aarjaneiro/planetmath huggingface dataset:
injective C^*-algebra homomorphism is isometric

injective

C

*

-algebra homomorphism is isometric

Theorem

-

Let

𝒜

and

ℬ

be

C

*

-

algebras

(

http://planetmath.org/CAlgebra

) and

Φ

:

𝒜

⟶

ℬ

an

injective

*-

homomorphism

. Then

∥

Φ

⁢

(

x

)

∥

=

∥

x

∥

and

σ

⁢

(

Φ

⁢

(

x

)

)

=

σ

⁢

(

x

)

for every

x

∈

𝒜

, where

σ

⁢

(

y

)

denotes the spectrum of the

element

y

.

Proof:

It suffices to prove the result for unital

C

*

-algebras, since the general case follows directly by considering the

minimal unitizations

of

𝒜

and

ℬ

. So we assume that

𝒜

and

ℬ

are unital and we will denote their

identity elements

by

e

, being clear from

context

which one is being used.

Let us first prove the second part of the theorem for

normal elements

x

∈

𝒜

. It is clear that

σ

⁢

(

Φ

⁢

(

x

)

)

⊆

σ

⁢

(

x

)

since if

x

-

λ

⁢

e

invertible

for some

λ

∈

𝒞

, then so is

Φ

⁢

(

x

)

-

λ

⁢

e

=

Φ

⁢

(

x

-

λ

⁢

e

)

. Suppose the

inclusion

is strict, then there is a non-zero

function

f

∈

C

⁢

(

σ

⁢

(

x

)

)

whose

restriction

to

σ

⁢

(

Φ

⁢

(

x

)

)

is zero (here

C

⁢

(

σ

⁢

(

x

)

)

denotes the

C

*

-algebra of

continuous functions

σ

⁢

(

x

)

⟶

ℂ

). Thus we have, by the

continuous functional calculus

, that

f

⁢

(

x

)

≠

0

and also that

Φ

⁢

(

f

⁢

(

x

)

)

=

f

⁢

(

Φ

⁢

(

x

)

)

=

0

by the continuous functional calculus and the result on this entry (

http://planetmath.org/CAlgebraHomomorphismsPreserveContinuousFunctionalCalculus

). Thus, we conclude that

Φ

is not injective and which is a

contradiction

. Hence we must have

σ

⁢

(

Φ

⁢

(

x

)

)

=

σ

⁢

(

x

)

.

Let

R

σ

⁢

(

z

)

denote the spectral radius of the element

z

. From the norm and spectral radius

relation

in

C

*

-algebras (

http://planetmath.org/NormAndSpectralRadiusInCAlgebras

) we know that, for an arbitrary element

x

∈

𝒜

, we have that

∥

x

∥

2

=

R

σ

⁢

(

x

*

⁢

x

)

Since the element

x

*

⁢

x

is normal, from the preceding paragraph it follows that

R

σ

⁢

(

x

*

⁢

x

)

=

R

σ

⁢

(

Φ

⁢

(

x

*

⁢

x

)

)

, and hence we conclude that

∥

x

∥

2

=

R

σ

⁢

(

x

*

⁢

x

)

=

R

σ

⁢

(

Φ

⁢

(

x

)

*

⁢

Φ

⁢

(

x

)

)

=

∥

Φ

⁢

(

x

)

∥

2

i.e.

∥

Φ

⁢

(

x

)

∥

=

∥

x

∥

.

Since

Φ

is

isometric

,

Φ

⁢

(

𝒜

)

is closed *-

subalgebra

of

ℬ

, i.e.

Φ

⁢

(

𝒜

)

is a

C

*

-subalgebra of

ℬ

, and it is

isomorphic

to

𝒜

. Using the

spectral invariance theorem

we conclude that

σ

⁢

(

x

)

=

σ

⁢

(

Φ

⁢

(

x

)

)

for every

x

∈

𝒜

.

□

Title

injective

C

*

-algebra homomorphism is isometric

Canonical name

InjectiveCalgebraHomomorphismIsIsometric

Date of creation

2013-03-22 18:00:35

Last modified on

2013-03-22 18:00:35

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L05

Generated on Fri Feb 9 21:43:19 2018 by

LaTeXML
maximal ideals of the algebra of continuous functions on a compact set

maximal ideals of the algebra of continuous functions on a compact set

Let

X

be a

compact

Hausdorff space

and

C

⁢

(

X

)

the

Banach algebra

of

continuous functions

X

⟶

ℂ

(with the sup norm).

