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

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

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

Moreover, we can define an involution in $L^1(\mathbb{R})$ by $f^*(x)=\overline{f(-x)}$ . With this involution $L^1(\mathbb{R})$ is Banach *-algebra.

Generalization to $L^1(G)$ .

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

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.

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, hence $\Delta_G (s) = 1$ for all $s \in G$ .

So in the commutative case the convolution product and involution are given, respectively, by \begin{eqnarray*} (f * g)(s) & = & \int_G f(t)g(s-t)\;d\mu(t)\\ f^*(s) & = & \overline{f(-s)} \end{eqnarray*}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 $\mathbb{C}(G)$ .

An equivalent construction

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 \begin{eqnarray*} (f * g)(s) & = & \int_G f(t)g(t^{-1}s)\;d\mu(t)\\ f^*(s) & = & \Delta_G(s^{-1})\overline{f(s^{-1})}\\ \|f\|_1 & = & \int_G |f|\; d\mu \end{eqnarray*}With this operations and norm, $C_c(G)$ has a normed *-algebra structure and $L^1(G)$ can be defined as its completion.