Fourier–Stieltjes algebra - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group. Its Fourier–Stieltjes algebra $B(G)$ is the set of all coefficient functions

u(x)=\langle \pi(x)\xi,\eta\rangle

arising from strongly continuous unitary representations $\pi$ of $G$ on complex Hilbert spaces and vectors $\xi,\eta$ . Direct sums show that these coefficients form a vector space, while tensor products show that they are closed under pointwise multiplication. With the norm transported from the dual of the full group $C^*$ -algebra, $B(G)$ is a commutative unital Banach algebra.

Dual realization and norm

Every coefficient $u$ determines a unique bounded functional $\omega_u$ on the full group $C^*$ -algebra $C^*(G)$ , characterized on integrable functions by

\omega_u(f)=\int_G f(x)u(x)\,dx.

The norm is $\lVert u\rVert_{B(G)}=\lVert\omega_u\rVert$ . Equivalently, it is the infimum of $\lVert\xi\rVert\lVert\eta\rVert$ over all coefficient realizations of $u$ . This identification is isometric and is central to Eymard's treatment Eymard, §§2.1–2.2.

Positive-definite functions and examples

The algebra $B(G)$ is the linear span of the continuous positive-definite functions on $G$ . The constant function $1$ is the coefficient of the trivial representation. If $G$ is locally compact abelian, Fourier–Stieltjes transforms identify $B(G)$ with the measure algebra on the Pontryagin dual. For a nonabelian group, $B(G)$ remains commutative because its multiplication is pointwise, even though the full group $C^*$ -algebra is generally noncommutative.

Relation to the Fourier algebra

The Fourier algebra $A(G)$ consists of coefficients of the left regular representation and is a closed ideal in $B(G)$ . Thus $B(G)$ uses all unitary representations, whereas $A(G)$ records the regular representation. The distinction can be substantial for noncompact groups and should not be suppressed by calling both spaces simply “Fourier transforms.”

References

Pierre Eymard, “L'algèbre de Fourier d'un groupe localement compact,” Bulletin de la Société Mathématique de France 92 (1964), 181–236. DOI record. Relevant: §§2.1–2.3 on $B(G)$ , its norm, and $A(G)$ .
Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: chapters on unitary representations and Fourier–Stieltjes transforms.