In this entry we are interested in identifying the

maximal ideals

and the

character space

of

C

⁢

(

X

)

. Since

C

⁢

(

X

)

is a Banach algebra with an identity element, there is a

bijective

correspondence between the character space of

C

⁢

(

X

)

and the set of maximal ideals of this

algebra

, given by

ϕ

⟷

K

⁢

e

⁢

r

⁢

ϕ

Hence, by identifying the character space of

C

⁢

(

X

)

we are able to identify its maximal ideals.

Theorem

1 -

Let

Δ

be the character space of

C

⁢

(

X

)

. For each

x

∈

X

let

e

⁢

v

x

∈

Δ

be the point-evaluation at

x

, i.e.

e

⁢

v

x

⁢

(

f

)

=

f

⁢

(

x

)

,

f

∈

C

⁢

(

X

)

Then the

mapping

x

⟼

e

⁢

v

x

is an

homeomorphism

between

Δ

and

X

.

Thus, the character space of

C

⁢

(

X

)

is homeomorphic to

X

via point-evaluations.

Now, the maximal ideals of

C

⁢

(

X

)

correspond to the kernels of the point-evaluation

functions

. The kernel of

e

⁢

v

x

, the point-evaluation at

x

, is just

{

f

∈

C

(

X

)

:

f

(

x

)

=

0

}

i.e., the functions that vanish at

x

.

Thus, each maximal ideal of

C

⁢

(

X

)

is just the set of functions that vanish in a given point.

0.1

Generalization to locally compact Hausdorff spaces

Now, let

X

be a

locally compact Hausdorff space

and

C

0

⁢

(

X

)

the space of continuous functions

X

⟶

ℂ

that

vanish at infinity

.

There is a

generalization

of Theorem 1 above that allows one to identify the character space of

C

0

⁢

(

X

)

, but since this algebra is not unital unless

X

is compact, we cannot identify its maximal ideals by the above method.

Theorem 2-

Let

Δ

be the character space of

C

0

⁢

(

X

)

. For each

x

∈

X

let

e

⁢

v

x

∈

Δ

be the point-evaluation at

x

, i.e.

e

⁢

v

x

⁢

(

f

)

=

f

⁢

(

x

)

,

f

∈

C

0

⁢

(

X

)

Then the mapping

x

⟼

e

⁢

v

x

is an homeomorphism between

Δ

and

X

.

Thus, the character space of

C

0

⁢

(

X

)

is also homeomorphic to

X

via point-evaluations.

Title

maximal ideals of the algebra of continuous functions on a compact set

Canonical name

MaximalIdealsOfTheAlgebraOfContinuousFunctionsOnACompactSet

Date of creation

2013-03-22 17:44:57

Last modified on

2013-03-22 17:44:57

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L05

Classification

msc 46J20

Classification

msc 46J10

Classification

msc 16W80

Synonym

character space of the algebra of continuous functions on a compact set

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

LaTeXML
minimal unitizations of algebras with additional structure

minimal unitizations of algebras with additional structure

Given a (non-unital)

algebra

(

http://planetmath.org/Algebra

) there is a procedure to add an unit to it (parent entry (

http://planetmath.org/

Unitization

)). When the algebra has some additional structure (topological structure, for example), it is often useful to endow the same structure on the minimal unitization of the algebra.

All the algebras are to be considered non-unital.

0.1

Topological Algebras

Let

𝒜

be a

topological algebra

algebra over a (topological) field

𝕂

. Let

𝒜

~

be its minimal unitization.

Then

𝒜

~

=

𝒜

⊕

𝕂

is a topological algebra with the product topology.

0.2

Normed and Banach Algebras

Let

𝒜

be a normed algebra over

𝕂

(

=

ℝ

or

ℂ

) with norm

∥

⋅

∥

. Let

𝒜

~

be its minimal unitization.

Then

𝒜

~

is a normed algebra under the norm

∥

⋅

∥

u

:

∥

a

+

λ

∥

u

=

∥

a

∥

+

|

λ

|

,

a

∈

𝒜

,

λ

∈

𝕂

Moreover, if

𝒜

is a

Banach algebra

, then

𝒜

~

is a Banach algebra with the norm

∥

⋅

∥

u

.

0.3

*-algebras

Let

𝒜

be a *-algebra over an involutory field (

http://planetmath.org/InvolutaryRing

)

𝕂

. Let

𝒜

~

be its minimal unitization.

