$L^1(G)$ 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

identified up to equivalence almost everywhere.

The convolution product of functions $f,g\in L^1(ℝ)$, given by

$$(f*g)(z)={\int }_{ℝ}f(x)g(z-x)𝑑x,$$

is a well-defined product in $L^1(ℝ)$, i.e. $f*g\in L^1(ℝ)$, that satisfies the inequality

$${\parallel f*g\parallel }_1\le {\parallel f\parallel }_1{\parallel g\parallel }_1.$$

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 $f^{*}(x)=\overline{f(-x)}$. 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 $\mu$ its left Haar measure. Consider the space $L^1(G)$ (http://planetmath.org/LpSpace) consisting of measurable functions $f:G⟶ℂ$ such that

identified up to equivalence almost everywhere.

The convolution product of functions $f,g\in L^1(G)$, given by

$$(f*g)(s)={\int }_{G}f(t)g(t^{-1}s)𝑑\mu (t),$$

is a well-defined product in $L^1(G)$, i.e. $f*g\in L^1(G)$, that satisfies the inequality

$${\parallel f*g\parallel }_1\le {\parallel f\parallel }_1{\parallel g\parallel }_1.$$

Therefore, with this convolution product, $L^1(G)$ is a Banach algebra.

An involution can also be defined in $L^1(G)$ by $f^{*}(s)={\mathrm{\Delta }}_G(s^{-1})\overline{f(s^{-1})}$, where ${\mathrm{\Delta }}_G$ 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 ${\mathrm{\Delta }}_G(s)=1$ for all $s\in G$.

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

$(f*g)(s)$ $=$ ${\displaystyle {\int }_G}f(t)g(s-t)𝑑\mu (t)$

$f^{*}(s)$ $=$ $\overline{f(-s)}$

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

$(f*g)(s)$ $=$ ${\displaystyle {\int }_G}f(t)g(t^{-1}s)𝑑\mu (t)$

$f^{*}(s)$ $=$ ${\mathrm{\Delta }}_G(s^{-1})\overline{f(s^{-1})}$

${\parallel f\parallel }_1$ $=$ ${\displaystyle {\int }_G}|f|𝑑\mu$

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