aarjaneiro/planetmath huggingface dataset:
Uniform Algebra

Uniform Algebra

Definition

.

A

commutative

, unital

Banach algebra

(

𝒜

,

∥

⋅

∥

)

is called

uniform Banach algebra

(for short: uB

algebra

) if for all

f

∈

𝒜

we have

∥

f

2

∥

=

∥

f

∥

2

In what follows we will show that the

Gelfand transform

Γ

𝒜

:

𝒜

↦

𝒜

^

of a commutative, unital Banach algebra

𝒜

is an isometry if and only if

𝒜

is a uniform Banach algebra.

Denote by

M

⁢

(

𝒜

)

the space of (

continuous

)

characters

on

𝒜

.
Recall that for all

f

∈

𝒜

the spectrum

σ

⁢

(

f

)

of

f

is identical with the range

f

^

⁢

(

M

⁢

(

𝒜

)

)

and the spectral radius

r

⁢

(

f

)

=

∥

f

^

∥

M

⁢

(

𝒜

)

=

lim

n

→

∞

⁡

∥

f

n

∥

1

n

Proposition

1.

A Banach algebra

𝒜

is uniform if and only if it’s Gelfand transform

Γ

𝒜

:

𝒜

→

𝒜

^

,

f

↦

f

^

is isometric.

Proof.

If

f

↦

f

^

is an isometry we have

∥

f

2

∥

=

∥

f

^

2

∥

M

⁢

(

𝒜

)

=

∥

f

^

∥

M

⁢

(

𝒜

)

2

=

∥

f

∥

2

.

Conversely

assume

∥

f

2

∥

=

∥

f

∥

2

for all

f

∈

𝒜

. Then by

induction

we have

∥

f

2

n

∥

=

∥

f

∥

2

n

for all

n

∈

ℕ

. Hence

∥

f

∥

=

∥

f

2

n

∥

1

2

n

→

∥

f

^

∥

M

⁢

(

𝒜

)

.
∎

The following

characterization

is also often given as the

definition

of a uB algebra.

Proposition 2.

A Banach algebra

𝒜

is uniform iff it is topologically and algebraically isomorphic to a closed, pointseparating

subalgebra

of

C

⁢

(

X

)

for

X

a

compact

Hausdorff space.

Proof.

Since

𝒜

^

separates the points of the compact, nonempty space

M

⁢

(

𝒜

)

we see that a uB algebra

𝒜

must have this

property

.

Conversely, let

𝒜

be a closed pointseparating subalgebra of

C

⁢

(

X

)

. Then clearly

∥

f

∥

X

2

=

∥

f

2

∥

X

for all

f

∈

𝒜

.
∎

References

(Gamelin 2005)

Theodore W. Gamelin

Uniform Algebras

, Oxford University Press, New Edition, 2005

Title

Uniform Algebra

Canonical

name

UniformAlgebra

Date of creation

2013-03-22 19:04:20

Last modified on

2013-03-22 19:04:20

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

5

Author

karstenb (16623)

Entry type

Definition

Classification

msc 46J40

Classification

msc 46J10

Generated on Fri Feb 9 21:42:35 2018 by

LaTeXML
L^1⁢(G) has an approximate identity

L

1

⁢

(

G

)

has an approximate identity

Let

G

be a

locally compact topological group

. In general, the Banach *-

algebra

L

1

⁢

(

G

)

(parent entry (

http://planetmath.org/L1GIsABanachAlgebra

)) does not have an identity element. In fact:

-

L

1

⁢

(

G

)

has an identity element if and only if

G

is discrete.

When

G

is discrete the identity element of

L

1

⁢

(

G

)

is just the Dirac delta, i.e. the function that takes the value

1

on the identity element of

G

and vanishes everywhere else.

Nevertheless,

L

1

⁢

(

G

)

has always an

approximate identity

.