Then

𝒜

~

is a *-algebra with

involution

given by:

(

a

+

λ

)

*

=

a

*

+

λ

¯

a

∈

𝒜

,

λ

∈

𝕂

0.4

Topological *-algebras, Normed *-algebras and Banach *-algebras

Let

𝒜

be a topological *-algebra over

ℂ

. Let

𝒜

~

be its minimal unitization.

Then

𝒜

~

is a topological *-algebra with the product topology and the involution defined above.

Also, if

𝒜

is a normed *-algebra (Banach -*algebra), then

𝒜

~

is also a normed *-algebra (Banach *-algebra) under the above involution and the norm

∥

⋅

∥

u

.

0.5

C*-algebras

Let

𝒜

be a

C

*

-algebra (

http://planetmath.org/CAlgebra

) with norm

∥

⋅

∥

. Let

𝒜

~

be its minimal unitization.

Then

𝒜

~

is

C

*

-algebra under the norm

∥

⋅

∥

L

:

∥

a

+

λ

∥

L

=

sup

∥

b

∥

=

1

⁡

∥

a

⁢

b

+

λ

⁢

b

∥

,

a

∈

𝒜

,

λ

∈

ℂ

This norm comes from regarding elements of

𝒜

~

as left on

𝒜

. The norm

∥

⋅

∥

L

is to the norm

∥

⋅

∥

u

.

Title

minimal unitizations of algebras with additional structure

Canonical name

MinimalUnitizationsOfAlgebrasWithAdditionalStructure

Date of creation

2013-03-22 17:46:29

Last modified on

2013-03-22 17:46:29

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Result

Classification

msc 46L05

Classification

msc 46K05

Classification

msc 46H05

Classification

msc 16W99

Classification

msc 16B99

Defines

minimal unitization of a topological algebra

Defines

minimal unitization of a Banach algebra

Defines

minimal unitization of a Banach-* algebra

Defines

minimal unitization of a

C

*

-algebra

Generated on Fri Feb 9 21:43:23 2018 by

LaTeXML
norm and spectral radius in C^*-algebras

norm and spectral radius in

C

*

-algebras

Let

𝒜

be a

C

*

-algebra (

http://planetmath.org/CAlgebra

). Let

R

σ

⁢

(

a

)

denote the

spectral radius

of the element

a

∈

𝒜

.

Theorem -

For every

a

∈

𝒜

we have that

∥

a

∥

=

R

σ

⁢

(

a

*

⁢

a

)

.

This result shows that the norm in a

C

*

-algebra has a purely nature. Moreover, the norm in a

C

*

-algebra is unique (in the sense that there is no other norm for which the algebra is a

C

*

-algebra).

Title

norm and spectral radius in

C

*

-algebras

Canonical name

NormAndSpectralRadiusInCalgebras

Date of creation

2013-03-22 17:38:41

Last modified on

2013-03-22 17:38:41

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L05

Related topic

HomomorphismsOfCAlgebrasAreContinuous

Related topic

CAlgebra

Generated on Fri Feb 9 21:43:25 2018 by

LaTeXML
ordering of self-adjoints

ordering of self-adjoints

Let

𝒜

be a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

). Let

𝒜

+

denote the set of

positive elements

of

𝒜

and

𝒜

s

⁢

a

denote the set of

self-adjoint elements

of

𝒜

.

Since

𝒜

+

is a proper convex cone (

http://planetmath.org/Cone5

) (see this entry (

http://planetmath.org/PositiveElement3

)), we can define a

partial order

≤

on the set

𝒜

s

⁢

a

, by setting

a

≤

b

if and only if

b

-

a

∈

𝒜

+

, i.e.

b

-

a

is

positive

.

Theorem

-

The

relation

≤

is a partial order relation on

𝒜

s

⁢

a

. Moreover,

≤

turns

𝒜

s

⁢

a

into an

ordered topological vector space

.

0.0.1

Properties:

•

a

≤

b

⇒

c

*

⁢

a

⁢

c

≤

c

*

⁢

b

⁢

c

for every

c

∈

𝒜

.

•

If

a

and

b

are

invertible

and

a

≤

b

, then

b

-

1

≤

a

-

1

.

•

If

𝒜

has an

identity element

e

, then

-

∥

a

∥

⁢

e

≤

a

≤

∥

a

∥

⁢

e

for every

a

∈

𝒜

s

⁢

a

.

•

-

b

≤

a

≤

b

⇒

∥

a

∥

≤

∥

b

∥

.

