# ${L}^{1}(G)$ 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 $f:\mathbb{R}\u27f6\u2102$ such that

$$ |

identified up to equivalence almost everywhere.

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

$$(f*g)(z)={\int}_{\mathbb{R}}f(x)g(z-x)\mathit{d}x,$$ |

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

$${\parallel f*g\parallel}_{1}\le {\parallel f\parallel}_{1}{\parallel g\parallel}_{1}.$$ |

Therefore, with the convolution product, ${L}^{1}(\mathbb{R})$ 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 $f:G\u27f6\u2102$ such that

$$ |

identified up to equivalence almost everywhere.

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

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

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

$(f*g)(s)$ | $=$ | ${\int}_{G}}f(t)g(s-t)\mathit{d}\mu (t)$ | ||

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

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 $\u2102(G)$ (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^{} $G\u27f6\u2102$ with compact support. We can endow this space with a convolution product, an involution and a norm by setting

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

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

${\parallel f\parallel}_{1}$ | $=$ | ${\int}_{G}}|f|\mathit{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 | ${L}^{1}(\mathbb{R})$ is a Banach *-algebra |

Defines | group algebra |