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[parent] $L^1(G)$ is a Banach *-algebra (Example)

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

$\displaystyle \Vert f\Vert _1 := \int_{\mathbb{R}} \vert f(x)\vert\; dx < \infty $
identified up to equivalence almost everywhere.

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

$\displaystyle (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
$\displaystyle \Vert f*g\Vert _1 \leq \Vert f\Vert _1\Vert g\Vert _1\;. $
Therefore, with the convolution product, $ L^1(\mathbb{R})$ is a Banach algebra.

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)$.

Let $ G$ be a locally compact topological group and $ \mu$ its left Haar measure. Consider the space $ L^1(G)$ consisting of measurable functions $ f:G \longrightarrow \mathbb{C}$ such that
$\displaystyle \Vert f\Vert _1 := \int_G \vert f\vert\; d\mu < \infty $
identified up to equivalence almost everywhere.

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

$\displaystyle (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
$\displaystyle \Vert f*g\Vert _1 \leq \Vert f\Vert _1\Vert g\Vert _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.

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

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

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)$.

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 (equivalent) 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

$\displaystyle (f * g)(s)$ $\displaystyle =$ $\displaystyle \int_G f(t)g(t^{-1}s)\;d\mu(t)$  
$\displaystyle f^*(s)$ $\displaystyle =$ $\displaystyle \Delta_G(s^{-1})\overline{f(s^{-1})}$  
$\displaystyle \Vert f\Vert _1$ $\displaystyle =$ $\displaystyle \int_G \vert f\vert\; d\mu$  

With this operations and norm, $ C_c(G)$ has a normed *-algebra structure and $ L^1(G)$ can be defined as its completion.



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See Also: dual group of $G$ is homeomorphic to the character space of $L^1(G)$

Also defines:  $L^1(\mathbb{R})$ is a Banach *-algebra, group algebra

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Cross-references: completion, *-algebra, operations, norm, support, compact, continuous functions, measure, Haar measure, equivalence classes, finite groups, commutative groups, group, commutative, algebras, modular function, measurable functions, left Haar measure, topological group, locally compact, Banach algebra, inequality, product, well-defined, product of functions, convolution, almost everywhere, equivalence, Borel measurable functions, Banach space
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This is version 12 of $L^1(G)$ is a Banach *-algebra, born on 2007-12-18, modified 2008-04-05.
Object id is 10146, canonical name is L1GIsABanachAlgebra.
Accessed 552 times total.

Classification:
AMS MSC: 22A10 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Analysis on general topological groups)
 22D05 (Topological groups, Lie groups :: Locally compact groups and their algebras :: General properties and structure of locally compact groups)
 43A20 (Abstract harmonic analysis :: $L^1$-algebras on groups, semigroups, etc.)
 44A35 (Integral transforms, operational calculus :: Convolution)
 46H05 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: General theory of topological algebras)
 46K05 (Functional analysis :: Topological algebras with an involution :: General theory of topological algebras with involution)
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