0.0.2

Remark:

The proof that

≤

is partial order makes no use of the self-adjointness . In fact,

𝒜

itself is an ordered topological vector space under the relation

≤

.

However, it turns out that this

ordering relation

provides its most usefulness when restricted to self-adjoint elements. For example, some of the above would not hold if we did not restrict to

𝒜

s

⁢

a

.

Title

ordering of self-adjoints

Canonical name

OrderingOfSelfadjoints

Date of creation

2013-03-22 17:30:37

Last modified on

2013-03-22 17:30:37

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L05

Generated on Fri Feb 9 21:43:27 2018 by

LaTeXML
positive element

positive element

Let

H

be a complex

Hilbert space

. Let

T

:

H

⟶

H

be a

bounded operator

in

H

.

Definition -

T

is said to be a

positive operator

if there exists a bounded operator

A

:

H

⟶

H

such that

T

=

A

*

⁢

A

where

A

*

denotes the

adjoint

of

A

.

Every positive operator

T

satisfies

the very strong condition

⟨

T

⁢

v

,

v

⟩

≥

0

for every

v

∈

H

since

⟨

T

⁢

v

,

v

⟩

=

⟨

A

*

⁢

A

⁢

v

,

v

⟩

=

⟨

A

⁢

v

,

A

⁢

v

⟩

=

∥

A

⁢

v

∥

2

≥

0

The

converse

is also true, although it is not so to prove. Indeed,

Theorem

-

T

is

positive

if and only if

⟨

T

⁢

v

,

v

⟩

≥

0

∀

v

∈

H

0.1

Generalization to

C

*

-algebras

The above notion can be generalized to

elements

in an arbitrary

C

*

-algebra (

http://planetmath.org/CAlgebra

).

In what follows

𝒜

denotes a

C

*

-algebra.

Definition -

An element

x

∈

𝒜

is said to be

positive

(and denoted

0

≤

x

) if

x

=

a

*

⁢

a

for some element

a

∈

𝒜

.

R

⁢

e

⁢

m

⁢

a

⁢

r

⁢

k

-

Positive elements

are clearly

self-adjoint

(

http://planetmath.org/InvolutaryRing

).

0.2

Positive spectrum

It can be shown that the positive elements of

𝒜

are precisely the

normal elements

of

𝒜

with a positive spectrum. We it here as a theorem:

Theorem -

Let

x

∈

𝒜

and

σ

⁢

(

x

)

denote its spectrum. Then

x

is positive if and only if

x

is and

σ

⁢

(

x

)

⊂

ℝ

0

+

.

0.3

Square roots

Positive elements admit a unique positive square root.

Theorem -

Let

x

be a positive element in

𝒜

. There is a unique

b

∈

𝒜

such that

•

b

is positive

•

x

=

b

2

.

The converse is also true (with

assumptions

): If

x

admits a square root then

x

is positive, since

x

=

b

2

=

b

⁢

b

=

b

*

⁢

b

0.4

The positive cone

Another interesting fact about positive elements is that they form a proper convex cone (

http://planetmath.org/Cone5

) in

𝒜

, usually called the

positive cone

of

𝒜

. That is stated in following theorem:

Theorem -

Let

a

,

b

be positive elements in

𝒜

. Then

•

a

+

b

is also positive

•

λ

⁢

a

is also positive for every

λ

≥

0

•

If both

a

and

-

a

are positive then

a

=

0

.

0.5

Norm closure

Theorem -

The set of positive elements in

𝒜

is norm closed.

Title

positive element

Canonical name

PositiveElement

Date of creation

2013-03-22 17:30:31

Last modified on

2013-03-22 17:30:31

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

8

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46L05

Classification

msc 47L07

Classification

msc 47A05

Synonym

positive

Defines

positive operator

Defines

positive cone

Defines

square root of positive element

Generated on Fri Feb 9 21:43:29 2018 by

LaTeXML
positive linear functional

positive linear functional

0.0.1

Definition

Let

𝒜

be a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

) and

ϕ

a

linear functional

on

𝒜

.

We say that

ϕ

is a

positive linear functional

on

𝒜

if

ϕ

is such that

ϕ

⁢

(

x

)

≥

0

for every

x

≥

0

, i.e. for every

positive element

x

∈

𝒜

.

0.0.2

Properties

Let

ϕ

be a positive linear functional on

𝒜

. Then

•

ϕ

⁢

(

x

*

)

=

ϕ

⁢

(

x

)

¯

for every

x

∈

𝒜

.