Theorem -

L

1

⁢

(

G

)

has an approximate identity

(

e

λ

)

λ

∈

Λ

. Moreover the approximate identity

(

e

λ

)

λ

∈

Λ

can be chosen to the following :

•

e

λ

is self-adjoint (

http://planetmath.org/InvolutaryRing

),

•

∥

e

λ

∥

1

=

1

,

•

e

λ

∈

C

c

⁢

(

G

)

where

C

c

⁢

(

G

)

stands for the space of

continuous functions

G

⟶

ℂ

with compact support.

Title

L

1

⁢

(

G

)

has an approximate identity

Canonical name

L1GHasAnApproximateIdentity

Date of creation

2013-03-22 17:42:40

Last modified on

2013-03-22 17:42:40

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

9

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46K05

Classification

msc 43A20

Classification

msc 22D05

Classification

msc 22A10

Defines

L

1

⁢

(

G

)

has an identity element iff

G

is discrete

Generated on Fri Feb 9 21:42:40 2018 by

LaTeXML
criterion for a Banach *-algebra representation to be irreducible

criterion for a Banach *-algebra representation to be irreducible

Theorem -

Let

𝒜

be a Banach *-

algebra

,

H

an

Hilbert space

and

I

the identity operator in

H

. A

representation

(

http://planetmath.org/BanachAlgebraRepresentation

)

π

:

𝒜

⟶

H

is

topologically irreducible

if and only if

π

⁢

(

𝒜

)

′

=

ℂ

⁢

I

, i.e. if and only if the

commutant

of

π

⁢

(

𝒜

)

consists of

scalar multiples

of the identity operator.

Proof :

(

⟹

)

As

π

⁢

(

𝒜

)

is selfadjoint,

π

⁢

(

𝒜

)

′

is a

von Neumann algebra

.

Suppose

π

⁢

(

𝒜

)

′

≠

ℂ

⁢

I

. Then the

dimension

of

π

⁢

(

𝒜

)

′

is greater than one.

It is known that

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

, so there is a

projection

P

∈

π

⁢

(

𝒜

)

′

such that

P

≠

0

and

P

≠

I

.

As

P

∈

π

⁢

(

𝒜

)

′

,

P

commutes with every operator

T

∈

π

⁢

(

𝒜

)

, that is

P

⁢

T

=

T

⁢

P

.

Thus

R

⁢

a

⁢

n

⁢

P

is an

invariant subspace

of every

T

∈

π

⁢

(

𝒜

)

. Therefore

π

is not an

irreducible representation

.

(

⟸

)

Conversely, suppose that

π

is not an irreducible representation. There exists a closed

π

⁢

(

𝒜

)

-invariant subspace different from

{

0

}

and

H

.

Let

P

be the projection onto that closed invariant subspace.

Invariance can be expressed as:

π

⁢

(

a

)

⁢

P

=

P

⁢

π

⁢

(

a

)

⁢

P

for every

a

∈

𝒜

. It follows that

P

⁢

π

⁢

(

a

)

=

(

π

⁢

(

a

)

*

⁢

P

)

*

=

(

π

⁢

(

a

*

)

⁢

P

)

*

=

(

P

⁢

π

⁢

(

a

*

)

⁢

P

)

*

=

P

⁢

π

⁢

(

a

*

)

*

⁢

P

=

P

⁢

π

⁢

(

a

)

⁢

P

=

π

⁢

(

a

)

⁢

P

for every

a

∈

𝒜

.

We conclude that

P

commutes with every element of

π

⁢

(

𝒜

)

, i.e.

P

∈

π

⁢

(

𝒜

)

′

.

Thus

π

⁢

(

𝒜

)

′

≠

ℂ

⁢

I

⁢

□

Title

criterion for a Banach *-algebra representation to be

irreducible

Canonical name

CriterionForABanachalgebraRepresentationToBeIrreducible

Date of creation

2013-03-22 17:27:43

Last modified on

2013-03-22 17:27:43

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

9

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46K10

Generated on Fri Feb 9 21:42:45 2018 by

LaTeXML
representations of Banach *-algebras are continuous

representations of Banach *-algebras are continuous

Theorem -

Let

π

:

𝒜

⟶

ℬ

⁢

(

H

)

be a representation (

http://planetmath.org/BanachAlgebraRepresentation

) of a Banach *-algebra

𝒜

on a

Hilbert space

H

. Then

π

is

bounded

(

http://planetmath.org/BoundedOperator

) and

∥

π

∥

≤

1

.

