# $L^{1}(G)$ is a Banach *-algebra

## 0.1 The Banach *-algebra $L^{1}(\mathbb{R})$.

Consider the Banach space $L^{1}(\mathbb{R})$ (http://planetmath.org/LpSpace), i.e. the space of Borel measurable functions $f:\mathbb{R}\longrightarrow\mathbb{C}$ such that

 $\|f\|_{1}:=\int_{\mathbb{R}}|f(x)|\;dx<\infty$

identified up to equivalence almost everywhere.

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

 $(f*g)(z)=\int_{\mathbb{R}}f(x)g(z-x)dx,$

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

 $\|f*g\|_{1}\leq\|f\|_{1}\|g\|_{1}\;.$

Therefore, with the convolution product, $L^{1}(\mathbb{R})$ is a Banach algebra.

Moreover, we can define an involution (http://planetmath.org/InvolutaryRing) in $L^{1}(\mathbb{R})$ by $f^{*}(x)=\overline{f(-x)}$. With this involution $L^{1}(\mathbb{R})$ 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 $f:G\longrightarrow\mathbb{C}$ such that

 $\|f\|_{1}:=\int_{G}|f|\;d\mu<\infty$

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

 $\|f*g\|_{1}\leq\|f\|_{1}\|g\|_{1}\;.$

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

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

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

 $\displaystyle(f*g)(s)$ $\displaystyle=$ $\displaystyle\int_{G}f(t)g(s-t)\;d\mu(t)$ $\displaystyle f^{*}(s)$ $\displaystyle=$ $\displaystyle\overline{f(-s)}$

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 $\mathbb{C}(G)$ (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 $G\longrightarrow\mathbb{C}$ with compact support. We can endow this space with a convolution product, an involution and a norm by setting

 $\displaystyle(f*g)(s)$ $\displaystyle=$ $\displaystyle\int_{G}f(t)g(t^{-1}s)\;d\mu(t)$ $\displaystyle f^{*}(s)$ $\displaystyle=$ $\displaystyle\Delta_{G}(s^{-1})\overline{f(s^{-1})}$ $\displaystyle\|f\|_{1}$ $\displaystyle=$ $\displaystyle\int_{G}|f|\;d\mu$

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 L1GIsABanachalgebra 2013-03-22 17:42:14 2013-03-22 17:42:14 asteroid (17536) asteroid (17536) 15 asteroid (17536) Example msc 46K05 msc 46H05 msc 44A35 msc 43A20 msc 22D05 msc 22A10 DualGroupOfGIsHomeomorphicToTheCharacterSpaceOfL1G ConvolutionsOfComplexFunctionsOnLocallyCompactGroups $L^{1}(\mathbb{R})$ is a Banach *-algebra group algebra