•

|

ϕ

⁢

(

x

*

⁢

y

)

|

2

≤

ϕ

⁢

(

x

*

⁢

x

)

⁢

ϕ

⁢

(

y

*

⁢

y

)

for every

x

,

y

∈

𝒜

. This is an analog of the Cauchy-Schwartz inequality

Let

ϕ

be a linear functional on a

C

*

-algebra

𝒜

with

identity element

e

. Then

•

ϕ

is positive if and only if

ϕ

is

bounded

(

http://planetmath.org/ContinuousLinearMapping

) and

∥

ϕ

∥

=

ϕ

⁢

(

e

)

.

0.0.3

Examples

•

Let

X

be a

locally compact Hausdorff space

and

C

0

⁢

(

X

)

the

C

*

-algebra of

continuous functions

X

⟶

ℂ

that

vanish at infinity

. Let

μ

be a

regular

Radon measure

on

X

. The linear functional

ϕ

defined by integration against

μ

,

ϕ

⁢

(

f

)

:=

∫

X

f

⁢

𝑑

μ

,

f

∈

C

0

⁢

(

x

)

is a positive linear functional on

C

0

⁢

(

X

)

. In fact, by the

Riesz representation theorem

(

http://planetmath.org/RieszRepresentationTheoremOfLinearFunctionalsOnFunctionSpaces

), all positive linear functionals of

C

0

⁢

(

X

)

are of this form.

Title

positive linear functional

Canonical name

PositiveLinearFunctional

Date of creation

2013-03-22 17:45:05

Last modified on

2013-03-22 17:45:05

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

11

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46L05

Generated on Fri Feb 9 21:43:31 2018 by

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proof of Gelfand-Naimark representation theorem

proof of Gelfand-Naimark representation theorem

\PMlinkescapephrase

representation

\PMlinkescapephrase

representations

Proof:

Let

𝒜

be a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

). We intend to prove that

𝒜

is

isometrically isomorphic

to a norm closed *-

subalgebra

of

B

⁢

(

H

)

, the algebra of

bounded operators

of a suitable

Hilbert space

H

.

Let

S

⁢

(

𝒜

)

denote the

state space

of

𝒜

. For every state

ϕ

∈

S

⁢

(

𝒜

)

the

Gelfand-Naimark-Segal construction

allows one to

construct

a representation (

http://planetmath.org/BanachAlgebraRepresentation

)

π

ϕ

:

𝒜

⟶

B

⁢

(

H

ϕ

)

of

𝒜

in a Hilbert space

H

ϕ

.

Now consider the

direct sum

(

http://planetmath.org/BanachAlgebraRepresentation

) of these representations

π

:=

⊕

ϕ

∈

S

⁢

(

𝒜

)

π

ϕ

. Recall that

π

is a representation

π

:

𝒜

⟶

B

⁢

(

⊕

ϕ

∈

S

⁢

(

𝒜

)

H

ϕ

)

of

𝒜

in the direct sum of the family of Hilbert spaces (

http://planetmath.org/DirectSumOfHilbertSpaces

)

{

H

ϕ

}

ϕ

∈

S

⁢

(

𝒜

)

.

We now prove that this representation is

injective

.

Suppose there exists

a

∈

𝒜

such that

π

⁢

(

a

)

=

0

. Then, for all

ϕ

∈

S

⁢

(

𝒜

)

,

π

ϕ

⁢

(

a

)

=

0

. Thus, by definition of

π

ϕ

,

ϕ

⁢

(

a

)

=

⟨

π

ϕ

⁢

(

a

)

⁢

ξ

ϕ

,

ξ

ϕ

⟩

=

0

where

ξ

ϕ

∈

H

ϕ

is the

cyclic vector

associated with

π

ϕ

. Since for every

ϕ

∈

S

⁢

(

𝒜

)

we have

ϕ

⁢

(

a

)

=

0

, we can conclude that

a

=

0

(see this entry (

http://planetmath.org/PropertiesOfStates

)), i.e.

π

is injective.

Since an injective *-

homomorphism

between

C

*

-algebras is isometric (

http://planetmath.org/InjectiveCAlgebraHomomorphismIsIsometric

), we conclude that

π

is also isometric. Hence

π

⁢

(

𝒜

)

is a closed *-subalgebra of

B

⁢

(

⊕

ϕ

∈

S

⁢

(

𝒜

)

H

ϕ

)

. Thus, we have proven that

π

is an isometric isomorphism between

𝒜

and a closed *-subalgebra of

B

⁢

(

H

)

, for a suitable Hilbert space

H

.

