its restriction to the subalgebra in question is the zero mapping, therefore not being a state.

0.2

Extension of States

Let

𝒜

be a

C

*

-algebra and

ℬ

⊂

𝒜

a

C

*

-subalgebra (not necessarily unital).

Theorem

1 -

Every state

ϕ

of

ℬ

admits an

extension

to a state

ϕ

~

of

𝒜

. Moreover, every

pure state

ϕ

of

ℬ

admits an extension to a pure state

ϕ

~

of

𝒜

.

Theorem 2 -

The set of extensions of a state

ϕ

of

ℬ

is a

compact

and

convex subset

of

S

𝒜

, the of

𝒜

endowed with the weak-*

topology

.

Title

extension and restriction of states

Canonical name

ExtensionAndRestrictionOfStates

Date of creation

2013-03-22 18:09:35

Last modified on

2013-03-22 18:09:35

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L30

Related topic

State

Generated on Fri Feb 9 21:44:06 2018 by

LaTeXML
properties of states

properties of states

Let

𝒜

be a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

) and

x

∈

𝒜

.

Let

S

⁢

(

𝒜

)

and

P

⁢

(

𝒜

)

denote the state (

http://planetmath.org/State

) space and the

pure state

space of

𝒜

, respectively.

0.1

States

The space is sufficiently large to reveal many of elements of a

C

*

-algebra.

Theorem 1-

We have that

•

S

⁢

(

𝒜

)

separates points, i.e.

x

=

0

if and only if

ϕ

⁢

(

x

)

=

0

for all

ϕ

∈

S

⁢

(

𝒜

)

.

•

x

is

self-adjoint

(

http://planetmath.org/InvolutaryRing

) if and only if

ϕ

⁢

(

x

)

∈

ℝ

for all

ϕ

∈

S

⁢

(

𝒜

)

.

•

x

is

positive

if and only if

ϕ

⁢

(

x

)

≥

0

for all

ϕ

∈

S

⁢

(

𝒜

)

.

•

If

x

is normal (

http://planetmath.org/InvolutaryRing

), then

ϕ

⁢

(

x

)

=

∥

x

∥

for some

ϕ

∈

S

⁢

(

𝒜

)

.

0.2

Pure states

The pure state space is also sufficiently large to the of Theorem 1. Hence, we can replace

S

⁢

(

𝒜

)

by

P

⁢

(

𝒜

)

, or by any other family of

linear functionals

F

such that

P

⁢

(

𝒜

)

⊂

F

⊂

S

⁢

(

𝒜

)

, in the previous result.

Theorem 2 -

We have that

•

P

⁢

(

𝒜

)

separates points, i.e.

x

=

0

if and only if

ϕ

⁢

(

x

)

=

0

for all

ϕ

∈

P

⁢

(

𝒜

)

.

•

x

is if and only if

ϕ

⁢

(

x

)

∈

ℝ

for all

ϕ

∈

P

⁢

(

𝒜

)

.

•

x

is positive if and only if

ϕ

⁢

(

x

)

≥

0

for all

ϕ

∈

P

⁢

(

𝒜

)

.

•

If

x

is , then

ϕ

⁢

(

x

)

=

∥

x

∥

for some

ϕ

∈

P

⁢

(

𝒜

)

.

-

Every

multiplicative linear functional

on

𝒜

is a pure state.

Title

properties of states

Canonical name

PropertiesOfStates

Date of creation

2013-03-22 17:45:24

Last modified on

2013-03-22 17:45:24

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L30

Classification

msc 46L05

Generated on Fri Feb 9 21:44:11 2018 by

LaTeXML
state space is non-empty

state space is non-empty

In this entry we prove the

existence

of states for every

C

*

-

algebra

(

http://planetmath.org/CALgebra

).

Theorem

-

Let

𝒜

be a

C

*

-algebra. For every

self-adjoint

(

http://planetmath.org/InvolutaryRing

)

element

a

∈

𝒜

there exists a state

ψ

on

𝒜

such that

|

ψ

⁢

(

a

)

|

=

∥

a

∥

.

Proof :

We first consider the case where

𝒜

is unital (

http://planetmath.org/Ring

), with

identity element

e

.

Let

ℬ

be the

C

*

-subalgebra generated by

a

and

e

. Since

a

is self-adjoint,

ℬ

is a comutative

C

*

-algebra with identity element.

Thus, by the

Gelfand-Naimark theorem

,

ℬ

is

isomorphic

to

C

⁢

(

X

)

, the space of

continuous functions

X

⟶

ℂ

for some

compact set

X

.

Regarding

a

as an

element of

C

⁢

(

X

)

,

a

attains a maximum at a point

x

0

∈

X

, since

X

is compact. Hence,

∥

a

∥

=

|

a

⁢

(

x

0

)

|

.

The evaluation

function

at

x

0

,

e

⁢

v

x

0

:

C

⁢

(

X

)

⟶

ℂ

e

⁢

v

x

0

⁢

(

f

)

:=

f

⁢

(

x

0

)

is a

multiplicative linear functional

of

C

⁢

(

X

)

. Hence,

∥

e

⁢

v

x

0

∥

=

1

and also

|

e

⁢

v

x

0

⁢

(

a

)

|

=

|

a

⁢

(

x

0

)

|

=

∥

a

∥

.

We can now extend

e

⁢

v

x

0

to a

linear functional

ψ

on

𝒜

such that

∥

ψ

∥

=

∥

e

⁢

v

x

0

∥

=

1

, using the

Hahn-Banach theorem

.

Also,

ψ

⁢

(

e

)

=

e

⁢

v

x

0

⁢

(

e

)

=

1

and so

ψ

is a norm one

positive linear functional

, i.e.

ψ

is a state on

𝒜

.

Of course,

ψ

is such that

|

ψ

⁢

(

a

)

|

=

|

e

⁢

v

x

0

⁢

(

a

)

|

=

∥

a

∥

.

In case

𝒜

does not have an identity element we can consider its

minimal unitization

𝒜

~

. By the preceding there is a state

ψ

~

on

𝒜

~

satisfying the required . Now, we just need to take the

restriction

(

http://planetmath.org/RestrictionOfAFunction

) of

ψ

~

to

𝒜

and this restriction is a state in

𝒜

satisfying the required .

□

Title

state space is non-empty

Canonical name

StateSpaceIsNonempty

Date of creation

2013-03-22 17:45:14

Last modified on

2013-03-22 17:45:14

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

14

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L30

Classification

msc 46L05

Generated on Fri Feb 9 21:44:13 2018 by

LaTeXML