L1 group algebra - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group with a fixed left Haar measure. The $L^1$ group algebra is the $L^1$ space $L^1(G)$ , equipped with group convolution

(f*g)(x)=\int_G f(y)g(y^{-1}x)\,dy

and the convolution involution

f^*(x)=\Delta(x^{-1})\overline{f(x^{-1})},

where $\Delta$ is the modular function. With the $L^1$ -norm these operations make $L^1(G)$ a Banach $*$ -algebra, and $\|f*g\|_1\leq\|f\|_1\|g\|_1$ . Its multiplication therefore records the group law, not merely the underlying measure space.

Haar-measure conventions

Replacing the left Haar measure by a positive scalar multiple gives an isometrically isomorphic algebra after the corresponding rescaling of functions. The modular factor in the involution is essential for a nonunimodular group. With right Haar measure or a different convolution convention, the displayed formulas change; convolution and involution must always be matched.

Representations and C-star completions

Every strongly continuous unitary representation $U$ of $G$ has an integrated form

U(f)=\int_G f(x)U_x\,dx,

which is a nondegenerate contractive $*$ -representation of $L^1(G)$ . The supremum of the resulting operator norms produces the full group $C^*$ -norm, while the left regular representation produces the reduced group $C^*$ -norm. Thus the two group $C^*$ -algebras are completions of the same convolution algebra Folland, Chapters 2–3 and 7.

Units and approximate units

If $G$ is discrete and Haar measure is counting measure, the point mass at the identity is an identity for $L^1(G)$ . For a nondiscrete group, that point mass is not an $L^1$ -function, and $L^1(G)$ has no identity. Nevertheless, it has contractive approximate identities concentrated in shrinking neighborhoods of the group identity.

Examples

For a discrete group, $L^1(G)=\ell^1(G)$ and convolution is a sum. For $G=\mathbb R^n$ , the algebra is the usual convolution algebra of integrable functions, the modular function is $1$ , and Fourier transformation converts convolution into pointwise multiplication. The algebra is commutative exactly when $G$ is abelian.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapters 2–3 on Haar convolution and integrated representations.
Jacques Dixmier, C-Algebras*, North-Holland, 1977. Publisher record. Relevant: the chapters on involutive Banach algebras and unitary representations of locally compact groups.