Title

representations of Banach *-algebras are

continuous

Canonical name

RepresentationsOfBanachalgebrasAreContinuous

Date of creation

2013-03-22 17:27:41

Last modified on

2013-03-22 17:27:41

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46K10

Generated on Fri Feb 9 21:42:47 2018 by

LaTeXML
H⁢i⁢l⁢b category of Hilbert spaces

H

⁢

i

⁢

l

⁢

b

category of Hilbert spaces

Definition 0.1

.

The

category

H

⁢

i

⁢

l

⁢

b

f

of

finite-dimensional

Hilbert spaces

is defined as the category whose objects are all finite-dimensional Hilbert spaces

ℋ

f

, and whose

morphisms

are

linear maps

between

ℋ

f

spaces.
The

isomorphisms

in

ℋ

⁢

i

⁢

l

⁢

b

f

are all

isometric isomorphisms

.

Furthermore, one also has the following, general definition for any Hilbert space.

Definition 0.2

.

The

category

H

⁢

i

⁢

l

⁢

b

of Hilbert spaces

is defined as the category whose objects are all Hilbert spaces

ℋ

, and whose morphisms are linear maps between

ℋ

spaces.
The

isomorphisms

in

ℋ

⁢

i

⁢

l

⁢

b

are all isometric isomorphisms.

Remark 0.1

.

The category of

ℋ

⁢

i

⁢

l

⁢

b

Hilbert spaces has

direct sums

and is a Cartesian category.

Title

H

⁢

i

⁢

l

⁢

b

category of Hilbert spaces

Canonical name

HilbCategoryOfHilbertSpaces

Date of creation

2013-03-22 18:25:10

Last modified on

2013-03-22 18:25:10

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

10

Author

bci1 (20947)

Entry type

Definition

Classification

msc 46K15

Classification

msc 46C05

Classification

msc 46C50

Classification

msc 46C15

Classification

msc 46E20

Classification

msc 18-00

Synonym

H

⁢

i

⁢

l

⁢

b

Related topic

DirectSumOfHilbertSpaces

Related topic

ClassificationOfHilbertSpaces

Related topic

IndexOfCategories

Defines

isomorphisms in

H

⁢

i

⁢

l

⁢

b

Defines

Hilbert space morphisms

Generated on Fri Feb 9 21:42:49 2018 by

LaTeXML
dual group of G is homeomorphic to the character space of L^1⁢(G)

dual group of

G

is homeomorphic to the character space of

L

1

⁢

(

G

)

Let

G

be a

locally compact

abelian

(

http://planetmath.org/AbelianGroup2

) group (

http://planetmath.org/TopologicalGroup

) and

L

1

⁢

(

G

)

its

group algebra

.

Let

G

^

denote the

Pontryagin dual

of

G

and

Δ

the

character space

of

L

1

⁢

(

G

)

, i.e. the set of multiplicative linear functionals of

L

1

⁢

(

G

)

endowed with the weak-*

topology

.

Theorem -

The spaces

G

^

and

Δ

are

homeomorphic

. The homeomorphism is given by

ω

⟼

ϕ

ω

,

ω

∈

G

^

where

ϕ

ω

∈

Δ

is defined by

ϕ

ω

⁢

(

f

)

:=

∫

G

f

⁢

(

s

)

⁢

ω

⁢

(

s

)

⁢

𝑑

μ

⁢

(

s

)

,

f

∈

L

1

⁢

(

G

)

Title

dual group of

G

is homeomorphic to the character space of

L

1

⁢

(

G

)