□

Title

proof of Gelfand-Naimark representation theorem

Canonical name

ProofOfGelfandNaimarkRepresentationTheorem

Date of creation

2013-03-22 18:01:17

Last modified on

2013-03-22 18:01:17

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Proof

Classification

msc 46L05

Related topic

GelfandNaimarkRepresentationTheorem

Generated on Fri Feb 9 21:43:33 2018 by

LaTeXML
quotients in C^*-algebras

quotients in

C

*

-algebras

Theorem

-

Let

𝒜

be a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

) and

ℐ

⊆

𝒜

a closed (

http://planetmath.org/ClosedSet

) ideal. Then the involution (

http://planetmath.org/InvolutaryRing

) in

𝒜

induces a

well-defined

involution in

𝒜

/

ℐ

and

𝒜

/

ℐ

is a

C

*

-algebra with this involution and the quotient norm.

Title

quotients

in

C

*

-algebras

Canonical

name

QuotientsInCalgebras

Date of creation

2013-03-22 17:41:39

Last modified on

2013-03-22 17:41:39

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L05

Related topic

NuclearCAlgebra

Generated on Fri Feb 9 21:43:40 2018 by

LaTeXML
special elements in a C^*-algebra and their spectral properties

special elements in a

C

*

-algebra and their spectral properties

Definition

-

Suppose

𝒜

is a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

). An element

x

∈

𝒜

is said to be:

•

normal

if

x

*

⁢

x

=

x

⁢

x

*

•

self-adjoint

if

x

*

=

x

•

unitary

if

𝒜

has an

identity element

e

and

x

*

⁢

x

=

x

⁢

x

*

=

e

•

positive

if

x

=

y

*

⁢

y

for some element

y

∈

𝒜

•

projection

if

x

*

=

x

and

x

2

=

x

•

partial isometry

if

x

*

⁢

x

and

x

⁢

x

*

are both projections

0.0.1

Properties of the special elements in terms of their spectrum

In the following

σ

⁢

(

x

)

denotes the

spectrum

of an element

x

and

R

σ

⁢

(

x

)

its

spectral radius

.

Theorem

1 -

Suppose

𝒜

is a

C

*

-algebra and

x

∈

𝒜

. If

x

is normal then

∥

x

∥

=

R

σ

⁢

(

x

)

Theorem 2 -

Suppose

𝒜

is a

C

*

-algebra and

x

∈

𝒜

.

•

If

x

is self-adjoint, then

σ

⁢

(

x

)

⊂

ℝ

.

•

If

x

is unitary, then

σ

⁢

(

x

)

⊂

∂

⁡

D

, where

D

⊂

ℂ

is the unit disk.

•

If

x

is positive, then

σ

⁢

(

x

)

⊂

ℝ

+

•

If

x

is a projection, then

σ

⁢

(

x

)

⊂

{

0

,

1

}

Theorem 3 -

Suppose

𝒜

is a

commutative

C

*

-algebra and

x

∈

𝒜

. Then

•

x

is self-adjoint if and only if

σ

⁢

(

x

)

⊂

ℝ

.

•

x

is unitary if and only if

σ

⁢

(

x

)

⊂

∂

⁡

D

, where

D

⊂

ℂ

is the unit disk.

•

x

is positive if and only if

σ

⁢

(

x

)

⊂

ℝ

+

•

x

is a projection if and only if

σ

⁢

(

x

)

⊂

{

0

,

1

}

Theorem 4 -

Suppose

𝒜

is a

C

*

-algebra and

x

is normal in

𝒜

. Then

•

x

is self-adjoint if and only if

σ

⁢

(

x

)

⊂

ℝ

.

•

x

is unitary if and only if

σ

⁢

(

x

)

⊂

∂

⁡

D

, where

D

⊂

ℂ

is the unit disk.

•

x

is positive if and only if

σ

⁢

(

x

)

⊂

ℝ

+

•

x

is a projection if and only if

σ

⁢

(

x

)

⊂

{

0

,

1

}

Title

special elements in a

C

*

-algebra and their spectral

properties

Canonical

name

SpecialElementsInACalgebraAndTheirSpectralProperties

Date of creation

2013-03-22 17:28:36

Last modified on

2013-03-22 17:28:36

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

11

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46L05

Defines

normal elements and spectral radius

Defines

spectrum of self-adjoint elements

Defines

spectrum of unitary elements

Defines

spectrum of projections

Defines

spectrum of positive elements

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

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