L^1⁢(G) is a Banach *-algebra

L

1

⁢

(

G

)

is a Banach *-algebra

0.1

The Banach *-algebra

L

1

⁢

(

ℝ

)

.

Consider the

Banach space

L

1

⁢

(

ℝ

)

(

http://planetmath.org/LpSpace

), i.e. the space of

Borel measurable functions

f

:

ℝ

⟶

ℂ

such that

∥

f

∥

1

:=

∫

ℝ

|

f

⁢

(

x

)

|

⁢

𝑑

x

<

∞

identified up to

equivalence

almost everywhere.

The

convolution product

of

functions

f

,

g

∈

L

1

⁢

(

ℝ

)

, given by

(

f

*

g

)

⁢

(

z

)

=

∫

ℝ

f

⁢

(

x

)

⁢

g

⁢

(

z

-

x

)

⁢

𝑑

x

,

is a well-defined

product

in

L

1

⁢

(

ℝ

)

, i.e.

f

*

g

∈

L

1

⁢

(

ℝ

)

, that

satisfies

the inequality

∥

f

*

g

∥

1

≤

∥

f

∥

1

⁢

∥

g

∥

1

.

Therefore, with the convolution product,

L

1

⁢

(

ℝ

)

is a

Banach algebra

.

Moreover, we can define an

involution

(

http://planetmath.org/InvolutaryRing

) in

L

1

⁢

(

ℝ

)

by

f

*

⁢

(

x

)

=

f

⁢

(

-

x

)

¯

. With this involution

L

1

⁢

(

ℝ

)

is Banach *-

algebra

.

0.2

Generalization to

L

1

⁢

(

G

)

.

Let

G

be a

locally compact topological group

and

μ

its

left Haar measure

. Consider the space

L

1

⁢

(

G

)

(

http://planetmath.org/LpSpace

) consisting of

measurable functions

f

:

G

⟶

ℂ

such that

∥

f

∥

1

:=

∫

G

|

f

|

⁢

𝑑

μ

<

∞

identified up to equivalence almost everywhere.

The convolution product of functions

f

,

g

∈

L

1

⁢

(

G

)

, given by

(

f

*

g

)

⁢

(

s

)

=

∫

G

f

⁢

(

t

)

⁢

g

⁢

(

t

-

1

⁢

s

)

⁢

𝑑

μ

⁢

(

t

)

,

is a well-defined product in

L

1

⁢

(

G

)

, i.e.

f

*

g

∈

L

1

⁢

(

G

)

, that satisfies the inequality

∥

f

*

g

∥

1

≤

∥

f

∥

1

⁢

∥

g

∥

1

.

Therefore, with this convolution product,

L

1

⁢

(

G

)

is a Banach algebra.

An involution can also be defined in

L

1

⁢

(

G

)

by

f

*

⁢

(

s

)

=

Δ

G

⁢

(

s

-

1

)

⁢

f

⁢

(

s

-

1

)

¯

, where

Δ

G

is the

modular function

of

G

.

With this product and involution

L

1

⁢

(

G

)

is a Banach *-algebra.

0.3

Commutative case: the group algebra.

The algebras

L

1

⁢

(

G

)

are

commutative

if and only if the group

G

is commutative.

Commutative groups are of course unimodular (

http://planetmath.org/UnimodularGroup2

), hence

Δ

G

⁢

(

s

)

=

1

for all

s

∈

G

.

So in the commutative case the convolution product and involution are given, respectively, by

(

f

*

g

)

⁢

(

s

)

=

∫

G

f

⁢

(

t

)

⁢

g

⁢

(

s

-

t

)

⁢

𝑑

μ

⁢

(

t

)

f

*

⁢

(

s

)

=

f

⁢

(

-

s

)

¯

and

L

1

⁢

(

G

)

is called the

group algebra

of

G

.

For

finite groups

, the group algebra defined as above coincides with the group algebra

ℂ

⁢

(

G

)

(

http://planetmath.org/GroupRing

).

0.4

An equivalent construction

In the construction of

L

1

⁢

(

G

)

presented above we are considering

equivalence classes

of measurable functions on

G

with respect to the Haar measure. To avoid this kind of

measure

theoretic considerations it is sometimes better to work with another () definition of

L

1

⁢

(

G

)

:

Let

C

c

⁢

(

G

)

be the space of

continuous functions

G

⟶

ℂ

with compact support. We can endow this space with a convolution product, an involution and a norm by setting

(

f

*

g

)

⁢

(

s

)

=

∫

G

f

⁢

(

t

)

⁢

g

⁢

(

t

-

1

⁢

s

)

⁢

𝑑

μ

⁢

(

t

)

f

*

⁢

(

s

)

=

Δ

G

⁢

(

s

-

1

)

⁢

f

⁢

(

s

-

1

)

¯

∥

f

∥

1

=

∫

G

|

f

|

⁢

𝑑

μ

With this

operations

and norm,

C

c

⁢

(

G

)

has a normed *-algebra and

L

1

⁢

(

G

)

can be defined as its

completion

.

Title

L

1

⁢

(

G

)

is a Banach *-algebra

Canonical name

L1GIsABanachalgebra

Date of creation

2013-03-22 17:42:14

Last modified on

2013-03-22 17:42:14

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

15

Author

asteroid (17536)

Entry type

Example

Classification

msc 46K05

Classification

msc 46H05

Classification

msc 44A35

Classification

msc 43A20

Classification

msc 22D05

Classification

msc 22A10

Related topic

DualGroupOfGIsHomeomorphicToTheCharacterSpaceOfL1G

Related topic

ConvolutionsOfComplexFunctionsOnLocallyCompactGroups

Defines

L

1

⁢

(

ℝ

)

is a Banach *-algebra

Defines

group algebra

Generated on Fri Feb 9 21:42:43 2018 by

LaTeXML