Canonical name

DualGroupOfGIsHomeomorphicToTheCharacterSpaceOfL1G

Date of creation

2013-03-22 17:42:49

Last modified on

2013-03-22 17:42:49

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46K99

Classification

msc 43A40

Classification

msc 43A20

Classification

msc 22D20

Classification

msc 22D15

Classification

msc 22D35

Classification

msc 22B10

Classification

msc 22B05

Related topic

L1GIsABanachAlgebra

Generated on Fri Feb 9 21:42:51 2018 by

LaTeXML
C^*-algebra homomorphisms have closed images

C

*

-algebra homomorphisms have closed images

Theorem

-

Let

f

:

𝒜

⟶

ℬ

be a *-

homomorphism

between the

C

*

-

algebras

(

http://planetmath.org/CAlgebra

)

𝒜

and

ℬ

. Then

f

has closed (

http://planetmath.org/ClosedSet

) image (

http://planetmath.org/

Function

), i.e.

f

⁢

(

𝒜

)

is closed in

ℬ

.

Thus, the image

f

⁢

(

𝒜

)

is a

C

*

-

subalgebra

of

ℬ

.

Proof:

The kernel of

f

,

Ker

⁢

f

, is a closed

two-sided ideal

of

𝒜

, since

f

is

continuous

(see this entry (

http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous

)). Factoring threw the quotient

C

*

-algebra

𝒜

/

Ker

⁢

f

we obtain an

injective

*-homomorphism

f

~

:

𝒜

/

Ker

⁢

f

⟶

ℬ

.

Injective *-homomorphisms between

C

*

-algebras are known to be isometric (see this entry (

http://planetmath.org/InjectiveCAlgebraHomomorphismIsIsometric

)), hence the image

f

~

⁢

(

𝒜

/

Ker

⁢

f

)

is closed in

ℬ

.

Since the images

f

~

⁢

(

𝒜

/

Ker

⁢

f

)

and

f

⁢

(

𝒜

)

coincide we conclude that

f

⁢

(

𝒜

)

is closed in

ℬ

.

□

Title

C

*

-algebra homomorphisms have closed images

Canonical name

CalgebraHomomorphismsHaveClosedImages

Date of creation

2013-03-22 17:44:37

Last modified on

2013-03-22 17:44:37

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

9

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L05

Synonym

image of

C

*

-homomorphism is a

C

*

-algebra

Generated on Fri Feb 9 21:43:02 2018 by

LaTeXML
C^*-algebras have approximate identities

C

*

-algebras have approximate identities

In this entry

≤

has three different meanings:

1.

- The

ordering of self-adjoint

elements

(

http://planetmath.org/OrderingOfSelfAdjoints

) of a given

C

*

-algebra (

http://planetmath.org/CAlgebra

).

2.

- The usual order (

http://planetmath.org/PartialOrder

) in

ℝ

.

3.

- The of a

directed set

taken as the domain of a given net.

It will be clear from the

context

which one is being used.

Theorem

-

Every

C

*

-algebra has an

approximate identity

(

e

λ

)

λ

∈

Λ

. Moreover, the approximate identity

(

e

λ

)

λ

∈

Λ

can be chosen to the following :

•

0

≤

e

λ

∀

λ

∈

Λ

•

∥

e

λ

∥

≤

1

∀

λ

∈

Λ

•

λ

≤

μ

⇒

e

λ

≤

e

μ

, i.e.

(

e

λ

)

λ

∈

Λ

is increasing.

For

separable

(

http://planetmath.org/Separable

)

C

*

-algebras the approximate identity can be chosen as an

increasing sequence

0

≤

e

1

≤

e

2

≤

…

of norm-one elements.

