Banach *-algebra representation

Banach *-algebra representation

Definition:

A

representation

of a Banach *-

algebra

𝒜

is a *-

homomorphism

π

:

𝒜

⟶

ℬ

⁢

(

H

)

of

𝒜

into the *-algebra of

bounded operators

on some

Hilbert space

H

.

The set of all representations of

𝒜

on a Hilbert space

H

is denoted

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

)

.

Special kinds of representations:

•

A

subrepresentation

of a representation

π

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

)

is a representation

π

0

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

0

)

obtained from

π

by restricting to a closed

π

⁢

(

𝒜

)

-

invariant subspace

(

http://planetmath.org/InvariantSubspace

)

1

1

by a

π

⁢

(

𝒜

)

- we a

subspace

which is

invariant

under every

operator

π

⁢

(

a

)

with

a

∈

𝒜

H

0

⊆

H

.

•

A representation

π

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

)

is said to be

nondegenerate

if one of the following

equivalent

conditions hold:

(a)

π

⁢

(

x

)

⁢

ξ

=

0

∀

x

∈

𝒜

⟹

ξ

=

0

, where

ξ

∈

H

.

(b)

H

is the closed

linear span

of the set of vectors

π

⁢

(

𝒜

)

⁢

H

:=

{

π

⁢

(

x

)

⁢

ξ

:

x

∈

𝒜

,

ξ

∈

H

}

•

A representation

π

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

)

is said to be

topologically irreducible

(or just ) if the only closed

π

⁢

(

𝒜

)

-invariant of

H

are the

trivial

ones,

{

0

}

and

H

.

•

A representation

π

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

)

is said to be

algebrically irreducible

if the only

π

⁢

(

𝒜

)

-invariant of

H

(not necessarily closed) are the trivial ones,

{

0

}

and

H

.

•

Given two representations

π

1

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

1

)

and

π

2

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

2

)

, the of

π

1

and

π

2

is the representation

π

1

⊕

π

2

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

1

⊕

H

2

)

given by

π

1

⊕

π

2

⁢

(

x

)

:=

π

1

⁢

(

x

)

⊕

π

2

⁢

(

x

)

,

x

∈

𝒜

.

More generally, given a family

{

π

i

}

i

∈

I

of representations, with

π

i

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

i

)

, their is the representation

⊕

i

∈

I

π

i

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

⊕

i

∈

I

H

i

)

, in the

direct sum of Hilbert spaces

⊕

i

∈

I

H

i

, such that

(

⊕

i

∈

I

π

i

)

⁢

(

x

)

:=

⊕

i

∈

I

π

i

⁢

(

x

)

is the

direct sum

of the family of bounded operators (

http://planetmath.org/DirectSumOfBoundedOperatorsOnHilbertSpaces

)

{

π

i

⁢

(

x

)

}

i

∈

I

.

•

Two representations

π

1

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

1

)

and

π

2

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

2

)

of a Banach *-algebra

𝒜

are said to be

unitarily equivalent

if there is a

unitary

U

:

H

1

⟶

H

2

such that

π

2

⁢

(

a

)

=

U

⁢

π

1

⁢

(

a

)

⁢

U

*

∀

a

∈

𝒜

•

A representation

π

∈

r

⁢

e

⁢

p

⁢

(

𝒜

,

H

)

is said to be if there exists a vector

ξ

∈

H

such that the set

π

⁢

(

A

)

⁢

ξ

:=

{

π

⁢

(

a

)

⁢

ξ

:

a

∈

𝒜

}

is dense (

http://planetmath.org/Dense

) in

H

. Such a vector is called a

cyclic vector

for the representation

π

.

Linked file: http://aux.planetmath.org/files/objects/9843/BanachAlgebraRepresentation.pdf

Title

Banach *-algebra representation

Canonical name

BanachalgebraRepresentation

Date of creation

2013-03-22 17:27:37

Last modified on

2013-03-22 17:27:37

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

23

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46H15

Classification

msc 46K10

Defines

subrepresentation

Defines

cyclic representation

Defines

cyclic vector

Defines

nondegenerate representation

Defines

topologically irreducible

Defines

algebrically irreducible

Defines

direct sum of representations

Defines

unitarily equivalent

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