Fourier algebra of a locally compact group - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group and let $\lambda$ be its left regular representation on $L^2(G)$ . The Fourier algebra $A(G)$ consists of the coefficient functions

u(x)=\langle\lambda(x)\xi,\eta\rangle, \qquad \xi,\eta\in L^2(G).

Its norm is

\|u\|_{A(G)} =\inf\{\|\xi\|_2\|\eta\|_2: u(x)=\langle\lambda(x)\xi,\eta\rangle\}.

With pointwise multiplication and this norm, $A(G)$ is a commutative Banach algebra of continuous functions vanishing at infinity. The norm is intrinsic, even though its definition uses coefficient factorizations.

Operator-algebraic interpretation

Let $VN(G)$ be the von Neumann algebra generated by $\lambda(G)$ . Each coefficient $u=u_{\xi,\eta}$ defines the normal functional

T\longmapsto\langle T\xi,\eta\rangle

on $VN(G)$ , and this identifies $A(G)$ isometrically with the canonical predual $VN(G)_*$ . Under this identification, pointwise multiplication in $A(G)$ is dual to the comultiplication $\lambda(x)\mapsto\lambda(x)\otimes\lambda(x)$ Eymard, §§2–3.

Abelian case

If $G$ is locally compact abelian with Pontryagin dual $\widehat G$ , then

A(G)=\{\widehat f:f\in L^1(\widehat G)\}, \qquad \|\widehat f\|_{A(G)}=\|f\|_{L^1(\widehat G)}.

Thus the Fourier algebra is the transform of the $L^1$ -algebra on the dual group. Except under a chosen self-duality, it should not be confused with $L^1(G)$ itself.

Relation to the Fourier–Stieltjes algebra

The Fourier–Stieltjes algebra $B(G)$ uses coefficient functions of all continuous unitary representations of $G$ , whereas $A(G)$ uses only the left regular representation. Hence $A(G)\subseteq B(G)$ . The smaller algebra is a closed ideal in $B(G)$ and is tailored to the regular representation and the group von Neumann algebra, rather than to the full group $C^*$ -algebra.

Examples

For a discrete group, functions in $A(G)$ still vanish at infinity, but $A(G)$ is generally not $\ell^1(G)$ . For a compact group, Peter–Weyl theory describes $A(G)$ through matrix coefficients of irreducible representations with a weighted trace-class norm. These examples show that the $A(G)$ -norm is representation-theoretic rather than a pointwise or supremum norm.

References

Pierre Eymard, “L’algèbre de Fourier d’un groupe localement compact,” Bulletin de la Société Mathématique de France 92 (1964), 181–236. Article and full text.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. Publisher record. Relevant: the duality and predual theory for von Neumann algebras.