Title

C

*

-algebras have approximate identities

Canonical name

CalgebrasHaveApproximateIdentities

Date of creation

2013-03-22 17:30:40

Last modified on

2013-03-22 17:30:40

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L05

Generated on Fri Feb 9 21:43:04 2018 by

LaTeXML
closed ideals in C^*-algebras are self-adjoint

closed ideals in

C

*

-algebras are self-adjoint

Theorem -

Every closed (

http://planetmath.org/ClosedSet

)

two-sided ideal

(

http://planetmath.org/IdealOfAnAlgebra

)

ℐ

of a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

)

𝒜

is

self-adjoint

(

http://planetmath.org/InvolutaryRing

), i.e.

if

x

∈

ℐ

then

x

*

∈

ℐ

.

Proof :

Let

ℐ

*

:=

{

a

*

:

a

∈

ℐ

}

.

Since

ℐ

is closed and the involution mapping is

continuous

, it follows that

ℐ

*

is also closed.

We claim that

ℐ

*

is also a of

𝒜

. To see this let

a

,

b

∈

ℐ

,

x

∈

𝒜

and

λ

∈

ℂ

. Then

•

a

*

+

λ

⁢

b

*

=

(

a

+

λ

¯

⁢

b

)

*

∈

ℐ

*

since

a

+

λ

¯

⁢

b

∈

ℐ

•

x

⁢

a

*

=

(

a

⁢

x

*

)

*

∈

ℐ

*

since

a

⁢

x

*

∈

ℐ

.

•

a

*

⁢

x

=

(

x

*

⁢

a

)

*

∈

ℐ

*

since

x

*

⁢

a

∈

ℐ

Let

ℬ

:=

ℐ

∩

ℐ

*

.

ℬ

is a

C

*

-

subalgebra

of

𝒜

(it is a norm-closed, involution-closed, subalgebra of

𝒜

).

It is known that every

C

*

-algebra has an

approximate identity

consisting of

positive elements

with norm less than

1

(see this entry (

http://planetmath.org/CAlgebrasHaveApproximateIdentities

)).

Let

(

e

λ

)

λ

∈

Λ

be an approximate identity for

ℬ

with the above :

1.

each

e

λ

is

positive

(hence self-adjoint) and

2.

∥

e

λ

∥

≤

1

⁢

∀

λ

∈

Λ

We now prove

ℐ

is self-adjoint:

Let

a

∈

ℐ

. We have that

∥

a

*

-

a

*

⁢

e

λ

∥

2

=

∥

(

a

*

-

a

*

⁢

e

λ

)

*

⋅

(

a

*

-

a

*

⁢

e

λ

)

∥

=

∥

(

a

-

e

λ

⁢

a

)

⋅

(

a

*

-

a

*

⁢

e

λ

)

∥

=

∥

(

a

⁢

a

*

-

a

⁢

a

*

⁢

e

λ

)

-

e

λ

⁢

(

a

⁢

a

*

-

a

⁢

a

*

⁢

e

λ

)

∥

≤

∥

a

⁢

a

*

-

a

⁢

a

*

⁢

e

λ

∥

+

∥

e

λ

∥

⋅

∥

a

⁢

a

*

-

a

⁢

a

*

⁢

e

λ

∥

≤

∥

a

⁢

a

*

-

a

⁢

a

*

⁢

e

λ

∥

+

∥

a

⁢

a

*

-

a

⁢

a

*

⁢

e

λ

∥

=

2

⁢

∥

a

⁢

a

*

-

a

⁢

a

*

⁢

e

λ

∥

Taking limits in both we obtain

lim

λ

⁡

∥

a

*

-

a

*

⁢

e

λ

∥

2

≤

lim

λ

⁡

2

⁢

∥

a

⁢

a

*

-

a

⁢

a

*

⁢

e

λ

∥

=

0

since

a

⁢

a

*

∈

ℐ

∩

ℐ

*

=

ℬ

and

(

e

λ

)

λ

∈

Λ

is an approximate identity for

ℬ

.

As

e

λ

∈

ℐ

we see that

a

*

⁢

e

λ

∈

ℐ

.

We conclude from the limit above that

a

*

is in the closure of

ℐ

. Therefore

a

*

∈

ℐ

.

Hence,

ℐ

is self-adjoint.

