$L^1(G)$ 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.