L1(G) is a Banach *-algebra

0.1 The Banach *-algebra L1().

Consider the Banach spaceMathworldPlanetmath L1() (http://planetmath.org/LpSpace), i.e. the space of Borel measurable functions f: such that


identified up to equivalence almost everywhere.

The convolution productPlanetmathPlanetmath of functions f,gL1(), given by


is a well-defined productPlanetmathPlanetmath in L1(), i.e. f*gL1(), that satisfies the inequality


Therefore, with the convolution product, L1() is a Banach algebraMathworldPlanetmath.

Moreover, we can define an involutionPlanetmathPlanetmath (http://planetmath.org/InvolutaryRing) in L1() by f*(x)=f(-x)¯. With this involution L1() is Banach *-algebra.

0.2 Generalization to L1(G).

Let G be a locally compact topological group and μ its left Haar measure. Consider the space L1(G) (http://planetmath.org/LpSpace) consisting of measurable functions f:G such that


identified up to equivalence almost everywhere.

The convolution product of functions f,gL1(G), given by


is a well-defined product in L1(G), i.e. f*gL1(G), that satisfies the inequality


Therefore, with this convolution product, L1(G) is a Banach algebra.

An involution can also be defined in L1(G) by f*(s)=ΔG(s-1)f(s-1)¯, where ΔG is the modular function of G.

With this product and involution L1(G) is a Banach *-algebra.

0.3 Commutative case: the group algebra.

The algebras L1(G) are commutativePlanetmathPlanetmathPlanetmath if and only if the group G is commutative.

Commutative groups are of course unimodular (http://planetmath.org/UnimodularGroup2), hence ΔG(s)=1 for all sG.

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

(f*g)(s) = Gf(t)g(s-t)𝑑μ(t)
f*(s) = f(-s)¯

and L1(G) is called the group algebraPlanetmathPlanetmath of G.

For finite groupsMathworldPlanetmath, the group algebra defined as above coincides with the group algebra (G) (http://planetmath.org/GroupRing).

0.4 An equivalent construction

In the construction of L1(G) presented above we are considering equivalence classesMathworldPlanetmathPlanetmath of measurable functions on G with respect to the Haar measure. To avoid this kind of measureMathworldPlanetmath theoretic considerations it is sometimes better to work with another () definition of L1(G):

Let Cc(G) be the space of continuous functionsPlanetmathPlanetmath G with compact support. We can endow this space with a convolution product, an involution and a norm by setting

(f*g)(s) = Gf(t)g(t-1s)𝑑μ(t)
f*(s) = ΔG(s-1)f(s-1)¯
f1 = G|f|𝑑μ

With this operationsMathworldPlanetmath and norm, Cc(G) has a normed *-algebra and L1(G) can be defined as its completion.

Title L1(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 L1() is a Banach *-algebra
Defines group algebra