□

Title

closed ideals in

C

*

-algebras are self-adjoint

Canonical name

ClosedIdealsInCalgebrasAreSelfadjoint

Date of creation

2013-03-22 17:30:42

Last modified on

2013-03-22 17:30:42

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

9

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46L05

Classification

msc 46H10

Generated on Fri Feb 9 21:43:09 2018 by

LaTeXML
equivalence of definitions of C^*-algebra

equivalence of definitions of

C

*

-algebra

In this entry, we will prove that the

definitions

of

C

*

algebra

given in the
main entry are

equivalent

.

Theorem 1

.

A

Banach algebra

A

with an antilinear involution

*

such that

∥

a

∥

2

≤

∥

a

*

⁢

a

∥

for all

a

∈

A

is a

C

*

-algebra.

Proof.

It follows from the

product

inequality

∥

a

⁢

b

∥

≤

∥

a

∥

⁢

∥

b

∥

that

∥

a

∥

2

≤

∥

a

*

⁢

a

∥

≤

∥

a

*

∥

⁢

∥

a

∥

.

Therefore,

∥

a

∥

≤

∥

a

*

∥

. Putting

a

*

for

a

, we also have

∥

a

*

∥

≤

∥

a

*

*

∥

=

∥

a

∥

. Thus, the involution is an isometry:

∥

a

∥

=

∥

a

*

∥

.
So now,

∥

a

∥

2

≤

∥

a

*

⁢

a

∥

≤

∥

a

∥

2

.

Hence,

∥

a

*

⁢

a

∥

=

∥

a

∥

2

.
∎

Theorem 2

.

A Banach algebra

A

with an antilinear involution

*

such that

∥

a

*

⁢

a

∥

=

∥

a

*

∥

⁢

∥

a

∥

is a

C

*

-algebra.

Title

equivalence of definitions of

C

*

-algebra

Canonical

name

EquivalenceOfDefinitionsOfCalgebra

Date of creation

2013-03-22 17:42:27

Last modified on

2013-03-22 17:42:27

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 46L05

Related topic

HomomorphismsOfCAlgebrasAreContinuous

Related topic

CAlgebra

Generated on Fri Feb 9 21:43:11 2018 by

LaTeXML
example of Banach algebra which is not a C^*-algebra for any involution

example of Banach algebra which is not a

C

*

-algebra for any involution

Consider the Banach algebra

𝒜

=

{

[

λ

⁢

I

n

A

0

λ

⁢

I

n

]

:

λ

∈

ℂ

,

A

∈

M

⁢

a

⁢

t

n

×

n

⁢

(

ℂ

)

}

with the usual

matrix operations

and

matrix norm

, where

I

n

denotes the

identity matrix

in

M

⁢

a

⁢

t

n

×

n

⁢

(

ℂ

)

.

Claim -

𝒜

is not a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

) for any

involution

*

.

To prove the above claim we will give a proof of a more general fact about

finite dimensional

C

*

-algebras, which clearly shows the for a Banach algebra to be a

C

*

-algebra for some involution.

Theorem -

Every finite dimensional

C

*

-algebra is

semi-simple

, i.e. its

Jacobson radical

is

{

0

}

.

Proof :

Let

ℬ

be a finite dimensional

C

*

-algebra. Let

a

be an element of

J

⁢

(

ℬ

)

, the Jacobson radical of

ℬ

.

J

⁢

(

ℬ

)

is an ideal of

ℬ

, so

a

*

⁢

a

∈

J

⁢

(

ℬ

)

.

The Jacobson radical of a finite dimensional algebra is

nilpotent

, therefore there exists

n

∈

ℕ

such that

(

a

*

⁢

a

)

n

=

0

. Then, by the

C

*

condition and the fact that

a

*

⁢

a

is selfadjoint,

0

=

∥

(

a

*

⁢

a

)

2

n

∥

=

∥

a

*

⁢

a

∥

2

n

=

∥

a

∥

2

n

+

1

so

a

=

0

and

J

⁢

(

ℬ

)

is trivial.

