Fourier–Stieltjes algebra

Summary

The Fourier–Stieltjes algebra $B(G)$ of a locally compact group $G$ is the algebra of coefficient functions $x \mapsto \ip{\pi(x)\xi}{\eta}$ of all continuous unitary representations $\pi$ of $G$. It is the linear span of the continuous positive-definite functions on $G$. It is a commutative unital Banach algebra under pointwise operations, the dual of the group C*-algebra. For abelian $G$ it is the image of the measure algebra of the dual group under the Fourier–Stieltjes transform, which explains the name.