□

We now prove the above claim.

Proof of the claim:

It is easy to see that

{

[

0

A

0

0

]

:

A

∈

M

⁢

a

⁢

t

n

×

n

⁢

(

ℂ

)

}

is the only

maximal ideal

of

𝒜

. Therefore the Jacobson radical of

𝒜

is not trivial.

By the theorem we conclude that there is no involution

*

that makes

𝒜

into a

C

*

-algebra.

□

Remark -

It could happen that there were no involutions in

𝒜

and so the above claim would be uninteresting. That’s not the case here. For example, one can see that

[

a

i

,

j

]

⟶

[

a

¯

2

⁢

n

+

1

-

j

,

2

⁢

n

+

1

-

i

]

defines an involution in

𝒜

(this is just the taken over the other

diagonal

of the matrix).

Title

example of Banach algebra which is not a

C

*

-algebra for any involution

Canonical name

ExampleOfBanachAlgebraWhichIsNotACalgebraForAnyInvolution

Date of creation

2013-03-22 17:25:54

Last modified on

2013-03-22 17:25:54

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Example

Classification

msc 46L05

Defines

finite dimensional

C

*

-algebras are semi-simple

Generated on Fri Feb 9 21:43:13 2018 by

LaTeXML
finite quantum group

finite quantum group

Definition 0.1

.

A

finite quantum group

Q

G

⁢

f

is a pair

(

ℍ

,

Φ

)

of a finite-dimensional

C

*

-

algebra

ℍ

with a comultiplication

Φ

such that

(

ℍ

,

Φ

)

is a Hopf

*

-algebra.

Note that one obtains the dual

Hopf algebra

of a

commutative

, finite quantum group via
Fourier

transformation

of the group’s

elements

.

References

1

Abe, E.,

Hopf Algebras

, Cambridge University Press, 1977.

2

Sweedler, M. E.,

Hopf Algebras

, W.A. Benjamin, inc., New York, 1969.

3

Kustermans, J., Van Daele, A., C*-

algebraic

Quantum Groups

arising from
Algebraic Quantum Groups,

Int. J. of Math.

8 (1997), 1067-1139.

4

Lance, E.C., An explicit description of the fundamental

unitary

for

S

⁢

U

⁢

(

2

)

q

,

Commun. Math. Phys.

164 (1994), 1-15.

Title

finite quantum group

Canonical name

FiniteQuantumGroup

Date of creation

2013-03-22 18:24:10

Last modified on

2013-03-22 18:24:10

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

17

Author

bci1 (20947)

Entry type

Definition

Classification

msc 46L05

Classification

msc 81R15

Classification

msc 81R50

Synonym

quantum group

Synonym

dual of a finite Hopf algebra

Related topic

CompactQuantumGroup

Related topic

HopfAlgebra

Related topic

GrassmanHopfAlgebrasAndTheirDualCoAlgebras

Related topic

CompactMatrixQuantumGroup

Defines

comultiplication in a quantum group

Defines

dual of a finite Hopf algebra

Generated on Fri Feb 9 21:43:15 2018 by

LaTeXML
Gelfand-Naimark representation theorem

Gelfand-Naimark representation theorem

The

Gelfand-Naimark representation theorem

is as follows:

Theorem 1.1

Every

C

*

-

algebra

is

isometrically isomorphic

to a norm closed *-

subalgebra

B

n

⁢

c

⁢

(

H

)

of an algebra

B

⁢

(

H

)

of

bounded operators

on some

Hilbert space

H

. In particular, every

finite dimensional

C

*

-algebra is

isomorphic

to a

direct sum of matrix

algebras.

Title

Gelfand-Naimark representation theorem

Canonical name

GelfandNaimarkRepresentationTheorem

Date of creation

2013-03-22 12:57:58

Last modified on

2013-03-22 12:57:58

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

12

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 46L05

Related topic

ProofOfGelfandNaimarkRepresentationTheorem

Generated on Fri Feb 9 21:43:17 2018 by

LaTeXML